Dispersion relation approach to the coulomb correction term of the proton-nucleus optical model potential

Volume 210. number 1,2

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18 August 1988

DISPERSION RELATION APPROACH TO THE COULOMB CORRECTION TERM OF THE P R O T O N - N U C L E U S OPTICAL MODEL POTENTIAL W. T O R N O W Department qf Physics, Duke University, Durham, NC 27706, USA and Triangle Universities Nuclear Laboratory, Duke Station, Durham, NC 27706, LISA

and J.P. D E L A R O C H E Service de Physique et Techniques Nuclbaires, Centre d'Etudes de Bruybres-le-Ch(ttel, F-91680 Bruybres-le-Chdtel, France and 7)'Jangle Universities Nuclear Laboratory, Duke Station, Durham, NC 27706, USA

Received 23 March 1988

it is shown that the dispersion relation which connects the real and imaginary central parts of the proton-nucleus optical model potential gives rise to a real Coulomb correction term AI~-, which consists of energy dependent surface and volume components. This feature is illustrated for proton scattering from 4°Caat energies below 40 MeV, where the surface component of AVe exhibits large variations with incident energy. Our findings are compared to a recent experimental determination of AVc,

As is well known, the nuclear part of the nucleonnucleus interaction felt by incident protons is different from that experienced by neutrons. This property is due to the Coulomb field Vc(r) and to the energy dependence of the nuclear part of the optical model potential UN(r, E ) . Thus the Coulomb correction term AUc(r, E) arises [ 1 ]. In general, AUc(r, E) is complex because UN (r, E) is complex [2 ]. According to Satchler's prescription [ 3 ], AUc.(r) is usually written as AUc(r, E) =

O U N ( r , E ) 12c. OE

(1)

Here Vc. is the value of Vc(r) averaged over the ordinary space coordinate r, assuming that the protons of incident energy E feel a uniform charge distribution. At the experimental level, the comparison between neutron and proton optical model potentials at similar bombarding energies provides information on both Coulomb and nuclear efffects. Since these effects usually are difficult to disentangle, it is only through nucleon scattering from T = 0 nuclei that 26

specific information on Coulomb effects (i.e. AUc) can be extracted. This information should be relatively accurate because charge-symmetry breaking effects in optical model potentials (OMPs) are found to be small [ 4 ] for these nuclei. However, it is questionable whether a detailed knowledge of AUc(r, E) has been achieved yet. For instance, the scattering analyses conducted by several authors [ 5-9] assume that the real part AVc (r, E) of the complex Coulomb correction term, eq. ( 1 ), has a volume radial shape. In addition, a constant strength was assigned to AVc(r, E), which is a consequence of an assumed linear energy dependence for the real central potential VN(r, E ) . These two features are in agreement with the early work of Perey [ 10] on the Coulomb correction term, but are at variance with results of O M P analyses recently performed by A l a r o n et al. [ 11 ] on nucleon scattering from 4°Ca. Based on a Fourier-Bessel technique which provides model-independent potential form factors, the analysis of Alar~on et al. at energies between 20 and 26 MeV (see fig. 1, solid curve) showed that the phenomenological, real Coulomb correction

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adopted in a recent O M P analysis of neutron scattering from 4°Ca [ 13 ]. For simplicity, the present work exploits the method adopted in ref. [ 13 ] to construct the absorptive potential up to infinity and to evaluate the D R term:

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(3)

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Fig. 1. Radial dependence of the real Coulomb correction term AVe(r) for p+4°Ca scattering. The continuous curve is from Alarqon et al. [ 11 ]. This curve is enclosed in a large error envelope which is not shown. The dashed curve was obtained in the present work. term has a radial shape consisting of surface and volume components. Although a large uncertainty has been assigned to this radial dependence, it seems interesting to investigate the origin of the surface component observed for the real Coulomb correction term

A Vc(r, E). The purpose of this letter is to show that the dipersion relation, as defined by Mahaux and Ng6 [12] for nucleon-nucleus interactions, provides a natural explanation for the observation of a surface contribution to AVc(r, E) at incident energies lower than, say, 40 MeV. Furthermore, it is shown that the real part of AUc(r, E) has quite intricate energy and radial dependences in that energy range. The impact of our finding on the O M P predictions for elastic scattering is illustrated for protons incident on the T = 0 nucleus 4°Ca. Following Mahaux and Ng6, the dispersion relation ( D R ) for nucleon potentials is written

V(r, E) = V.v(r, E) + P -j W(r, E') dE' E'-E '

(2)

where P means principal value, W(r, E' ) is the absorptive potential evaluated at the energy E ' and Vr~v(r, E) is the local equivalent of the non-local Hartree-Fock potential. Under the assumption that W(r, E' ) is symmetric around E' =ev (eF is the Fermi energy), eq. (2) takes on a specific form defined and

and to determine the Hartree-Fock component. Here, the proton-a°Ca absorption W(r, E' ) was used rather than that for neutrons in order to account for the contribution of the imaginary Coulomb correction term AWe(r, E) to AV(r, E). Values for W(r, E' ) below 80 MeV were taken from ref. [9], and ev was set to - 8 . 5 MeV. The use of a dispersion relation in O M P scattering studies requires some caution because eq. (2), valid both for positive and negative energies, is tentatively parametrized for E > 0 only. For instance, there is no guarantee that VHF, as determined for E > 0, is also valid for E < 0 . Therefore, the potential labelled here as VHv represents only an estimate of the actual Hartree-Fock potential [14] at positive energies. For simplicity, the estimate adopted here for VH~ is that o f ref. [ 13] for neutron scattering from 4°Ca. This potential has a Woods-Saxon radial shape f ( r , aHF, RHv), and is defined below 40 MeV as

VHF(r, E) = ( I7o --c~E)f( r, ally, RHV ) ,

(4)

where c~= 0.2, a n F = 0 . 6 8 fm, RHV= 1.25A ~/3 fm and Vo= 45.2 MeV. Small changes to VHV do not significantly alter the results of the present study. The surface WD(E) and volume W v ( E ) components of the proton-nucleus absorptive potentials display energy dependences which are essentially similar to those for neutrons [ 9 ]. Therefore, the calculated surface and volume components of the D R correction term A V eq, (3) resemble those shown in the lower panel of fig. 1 in ref. [ 13 ]. Here the surface term A VD for protons shows an extremum at E = 18 MeV, and flips its sign at an incident energy of 41 MeV. On the other hand, the volume component A Vv is identical to that determined earlier for neutrons [131. Using eqs. (2) and (3), the real part o f the proton O M P can be expressed as 27

Volume 210, n u m b e r 1,2

PHYSICS LETTERS B

V(r, E) = VUF(r, E) + A Vv (r, E) + A VD(r, E) +AK:(r, E ) + Vc(r),

(5)

In this approach, the Coulomb correction term A Vc (r, E) includes three components: 0

A Vc,uF(r, E) = - ~

[ VHF(r, E) ] Vc,

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AVc,v(r,E)=- ~ [AVv(E) ]f(r, av,Rv)Vc, AVc.D(r,E) =+~[AVD(E)]4aD

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f(r, aD, RD)ffC.

(6)

The values for the radii and diffusenesses of the absorptive potentials can be found in refs. [ 9,13 ]. We used Vc = 8,07 MeV. The energy dependence of the three terms given in eq. (6) is displayed in fig. 2. As can be seen, marked differences exist between these curves. For instance, the component related to the Hartree-Fock potential (dot-dashed curve) has a constant value. This behavior reflects the linear energy dependence given in eq. (4). On the other hand, the volume component A Vc,v (dashed curve ) has a smooth energy dependence and

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remains positive up to approximately 70 MeV. In contrast, the surface term AVc.D (solid curve) exhibits rapid energy variations and flips its sign near E = 18 MeV. It is evident that, except for the zero crossing at 18 MeV, AV
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Volume 210, number 1,2

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18 August 1988

104

approach, since the difference 8 Vc (r, E) between the nuclear parts of the proton and neutron potentials is given for 4°Ca by

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(7)

where the indices p and n refer to the D R terms for proton and neutron potentials, respectively. Eq. (7) is based on the assumption that the potentials VHV(r, E) as well as Wv(r, E) are identical for proton and neutron scattering (see above). The difference shown in brackets on the right side of eq. (7) represents the D R term which originates from the imaginary Coulomb component AWe(r, E) of the proton optical model potential. Since this component is expected to be small [2], a property well established for 4°Ca from either conventional [7,8] or model independent [11] O M P analyses, it is reasonable to assume

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6Vc(r, E) ~ AVe(r, E ) . This approximation ofeq. (7) together with the large uncertainty attached to the result of ref. [ 11 ] makes it possible to directly compare the D R based predictions for AVe(r, E) to the Coulomb correction term 6 Vu (r, E) determined experimentally by Alar¢on et al. This comparison is shown in fig. 1 where the continuous curve is taken from ref. [ 11 ]. On the other hand, the dashed curve represents the present estimate of AVc ( r, E) at the mid-point energy (E = 23.5 MeV) of the interval in which the measurements of ref. [ l 1 ] were performed. Except for radial distances shorter than 2 fm, where the model independent analyses provide large uncertainties, there is fair agreement between the two curves. In particular, they both display a peak at the surface near r ~ 4 fm. The impact of the present study on O M P predictions is illustrated in fig. 4 for elastic scattering cross section d(0) and analyzing power A, (0) data at 26.3 MeV [ 15,16 ], the incident energy for which AVc,x~ reaches an extremum. The new coupled-channel (CC) calculations (solid curves) used the same coupling scheme and potentials as in a recent work [ 13 ], and in addition include AVc(r, E ) from the present investigation. As can be seen, the new CC analysis provides a better description of the measurements at angles below 120 ° than those of ref. [9] (dashed

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Fig. 4. Comparison between differential cross section (top part) and analyzing power (bottom part) data for proton scattering [ 15,16] and fits using the conventional Coulomb correction term AVc of Honor6 et al. [9] (dashed curves) and the present Coulomb correction approach includingthe surface term Al~.o (solid curves).

curves) which treat all parts of the real central potential as pure volume terms. In summary, it has been shown that the application of the dispersion relation to proton scattering leads to a quite complicated, energy dependent real Coulomb correction term at low incident energies. Illustrations given for the p-4°Ca system indicate that this term is a superposition of surface and volume components, thus explaining results obtained by Alargon et al. from a model-independent analysis of (p, p) and (n, n) scattering data near 26 MeV. It has also been verified that the use of the dispersion relation in O M P calculations improves the description o f elastic scattering observables measured at this incident energy over that based on a conventional treatment of the real central potential. As a consequence of our study, modifications to results obtained on charge-symmetry breaking in the nuclear mean field [15] are expected. Finally, the present approach is quite general, whether it involves a spherical or a 29

Volume 210, number 1,2

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coupled-channel OMP formalism, and should apply to i n c i d e n t c h a r g e d p a r t i c l e s a b s o r b e d at t h e n u c l e a r surface, w h i c h is m o s t likely to h a p p e n at i n c i d e n t e n e r g i e s n e a r or b e l o w t h e C o u l o m b b a r r i e r . T h e a u t h o r s w i s h to t h a n k G. W e i s e l f o r h i s assist a n c e in p r o d u c i n g c u r v e s s h o w n i n t h e m a n u s c r i p t . W e w o u l d like t o t h a n k P r o f e s s o r R.L. W a l t e r for many inspiring discussions. This work was supp o r t e d in p a r t b y t h e U S D e p a r t m e n t o f Energy, Ofrice o f H i g h E n e r g y a n d N u c l e a r P h y s i c s , u n d e r Contract No. DE-AC05-76ER01067.

References [ 1 ] A.M. Lane, Rev. Mod. Phys. 29 (1957) 191. [2] J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C 15 (1977) 10;C16 (1977) 80. [3]G.R. Satchler, in: lsospin in nuclear physics, ed. D.H. Wilkinson (North-Holland, Amsterdam, 1969) ch. 9, p. 411.

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[4 ] S.M. Austin, in: Use of the optical model for the calculation of neutron cross sections below 20 MeV (OECD, Paris, 1986) p. 53, and references therein. [5] J. Rapaport, J.D. Carlson, D. Bainum, T.S. Cheema and R.W. Finlay, Nucl. Phys. A 286 (1977) 232. [6] R.P. DeVito, S.M. Austin, W. Sterrenburg and U.E.P. Berg, Phys. Rev. Lett. 47 (1981) 628. [7] W. Tornow, E. Woye, G. Mack, C.E. Floyd, K. Murphy, P.P. Guss, S.A. Wender, R.C. Byrd, R.L. Walter, T.B. Clegg and H, Leeb, Nucl. Phys. A 385 (1982) 373. [8] J. Rapaport, Phys. Rep. 87 (1982) 25. [9] G.M. Honor6, W. Tornow, C.R. Howell, R.S. Pedroni, R,C. Byrd, R.L. Walter and J.P. Delaroche, Phys. Rev. C 33 (1986) 1129. [ 10] F.G. Perey, Phys. Rev. 131 (1963) 745. [ 11 ] R. Alar~on, J. Rapaport and R.W. Finlay, Nucl. Phys. A 462 (1987) 413; R. Alar¢on, Ph.D. thesis, Ohio University (1986), unpublished. [ 12 ] C. Mahaux and H. Ng6, Nucl. Phys. A 378 (1982) 205. [ 13 ] J.P. Delaroche and W. Tornow, Phys. Lett. B 203 ( 1988 ) 4. [ 14] C. Mahaux and R. Sartor, submitted to Nucl. Phys. A. [ 1 5 ] K.H. Bray, K.S. Jayaraman, G.A. Moss, W.T.H. van Oers, D.O. Wells, and Y.I. Wu, Nucl. Phys. A 167 ( 1971 ) 57. [ 16 ] J.S. Winfield, S.M. Austin, R.P. DeVito, U.E.P. Berg, Zipeng Chen and W. Sterrenburg, Phys. Rev. C 33 (1986) 33.