Analytical expressions for the dispersive contributions to the nucleon-nucleus optical potential

Analytical expressions for the dispersive contributions to the nucleon-nucleus optical potential J. M. Quesada,*R. Capote,†A. Molina, and M. Lozano Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, E-41080 Sevilla, Spain J. Raynal 4 rue du Bief, 91380 Chilly-Mazarin, France 共Received 22 October 2002; published 16 June 2003兲 Analytical solutions of dispersion relations in the nucleon-nucleus optical model have been found for both volume and surface potentials. For the energy dependence a standard Brown-Rho function has been assumed for both the volume and surface imaginary contributions multiplied in this later case by a decreasing exponential function. The solutions are valid for any even value of the powers appearing in these functional forms. DOI: 10.1103/PhysRevC.67.067601 PACS number共s兲: 11.55.Fv, 24.10.Ht Mahaux and co-workers 关1–5兴have shown how the study of the nuclear mean field may benefit from the use of dispersion relations. These are mathematical expressions that link certain contributions to the real and imaginary components of the optical model potential 共OMP兲. The constraint imposed by these dispersion relations helps in reducing ambiguities in the construction of phenomenological potentials from fits to the experimental data. We refer specifically to the so-called dispersive contribution ⌬V, which adds dynamical content to the otherwise static 共and real兲Hartree-Fock potential term VHF . Under favorable conditions of analyticity in the complex Eplane, the real part ⌬Vcan be constructed from the knowledge of the imaginary part Wof the mean field on the real axis through the dispersion relation ⌬V共r,E兲⫽ P ␲ 冕 ⫺⬁ ⬁W共r,E⬘兲 E⬘⫺EdE⬘,共1兲 where we have explicitly indicated the radial and energy dependence of these quantities. Assuming that ⌬V(r,E⫽EF) ⫽0, where EFis the Fermi energy, Eq. 共1兲can also be written in the subtracted form ⌬V共r,E兲⫽ P ␲ 冕 ⫺⬁ ⬁W共r,E⬘兲 冉 1 E⬘⫺E⫺1 E⬘⫺EF 冊 dE⬘. 共2兲 This transformation is difficult to implement in practice if the geometry of the dispersive potential depends on the energy. To simplify the problem, however, the shapes of the different components of the OMP are usually assumed to be energy independent and they are expressed in terms of a WoodsSaxon function fWS or its derivative. In such case the radial functions factorize out of the integrals and the energy dependence is completely accounted for by two overall multiplicative strengths ⌬V(E) and W(E). Both of these factors contain, we note, volume and surface contributions. It is customary to represent the variation with energy of the volume and surface components of the imaginary potential by functional forms that are suitable for an optical model analysis that exploits dispersion relations. An energy dependence for the imaginary volume term has been suggested by Brown and Rho in studies of nuclear matter 关6兴, WV共E兲⫽AV 共E⫺EF兲n 共E⫺EF兲n⫹共BV兲n,共3兲 where AVand BVare constants. Brown and Rho proposed n⫽2, while Mahaux and Sartor 关2兴suggest, for the same expression, n⫽4. An energy dependence for the imaginarysurface term has also been investigated by Delaroche et al 关7兴, who use the form WS共E兲⫽AS 共E⫺EF兲m 共E⫺EF兲m⫹共BS兲mexp共⫺CS 兩 E⫺EF 兩 兲,共4兲 where m⫽2,4 and AS,BS,CSare constants. According to Eqs. 共3兲and 共4兲the imaginary part of the OMP turns out to be zero at E⫽EFand nonzero elsewhere. A more realistic parametrization of WV(E) and WS(E) forces these quantities to be zero in some interval around the Fermi energy. A reasonable range for such a region is measured by the average energy of the single-particle states Ep 关4兴and a new definition for the imaginary volume part of the OMP can thus be written as WV共E兲⫽ 再 0, EF⬍E⬍EP AV 共E⫺EP兲n 共E⫺EP兲n⫹共BV兲n,E⭓EP.共5兲 Likewise, for the surface term we have WS共E兲⫽ 再 0, EF⬍E⬍EP ASe⫺CS 兩 E⫺EP 兩 共E⫺EP兲m 共E⫺EP兲m⫹共BS兲m,E⭓EP. 共6兲 *Electronic address: [email protected] †Permanent address: Centro de Estudios Aplicados al Desarrollo Nuclear, Apartado Postal 100, Miramar, La Habana, Cuba. PHYSICAL REVIEW C 67, 067601 共2003兲 0556-2813/2003/67共6兲/067601共3兲/$20.00 ©2003 The American Physical Society67 067601-1 The symmetry condition W共2EF⫺E兲⫽W共E兲共7兲 defines the imaginary part of the OMP for energies below the Fermi energy. In a recent work we have presented a numerical solution of the dispersion integral relations between the real and the imaginary parts of the nuclear optical potential 关8兴. In this contribution we obtain analytic solutions of Eq. 共2兲for the particular functional form of the imaginary potential W(E) given above by Eqs. 共5兲–共7兲. Following Ref. 关9兴, we adopt a notation where the offset energy is E0⫽EP⫺EF, the excitation energy Ex⫽E⫺EFand introduce the convenient quantities E⫹⫽Ex⫹E0,E⫺⫽Ex⫺E0. For the surface potential WS(E) given by Eqs. 共4兲and 共7兲, we can write the dispersive integral 共2兲for even mas ⌬VS共E兲⫽Ex ␲ P 冕 ⫺⬁ ⬁WS共E⬘兲 共E⬘⫺E兲共E⬘⫺EF兲dE⬘ ⫽AS Ex ␲ P 冕 0 ⬁Umexp共⫺CSU兲 共Um⫹BS m兲共U⫺E⫺兲共U⫹E0兲dU ⫹AS Ex ␲ P 冕 0 ⬁Umexp共⫺CSU兲 共Um⫹BS m兲共U⫹E⫹兲共U⫹E0兲dU. 共8兲 The integrand can then be replaced by its expression in terms of poles and residues 关10兴, and therefore we can also write AS Ex ␲ Um 共Um⫹BS m兲共U⫿E⫿兲共U⫹E0兲 ⫽AS 1 ␲ 再 兺 j⫽1 mRes共pj兲 U⫺pj ⫹Res共⫿E⫾兲 U⫿E⫿ ⫹Res共E0兲 U⫹E0 冎 . 共9兲 In the previous expression pjare the mzeros of Um⫹BS mand Res(pj) represent their corresponding residues, pj⫽BSexp 冉 i2j⫺1 m ␲ 冊 ,共10兲 Res共pj兲⫽Ex m pj 共pj⫿E⫿兲共pj⫹E0兲.共11兲 Here ⫾E⫿and ⫺E0are the poles of U⫿E⫿and U⫹E0, whereas Res(⫾E⫿) and Res(⫺E0) are their residues, Res共⫾E⫿兲⫽⫾ 共E⫿兲m 共E⫿兲m⫹BS m,共12兲 Res共⫺E0兲⫽⫿ 共E0兲m 共E0兲m⫹BS m.共13兲 As was pointed out by Raynal 关10兴, the contribution of each complex pole pjto the surface dispersive integral 共8兲is 冕 0 ⬁Res共pj兲e⫺CSU U⫺pjdU⫽Res共pj兲e⫺CSpj 冕 ⫺CSpj ⬁exp共⫺z兲 zdz ⬅Res共pj兲e⫺CSpjE1共⫺CSpj兲,共14兲 where E1(z) is the exponential integral function E1关11兴. The contribution of the real poles corresponding to the second term in the right-hand side of Eq. 共9兲is 冕 0 ⬁Res共⫿E⫾兲e⫺CSU U⫿E⫿ dU ⫽Res共⫿E⫾兲e⫿CSE⫿P 冕 ⫿CSE⫿ ⬁exp共⫺x兲 xdx ⬅⫺Res共⫿E⫾兲e⫿CSE⫿Ei共⫾CSE⫿兲,共15兲 where Ei(x) is the exponential integral function Ei关11兴. Finally, the contributions from the third term on the right-hand side of Eq. 共9兲in integral 共8兲cancel. For the volume potential WV(E) given by Eqs. 共5兲and 共7兲, the dispersive integral for even ncan be written as ⌬VV共E兲⫽Ex ␲ P 冕 ⫺⬁ ⬁WV共E⬘兲 共E⬘⫺E兲共E⬘⫺EF兲dE⬘ ⫽AV Ex ␲ P 冕 0 ⬁Um 共Um⫹BV m兲共U⫺E⫺兲共U⫹E0兲dU ⫹AV Ex ␲ P 冕 0 ⬁Um 共Um⫹BV m兲共U⫹E⫹兲共U⫹E0兲dU. 共16兲 According to Eq. 共9兲, the contribution of each pole p共either real or imaginary兲in integral 共16兲diverges as 冕 0 ⬁dU U⫺p⫽lim U→⬁ ln共U⫺p兲⫺ln共⫺p兲.共17兲 Obviously, their sum is a finite quantity, which is calculated by taking its limit. We quote, below, exact expressions for the surface and the volume dispersive integrals for any even value of mand nin the potentials. These forms have no limitations regarding their range of validity. The dispersive contribution of the surface imaginary potential WS(E), according to Eqs. 共6兲 and 共7兲,is ⌬VS共E兲⫽AS ␲ 再 兺 j⫽1 m Zje⫺pjCSE1共⫺pjCS兲 ⫺Res共⫺E⫹兲eCSE⫹Ei共⫺CSE⫹兲 ⫺Res共E⫺兲e⫺CSE⫺Ei共CSE⫺兲 冎 ,共18兲 BRIEF REPORTS PHYSICAL REVIEW C 67, 067601 共2003兲 067601-2 where Zjcomes from the sum of the residues Res(pj) in the two integrals 共8兲and is given by Zj⫽Ex m pj共2pj⫹E⫹⫺E⫺兲 共pj⫹E0兲共pj⫹E⫹兲共pj⫺E⫺兲.共19兲 For the dispersion relation corresponding to the volume imaginary potential WV(E), calculated with Eqs. 共5兲and 共7兲, the real contribution yields ⌬VV共E兲⫽⫺ AV ␲ 再 兺 j⫽1 n Zjln共⫺pj兲⫹Res共⫺E⫹兲lnE⫹ ⫹Res共E⫺兲ln 兩 E⫺ 兩 冎 ,共20兲 where pjis calculated according to Eq. 共10兲using BVand n instead of BSand mand Zjis calculated by Eq. 共19兲using n instead of m. A computer code to calculate the analytical expressions 共18兲and 共20兲has been recently published by the authors 关12兴. In conclusion, we have found analytical solutions of the dispersion relations for the volume and surface terms of the OMP when they are parametrized in the form given by expressions 共5兲–共7兲. The formulas are compact and easy to implement in current codes for the optimum parameter search defining the nucleon-nucleus OMP. Usually these searches for elastic scattering data are performed by adjusting simultaneously the real and imaginary parts of the OMP. In particular, the fact of having available a functional form of ⌬V(E) in terms of the parameters that define the imaginary potentials, makes it possible to implement a convenient alternative to the ordinary search procedures by adjusting only volume and surface real parts of the OMP. This work was supported by Junta de Andalucı ´a and the Spanish CICYT under Contract Nos. PB1998-1111, FPA2001-0144-C05-03, and FPA2001-4960-E and by the European Union under Contract No. FKIW-CT-2000-00107. 关1兴C. Mahaux and H. Ngo ˆ, Nucl. Phys. A431, 486 共1984兲. 关2兴C. Mahaux and R. Sartor, Nucl. Phys. A468, 193 共1987兲. 关3兴C.H. Johnson, D.J. Horen, and C. Mahaux, Phys. Rev. C 36, 2252 共1987兲. 关4兴C. Mahaux and R. Sartor, Nucl. Phys. A528, 253 共1991兲. 关5兴C. Mahaux and R. Sartor, in Advances in Nuclear Physics, edited by J.W. Negele and E. Vogt 共Plenum, New York, 1991兲, Vol. 20. 关6兴G.E. Brown and M. Rho, Nucl. Phys. A372, 397 共1981兲. 关7兴J.P. Delaroche, Y. Wang, and J. Rapaport, Phys. Rev. C 39, 391 共1989兲. 关8兴R. Capote, A. Molina, and J.M. Quesada, J. Phys. G 27, B15 共2001兲. 关9兴J.M. VanderKam, G.J. Weisel, and W. Tornow, J. Phys. G 26, 1787 共2000兲. 关10兴J. Raynal, Proceedings of the Specialists’ Meeting on the Nucleon Nucleus Optical Model up to 200 MeV, Bruye `res-leCha ˆtel, 1996, available at http://www.nea.fr/html/science/ om200/ 关11兴M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55 共National Bureau of Standards, Washington/Dover, New York, 1968兲. 关12兴J.M. Quesada, R. Capote, A. Molina, and M. Lozano, Comput. Phys. Commun. 153,97共2003兲. BRIEF REPORTS PHYSICAL REVIEW C 67, 067601 共2003兲 067601-3