Dispersion relations in the nuclear optical model

Transcription

1 Computer Physics Communications 153 (2003) Dispersion relations in the nuclear optical model J.M. Quesada a,,r.capote a,b,1,a.molina a,m.lozano a a Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Facultad de Física, Apdo 1065, Sevilla, Spain b Departamento de Física Aplicada, Universidad de Huelva, Huelva, Spain Received 19 November 2002; received in revised form 23 December 2002 Abstract We have developed a code to calculate integrals arising from the dispersion relation between the real and the imaginary parts of the nuclear optical model potential (OMP). Both, analytical solution for the most common dispersive OMP, and the general numerical solution, are included. In the numerical integration, fast convergence is achieved by means of the Gauss Legendre integration method, which offers accuracy, easiness of implementation and generality for dispersive optical model calculations. The numerical method is validated versus analytical solution. The use of this package in the OMP parameter search codes allows for an efficient and accurate dispersive analysis Elsevier Science B.V. All rights reserved. PROGRAM SUMMARY Title of the program: DOM Catalogue identifier: ADRO Program Summary URL: Program obtainable from: CPC Program Library, Queen s University of Belfast, N. Ireland License provisions: None Reference to original program: None Computers: PCs (Pentium class and above) Installation: Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Spain Operating system or monitors under which the program has been tested: COMPAQ-DIGITAL UNIX, Microsoft WINDOWS and LINUX Programming language used in the new version: FORTRAN 77 Memory required to execute with typical data: 250 Kbyte No. of bits in a word: 32 or 64 depending on the computer No. of lines in distributed program, including test data, etc.: 537 (source) + 6 (parameter input) + N (energy points input) + N (analytical integral) + N (numerical integral). N is the number of requested energy points This program can be downloaded from the CPC Program Library under catalogue identifier: * Corresponding author. address: quesada@us.es (J.M. Quesada). 1 Permanent address: Centro de Estudios Aplicados al Desarrollo Nuclear, Ap. 100, Miramar, La Habana, Cuba /03/$ see front matter 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 98 J.M. Quesada et al. / Computer Physics Communications 153 (2003) No. of bytes in distributed program, including test data, etc.: 9214 Distribution format: zip file Keywords: Optical model, dispersion relations, real and imaginary potentials Nature of physical problem An analytical solution of dispersion integral relations, linking the real and imaginary part of the nuclear OMP, has been derived for the most common functional forms. The general numerical solution is also presented. Method of solution The decomposition of the integrand function using his analytical properties has been used to calculate the dispersive integral in the complex plane. The numerical solution is based on Gauss Legendre quadrature formulae. Restrictions on the complexity of the problem None. If the dispersive integral of the imaginary potential does not have analytical solution, the general numerical solution can be used. Typical running time A few seconds in Pentium class computer. LONG WRITE-UP 1. Introduction A significant contribution to the optical model theory during the last two decades can be considered the work of Mahaux and co-workers on dispersive optical-model analysis [1 5]. The unified description of nuclear mean field in dispersive optical-model is accomplished by using a dispersion relation (DR), which links the real and absorptive terms of the optical model potential (OMP). The additional constraint imposed by DR helps to reduce the ambiguities in deriving phenomenological OMP parameters from the experimental data. In the following we assume that imaginary potential geometry is energy independent. In this case the whole radial dependence is included within the radial form factors, therefore we can deal with pure energy dependent quantities. In a dispersion relation treatment, the real central potential strength consists of a term which varies slowly with energy E, the so-called Hartree Fock (HF) term, V HF (E), plus a correction term, V (E), whichis calculated using a dispersion relation. The depth of the dispersive term of the potential V (E) can be written as V (E) = P π W(E ) E E de. Assuming that V (E) vanish at the Fermi energy E F, Eq. (1) can be written in the following form: V (E) = P π ( W(E ) 1 E E 1 E E F ) de. (2) If the function W(E) is symmetric respect to the Fermi energy, Eq. (2) can be expressed in a form which is stable under numerical treatment for E>0 [6], namely: V (E) = 2 π (E E F ) E F W(E ) W(E) (E E F ) 2 (E E F ) 2 de. (3) The dispersive term V (E) is divided into two terms V V (E) and V S (E), which arise through dispersion relations (1) from the volume W V (E) and surface W S (E) imaginary potentials, respectively. The real volume V V (E) and surface V S (E) central parts of the dispersive OMP are given by (1)

3 J.M. Quesada et al. / Computer Physics Communications 153 (2003) V V (E) = V HF (E) + V V (E), V S (E) = V S (E). (4) As was discussed in Refs. [9,13], it is useful to represent the variation of the surface W S (E) and volume W V (E) absorptive potential depths with energy in functional forms suitable for the dispersive optical model analysis. In this work we define the volume imaginary part of the OMP as: { 0 EF <E<E P, W V (E) = (E E A P ) n V (E E P ) n +(B V ) n E E (5) P and likewise for surface absorption. { 0 EF <E<E P, W S (E) = A S e C S E E P (E E P ) m (E E P ) m +(B S ) m E E P. (6) The symmetry condition W(2E F E) = W(E) (7) is used to define imaginary part of the OMP for energies below the Fermi energy. A V, B V, A S,B S and C S are undetermined constants, n and m must be even (otherwise the dispersive integral (1) is zero) and E P is the average energy of the single-particle states [4]. 2. High energy behavior of the volume absorption The assumption that the imaginary potential W(E) is symmetric about E = E F (according to Eq. (7)) is plausible for small values of E E F, however as was pointed out by Mahaux and Sartor [4] this approximate symmetry no longer holds for large values of E E F. In fact the influence of the non-locality of the imaginary part of the microscopic mean field will produce an increase of the empirical imaginary part W(E) at large positive nd approaches zero at large negative E [1,7]. Following Mahaux and Sartor [4], we assume that the absorption strengths are not modified below some fixed energy.theyused = 60 MeV, however this value is fairly arbitrary [4] and we could use it as fitting parameter [11]. Let us denote W V (E) as the approximate form of the non-local volume imaginary potential to be used in the dispersive integral. We can write [5] W V (E) = W V (E) [1 (E F E ) 2 ] (E F E ) 2 + Ea 2 for E<E F, (8) [ E (E F + ) W 3/2 V (E) = W V (E) + α V + 3 ] (EF + ) for E>E F +. (9) 2E 2 These functional forms are chosen in such a way that the function and its first derivative are continuous at E = E F ±. At large positive energies nucleons feel the hard core repulsive region of the nucleon nucleon interaction and W V (E) diverges like α V E. Using a model of a dilute Fermi gas hard-sphere the coefficient αv can be estimated to be equal to 1.65 MeV 1/2 [4], assuming that the Fermi impulse k F,V at the nuclear centre is equal to 1.36 fm 1 and the radius of the repulsive hard core is equal to 0.4 fm. On the contrary, at large negative energies the volume absorption decreases and goes asymptotically to zero. The equivalent to the non-local imaginary volume absorption potential W V (E) and the symmetric imaginary absorption potential W V (E) are represented by solid and dotted lines respectively in the lower panel of Fig. 1. For the surface imaginary potential W S (E) the nonlocality is considered by the equation similar to Eq. (9), because the surface potential already goes to zero at negative energies. In this case the coefficient α S can be estimated to be equal to MeV 1/2 [4], assuming that the Fermi impulse k F,S associated with the phenomenological value of nuclear density at the nuclear surface is equal to 0.71 fm 1.

4 100 J.M. Quesada et al. / Computer Physics Communications 153 (2003) Fig. 1. Dependence upon energy of the dispersive volume contribution of the real central potential of the n + 27 Al mean field. The dotted curve corresponds to Eq. (5), in which it is assumed that the imaginary part is symmetric about the Fermi energy. The thick solid curves correspond to the asymmetric model, considering non-local behavior of the imaginary volume absorption above certain energy following Eqs. (8) and (9). Thin dashed line corresponds to the Fermi energy. 3. Method of solution 3.1. Analytical solution of dispersion relations The analytical solutions of the dispersion relation integrals given by Eq. (2) for usually employed functional forms of the imaginary potential, W(E), used in DOM analyses (Eqs. (5) (7)) were recently derived [12,13]: V V (E) = A { n } Zj n π ln( pj n ) + R n + ln E + + R n ln E, (10) j=1 V S (E) = A { m Zj m π e pm j C ( E 1 p m j C ) } R+ m ece + Ei( CE + ) R m e CE Ei(CE ), (11) j=1 where the pj n are the n zeroes of (U n + B n ),thatis: ( pj n = B exp i 2j 1 ) π (12) n and E 1 (z) and Ei(x) are the Exponential Integral functions [10] and the following notations were introduced: E 0 = E P E F ; E x = E E F ; E ± = E x ± E 0 ; R n ± = En ± E n ± + Bn ; Zn j = E x n p n j (2pn j + E + E ) (p n j + E 0)(p n j + E +)(p n j E ). The asymmetric form of the volume imaginary potential W V (E) (Eqs. (8) and (9)) results in a dispersion relation that must be calculated directly from Eq. (2) and separates into three additive terms [8]. Therefore, we write the dispersive correction for the imaginary potential in the form Ṽ V (E) = V V (E) + V < (E) + α V V > (E), (13)

5 J.M. Quesada et al. / Computer Physics Communications 153 (2003) where V V (E) is the dispersive correction due to the symmetric imaginary potential of Eq. (5) and the terms V < (E) and V > (E) are the dispersive corrections due to the asymmetric terms of Eqs. (8) and (9), respectively. For the surface imaginary potential the dispersive correction can be written in the form Ṽ S (E) = V S (E) + α S V > (E). (14) For a given volume imaginary potential W V (E), the following exact V < (E) contribution is obtained V < (E) = AE { x Res [ } ] f(z),z j ln(ea E F z j ), (15) π j where Res[f(z),z j ] means the residue of the function f(z)at pole z = z j. The expression of the function f(z)is the following: (z + 2E F E P ) n (z + E F ) 2 f(z)= [(z + 2E F E P ) n + B n ][(z + E F ) 2 + Ea 2](z + E)(z + E F ). (16) The number of complex poles z j is equal n + 2. Their values are the following: z 1,2 = E F ± i, ( z j = E P 2E F + B exp i 2j 1 n ) (17) π, j = 1,...,n. (18) There are also two real poles at z = nd z = E F, therefore the sum in Eq. (15) is over n + 4terms. A very good approximation to the exact V < (E) contribution can be derived taking into account that the volume imaginary potential W V (E) for E<(E F ) increases in magnitude by less than 10% of its maximum value at E = E F. Therefore Eq. (8) can be approximated by (E F E ) W 2 V (E) = W V (E) W V (E F ) (E F E ) 2 + Ea 2 for E<E F. (19) For a given volume imaginary potential W V (E), the following expression for the dispersive contribution at negative energies is obtained V V (E) = W V (E F ) E { x ln π Ea + 2 E x + (E x + 2 ) ln π 2E x 2[Ea 2 + ( + E x ) 2 ( + E x ) 2 } ln( + E x ) ] E x [Ea 2 + ( + E x ) 2. (20) ] Using the notation E L = E F +,the V > (E) contribution for both surface and volume correction, due to the nonlocality at positive energies, is exactly: V > (E) = 1 [ EF π EL E F arctan E L E F + E3/2 L ln E ] a 2E F E L E ln E E+ EL + 3 E L EL 2 ln E E L + E3/2 L 2E ln E L E E L for E>E L, 3 EL ln 24/3 E L for E = E L, π E ln E+ EL EL + 3 E L L for E L >E>0, E 2E ln E L E L E 3 EL 2 + ln E L EL for E = 0, 2 ln E L E + E3/2 E arctan 2 E L E E L E + 3 E L 2 ln E L E + E3/2 L 2E ln E L E L E for E<0.

6 102 J.M. Quesada et al. / Computer Physics Communications 153 (2003) General numerical solution of dispersion relations Recently we have developed a general numerical solution for calculation of the DR integral (3), based on Gauss Legendre integration method (GLIM) [9]. The integral in a finite interval (a, b) can be converted by a linear transformation to the integral in the interval ( 1, 1) and then approximately calculated according to the following formula: 1 1 f(x)dx = i=1 Wi N f ( Xi N ), where W N i and X N i are weights and abscissas corresponding to the N point Gauss Legendre integration method [10]. In this work we were using N = 10 points. Beyond some energy cut-off E cut, such that E cut E, the dispersive integral can be very well approximated by assuming that the numerator of the integrand becomes constant-valued. In this case we obtain the following relation: V res (E) = 2 π (E E F ) E cut (21) W(E ) W(E) (E E F ) 2 (E E F ) 2 de W(E) E cut E log. (22) π E cut + E 2E F Therefore we can apply GLIM from E F up to E cut for a finite interval. This option was shown to be more accurate for a given problem [9]. The method is suitable for practically any type of DR integrand function. Moreover the implementation of this method in optical model parameter search code is straightforward. We compare GLIM results with the results of the analytical integration in Tables 1 and 2 for V V and V S, respectively. Table 1 Comparison between analytical V V (E) of Eq. (10) and GLIM results for n = 2, 4, 6 and 8. All values are given in MeV. Diff. denotes the error of the numerical integration. DOM parameters for the calculation are taken from Ref. [11] E n= 2 n = 4 n = 6 n = 8 V V Diff. V V Diff. V V Diff. V V Diff D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D 05

7 J.M. Quesada et al. / Computer Physics Communications 153 (2003) Table 2 Comparison between analytical V S (E) of Eq. (11) and GLIM results for m = 2, 4, 6 and 8. All values are given in MeV. Diff. denotes the error of the numerical integration. DOM parameters for the calculation are taken from Ref. [11] E m= 2 m = 4 m = 6 m = 8 V S Diff. V S Diff. V S Diff. V S Diff D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D 06 The parameters of the dispersive OMP used in our tests were recently derived by the authors in Ref. [11] to describe neutron scattering on 27 Al up to 250 MeV incident energy (see Table II, [11]). For the nucleus 27 Al, the experimental value of the Fermi energy E F derived from mass differences is equal to MeV and the offset energy E P was taken as 5.66 MeV. 4. Program organization 4.1. Structure of the deck The structure of the code is the following: (i) Main program DOM This a driver program which defines the functional form to be used for the imaginary OMP, then performs the parameter initialization and calls either DOM_Int to calculate dispersive integrals numerically, or DOM_Int_Wv and DOM_int_Ws for the analytical integration. Finally it produces the output files containing the table of results for the whole positive energy grid. Any functional form for W V (E) and W S (E) different of those presented here could be used. Exactly the same type and number of the function arguments must be preserved. (ii) routines: DOM_int Computes the dispersive integral by GLIM for both volume and surface terms. Delta_WV Computes the integrand of the volume term of the dispersive integral according to Eq. (3). (Called by DOM_Int for GLIM.)

8 104 J.M. Quesada et al. / Computer Physics Communications 153 (2003) WV Computes W V according to Eqs. (5) and (7). (Called by Delta_WV.) Delta_WD Computes the integrand of the surface term of the dispersive integral according to Eq. (3). (Called by DOM_Int for GLIM.) WD Computes W S according to Eqs. (6) and (7). (Called by Delta_WD.) DOM_int_Wv Computes the analytical dispersive integral for volume imaginary term, Eq. (10). DOM_int_Ws Computes the analytical dispersive integral for surface imaginary term, Eq. (11). ZFI Computes the complex function exp(z)e1(z), needed for calculation of Eq. (11) as proposed by J. Raynal [14]. EIn Computes the real exponential integral function Ei(x), needed for calculation of Eq. (11). DOM_int_T1 Computes the analytical dispersive integral V < according to Eq. (20). DOM_int_T2 Computes the analytical dispersive integral V > according to Eq. (21) Input cards The program requires two input files, energy grid file (energy.inp in the test run) and OMP parameter file (OMP_parameter.inp in the test run). In the first file the energy grid must be written in one column format. The second file requires 6 lines of input in free format according to the following structure: 1st line: Description line 2nd line: E F,E P,n,m The Fermi energy E F and the average single particle levels energy E P and the values of the even exponents n and m, defining volume and surface imaginary potentials. 3rd line: Description line 4th line: A V,B V,A S,B S,C S 5th line: Description line 6th line: Energy grid filename The parameter input file should be redirectioned to the standard input. The constants A V, B V, A S, B S and C S define the volume and surface terms according to Eqs. (5) and (6), respectively. Depending on the functional form of the imaginary part of the OMP the user must modify the main program to set parameter values. If user-defined imaginary OMP function is used, then corresponding DELTA_WV and WV functions (for the imaginary volume OMP case) must be supplied by the user. Corresponding functions included within the package can be taken as example Output The results are presented in two files DVanalytical.dat and DVnumerical.dat. DVanalytical.dat This file has a seven column structure: 1st column:energy 2rd column: V V of Eq. (10) 3th column: V S of Eq. (11) 4th column: V V + V < + α V V > 5th column: V S + α S V > 6th column: V < 7th column: α V V >

9 J.M. Quesada et al. / Computer Physics Communications 153 (2003) DVnumerical.dat Identical structure than the former file but with corrections V V and V S numerically calculated with GLIM. 5. Conclusions We have implemented an accurate and general method for calculating dispersive contribution to the real part of the optical model potential. The method is very fast and thanks to its computational simplicity should be easy to implement in current generation of the optical model parameter search codes. We stress that any energy functional representation of the imaginary part of the OMP can be treated in our approach, either by numerical integration or using compact analytical solutions. These properties make the described code very useful for introduction of the dispersive optical model relation in large scale OMP nuclear data calculations. Acknowledgements This work was supported by CICYT under contracts PB , FPA C05-03 and FPA nd by the European Union under contract FIKW-CT References [1] C. Mahaux, H. Ngo, Nucl. Phys. A 431 (1984) 486. [2] C. Mahaux, R. Sartor, Nucl. Phys. A 468 (1987) 193. [3] C.H. Johnson, D.J. Horen, C. Mahaux, Phys. Rev. C 36 (1987) [4] C. Mahaux, R. Sartor, Nucl. Phys. A 258 (1991) 253. [5] J.W. Negele, E. Vogt (Eds.), Advances in Nuclear Physics, Plenum, New York, [6] J.P. Delaroche, Y. Wang, J. Rapaport, Phys. Rev. C 39 (1989) 391. [7] C. Mahaux, R. Sartor, Nucl. Phys. A 458 (1986) 25. [8] J.M. van der Kam, G.J. Weisel, W. Tornow, J. Phys. G: Nucl. Part. Phys. 26 (2000) [9] R. Capote, A. Molina, J.M. Quesada, J. Phys. G: Nucl. Part. Phys. 27 (2001) B15. [10] M. Abramowitz, I. Stegun, in: Handbook of Mathematical Functions, in: Applied Mathematics Series, Vol. 55, Dover Publications, New York, [11] A. Molina, R. Capote, J.M. Quesada, M. Lozano, Phys. Rev. C 65 (2002) [12] J. Raynal, Proceedings of the Specialists Meeting on the Nucleon Nucleus Optical Model up to 200 MeV, Bruyères-le-Châtel, 1996; Available at [13] J.M. Quesada, R. Capote, J. Raynal, A. Molina, M. Lozano, Submitted to Phys. Rev. C, October 2002, LANL e-print: nucl-th/ [14] J. Raynal, Private communication, Dec