The trajectories of S-matrix poles are calculated in the finite-range
phenomenological potential introduced recently by P. Salamon and T. Vertse
(SV). The trajectories of the resonance poles in this SV potential are compared
to the corresponding trajectories in a cut-off Woods-Saxon (WS) potential for
l>0. The dependence on the cut-off radius is demonstrated. The starting points
of the trajectories turn out to be related to the average ranges of the two
terms in the SV potential

A. Rácz

Gy. Bencze

P. Salamon

T. Vertse

Physical Review C

English

arXiv

Crossref

Trajectories of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>-matrix poles in a new finite-range potential

Rácz, Anett

Salamon, Péter

Vertse, Tamás

DEA University of Debrecen Electronic Archive

Trajectories of S-matrix poles in a new finite-range potentialA. Ra´cz1, P. Salamon1,2 and T. Vertse1,21University of Debrecen, Faculty of Informatics,PO Box 12, H–4010 Debrecen, Hungary2Institute of Nuclear Research of the Hungarian Academy of Sciences,Debrecen, PO Box 51, H–4001, Hungary(Dated: August 2, 2011)AbstractThe trajectories of S-matrix poles are calculated in the finite-range phenomenological potentialintroduced recently by P. Salamon and T. Vertse [1] (SV). The potential is similar to a Woods-Saxon(WS) interaction, but it is exactly zero beyond a radius, without any cut-off. The trajectories ofthe resonance poles in this SV potential are compared to the corresponding trajectories in a cut-offWS potential for l > 0 . The dependence on the cut-off radius is demonstrated. The startingpoints of the trajectories turn out to be related to the average ranges of the two terms in the SVpotential.PACS numbers: 21.10.Pc,21.30.-x,21.60.CsKeywords: S-matrix1Solutions, of the Schro¨dinger equations, that belong to poles of the scattering matrixare often used in reaction calculations and also in the description of weakly bound states.Certain pole solutions can be used as single particle basis functions in a so called Berggrenrepresentation [2], and they are useful in the theoretical description of slightly bound orunbound states in the framework of the Gamow shell model [3] (GSM). The GSM is apowerful tool for the description of the new nuclei produced by the radioactive beam facilitiesrecently. For the numerical calculation of the S-matrix poles in the Woods-Saxon (WS)potential a new method was suggested recently [4].The WS potential form is often used in reaction calculations. Although the WS potentialis considered a finite-range potential, its value becomes zero only at infinite distances. Theradial Schro¨dinger equation with this potential has analytical solutions [5] only for l =0. Therefore, we have to calculate its solutions (eigenvalues and eigenfunctions) by usingnumerical methods for the integration. The typical way of the solution is that we integratethe radial equation numerically in an interval r ∈ [0, Rmatch] of the radial distance r wherethe nuclear potential is large, and we match the calculated solution at a distance r = RmatchRef. [9], or at two different distances in Ref. [4] to the solution of the asymptotic differentialequation, which is the Coulomb or the Ricatti-Hankel differential equation. Thus, in thissolution we effectively use a modified version of the WS potential in which the potential iscut to zero at a finite distance Rmax artificially. The cut-off WS potential is the following:V WS(r) = V0fWS(r) , (1)where the radial shape isfWS(r) = −11+er−Ra, if r < Rmax0 , if r ≥ Rmax .(2)The solution and, consequently, the resulting pole position does depend on the value of thecut-off radius Rmax. Therefore, the cut-off radius Rmax is a parameter of the WS form inEq.(2), in addition to its depth V0, radius R and diffuseness a. In publications, however,the value of Rmax is very seldom specified because most physicists think that the cut hasnegligible effect on the calculated quantities if Rmax is large and thus the jump at the cut-offis quite small.In a recent work by P. Salamon and T. Vertse [1] it was discussed that the positions ofthe broad resonances might be very sensitive to the Rmax value and the authors proposed an2alternative to the cut-off WS potential. The advantage of the proposed new potential (SV)is that its range is finite in the strict sense since it is exactly zero at a value ρmax. Frommathematical point of view the SV potential is nicer than the cut-off WS potential since itsderivative exists everywhere in contrast to the latter. Indeed the shape of the SV potentialbecomes zero smoothly, and even its derivatives become zero at and beyond r = ρmax.The objective of this Brief Report is to show the implications of the tail of the WSpotential for the positions of the S-matrix poles, and their remedy achieved by the SVpotential.The sensitivity to Rmax can be seen if we follow the position of the S-matrix poles in thecomplex wave number plane (k-plane) as the strength of the potential V0 is changed. Thesecurves are the so called pole trajectories. The shapes of the pole trajectories do depend onthe radial shape of the potential concerned. The pole trajectories in a square well/barrierpotential had been studied by Nussenzweig [6] long time ago. In a square well the analyticalform of the solution is known inside the potential radius Rsq as well, and the pole positionsare obtained as roots of a simple transcendental equation.Certain features of the pole trajectories in the square well remain valid for realistic (dif-fuse) potentials if their range is finite. Other features however do depend on the radialshape. The pole trajectories of the cut-off WS form have not been studied so extensivelyas those of the square well. In a work by J. Bang et al. [7] the trajectories of the resonantpoles were given for a WS potential cut off at a fixed Rmax value, but the Rmax dependenceof the trajectories was not explored.We have now calculate the pole trajectories in cut-off WS potentials with different cut-offradii, and compare them to the similar trajectories in the SV potentials. We present thecalculations for l > 0 only since in the l = 0 case we have no centrifugal barrier to producenarrow resonances, and therefore this case is of reduced importance.The radial shape of the potentials is fixed, and only the strength is varied along a poletrajectory. For the sake of comparison, we rewrite the original SV potential in Ref. [1] intothe formV SV (r) = V0fSV (r) , (3)where fSV (r) is taken a linear combinationfSV (r, c1, ρ0, ρ1) = fρ0(r) + c1f′ρ1(r) . (4)3Wherefρ(r) = −er2r2−ρ2 , if r < ρ0 , if r ≥ ρ ,(5)and f ′ρ(r) is its derivative:f ′ρ(r) = − 2rρ2(r2−ρ2)2 er2r2−ρ2 , if r < ρ0 , if r ≥ ρ .(6)Since both terms in Eq.(4) have finite ranges, the SV potential of Eq.(3) has a finite-rangeof ρmax = max{ρ0, ρ1}, i.e., fSV (r ≥ ρmax) = 0 . With the usual choices ( ρ0 > ρ1), therange is ρmax = ρ0.In order to make the shape of the SV potential similar to the fWS(r) shape, we adjustthe range parameters ρ0 , ρ1 and the weight c1 so as to minimize the integral of the squareddifferences:∆(ρ0, ρ1, c1) =∫ ρmax0[fSV (r, c1, ρ0, ρ1)− fWS(r)]2dr . (7)In general, the Rmax value is large (ρmax < Rmax). The fitting fixes the geometrical param-eters of the SV potential (ρ0, ρ1, c1), so that fSV (r) depends only on R and a of the WSshape fWS(r) and does not depend on Rmax. The quality of the fit can be characterized bythe minimum ∆min of ∆.We consider potentials whose shapes are suitable for describing 16O and 208Pb. Wecompare the radial shapes of the best fit SV potentials and the original WS potentials inFig. 1. One can see that the shapes of the WS forms are reproduced reasonably well bothinside and in the surface regions. There are, however, considerable differences in the tailregion, where the SV shape goes to zero faster than the WS shape.In Table I we summarize the geometrical parameters of the WS and SV potentials usedin this work. The parameters of the WS potential for oxygen and for lead were taken fromJ. Bang et al. [7] and from P. Curutchet et al. [8], respectively. The spin-orbit terms wereneglected for the sake of simplicity. The SV potential parameters in Table I are the resultsof the minimization of the deviation ∆ in Eq.(7). The values of ∆min are also given in theTable.We can notice that the shape of the SV potential in Eq.(3) has three parameters (c1, ρ0,ρ1) just as the WS shape (R, a, Rmax), but the SV potential has a natural finite-range: ρmax.4We can observe in Table I that the average values of the two ranges ρ¯ show some similarityto the radius parameter R of the WS potential but ρ¯ is larger in both cases.We shall only present trajectories for the decaying resonances since the trajectory of acapturing resonance is simply the mirror image of the trajectory of a decaying resonance.The pole energies and radial wave functions have been calculated by a modified versionof the computer code GAMOW [9]. The node number n starts with n = 1, by convention.This n is the actual radial node number in the limit of large potential depth when the polebecomes a bound-state pole.For conveniences we start the trajectories with a small V0 at a small positive strength,V0 = 5 keV. We denote the wave number of the pole belonging to V0 = 5 keV by k0n,l. Afterfinding k0n,l we increment the V0 value by small steps repeatedly until the resonance polereaches the imaginary k-axis. For a high l value or for a large node number the imaginaryk-axis is reached at a very large V0 value, which is much deeper than the realistic potentialwell.Nussenzweig [6] found that for a square well l = 0 the pole trajectories approach theRe(kn) =npiRsqlines with the depth approaching to zero. For l = 1 the starting values of thepole trajectories approach the Re(kn) =(2n−1)pi2Rsqvalues.Bang et al. [7] calculated pole trajectories in a cut-off WS potential for 16O with a cut-offradius Rmax = 9.4 fm for l = 0 and l = 1. Figs. 5 and 6 of that work suggest that the densityof the l = 0 and l = 1 poles are close to the densities of the square well poles belonging toRmax = Rsq, where Rsq denotes the radius of the square well. But the shapes of the cut-offWS trajectories were quite different from the shapes of the square well trajectories.In Fig. 2 we present the trajectories of the n = 2, l = 1 resonance pole for 16O for WSpotentials cut off at different distances Rmax. The curve with Rmax = 9.4 fm resembles theshape of the n = 2 curve in Fig. 6 of Ref. [7]. It should not be the same since we neglectedthe spin-orbit term of the potential. We can see that for different Rmax values the real partsof the starting points k02,1 are different. The larger the Rmax value, the smaller Re(k02,1) isobtained.We define a sort of pole density at the starting points belonging to V0 = 5 keV as thenumber of Re(k0n,l) values of the same l, falling into a Re(k) interval of unit length. Thus,the smaller is the distance between the starting values with node number n and n + 1, thehigher is the pole density in a given partial wave l. The density of the resonance poles in WS5potential increases with increasing cut-off radii. This is a typical feature of the trajectoriescalculated in cut-off WS potentials for all l and n values.Now we turn to studying trajectories in SV potential. In Fig. 3 one can see the trajectoriesof the l = 1 resonance pole for the SV potential for different n values. The vertical linesin the figure denote the k-values for which Re(k) = (2n−1)pi2ρ¯. The starting points of thetrajectories with different n values are close to the corresponding vertical lines of the figure.Therefore, we conclude that for the SV potential the average value of the ranges of thetwo terms play the same role as the cut-off radius Rmax of the WS potential. This rule isalso applies to the l = 2 case, which is shown in Fig. 4. However, in this case l is even,therefore the vertical lines correspond to the k values with Re(k) = npiρ¯. For l = 0 the realparts of the starting k00,n values are similar to those of the l = 2 k-trajectories (Re(k02,n)).The shapes of the trajectories are however different, because for l = 0, the decaying andcapturing resonance poles meet at the imaginary k-axis well below the origin.For 208Pb the pole trajectories in the WS potential depend on the Rmax value in thesame way as observed for 16O. The larger the Rmax value, the smaller is the abscissa of thestarting point of the trajectory.For the SV potential the pole trajectories are very similar to the those for 16O, therefore,we only show the l = 2 case with different n values.In Fig. 5 we show the resonance pole trajectories for l = 2 and for n = 1, 2, 3, 4 values.The curves are similar to the ones in Fig. 4. The starting points of the trajectories k0n,2 areclose to the values denoted by vertical lines at Re(k) = npiρ¯with ρ¯ = 9.644 fm.To sum up, a common feature of the trajectories in the three finite-range potentials(square well, cut-off WS and SV types) is that the density of the resonant poles is influencedby the range of the potentials. The rule derived by Nussenzweig [6] for square well is thatfor V0 → 0 the the distance of the consecutive resonance poles Re(kn+1,l − kn,l) tend to piRsqfor any l value. Bang et. al. [7] conjectured that this rule applies for cut-off WS if wereplace the Rsq by the cut-off radius Rmax. They also conjectured that for V0 → 0 the l = 0poles approach the Re(kn,0) =npiRmaxand the l = 1 poles approach the Re(kn,1) =(2n−1)pi2Rmax.Our results in Fig. 2 support these conjecture qualitatively.In the SV potential case the shapes of the trajectories are different from that of the WSpotential and the starting k-value depends on the average of the ranges of the two terms,rather than the range, i.e., on the value of ρmax. Since ρ¯ is close to the radius R of the WS6form, the distance between the starting Re(k)-values for consecutive n values is governedby the radius of the WS potential to which the parameters of the SV potential were fitted,and not by the cut-off radius Rmax.When we calculate the pole energies numerically, we have to take the matching radiusRmatch, or both values of the matching distances used in Ref. [4] larger than the ranges ofthe potentials (Rmax and ρmax).The dependence of the trajectory on the Rmax value points to the importance of specifyingthe value of the Rmax used in the calculations in order to avoid uncertainties in the polepositions. The uncertainty is appreciable for broad resonances.Contrary to common belief, this strong Rmax dependence in the cut-off WS potentialcannot be diminished by increasing the value of the cut-off radius.The authors are grateful to R. G. Lovas for valuable discussions. This work was sup-ported by the TA´MOP 4.2.1./B-09/1/KONV-2010-0007/IK/IT project. The project is im-plemented through the New Hungary Development Plan co-financed by the European SocialFund, and the European Regional Development Fund.[1] P. Salamon and T. Vertse, Phys. Rev. C 77, 037302 (2008).[2] T. Berggren, Nucl. Phys. A109, 265 (1968).[3] N. Michel, W. Nazarewicz, M. Ploszajczak and T. Vertse, J. Phys. G: Nucl. Part. Phys. 36,013101 (2009).[4] A. M. Mukhamedzhanov, B. F. Irgaziev, V. Z. Goldberg, Yu. V. Orlov and I. Qazi, Phys. Rev.C 81, 054314 (2010).[5] Gy. Bencze, Commentationes Physico-Mathematicae 31, 4 (1966).[6] H. M. Nussenzveig, Nucl. Phys. 11, 499 (1959).[7] J. Bang, S. N. Ershov, F. A. Gareev, and G. S. Kazacha, Nucl. Phys. A339, 89 (1980).[8] P. Curutchet,T. Vertse and R. J. Liotta, Phys. Rev. C 39, 1020 (1989).[9] T. Vertse, K. F. Pa´l, Z. Balogh, Comput. Phys. Commun. 27, 309 (1982).70 5 10 15r     [fm]-1-0.8-0.6-0.4-0.20208Pb with WS208Pb with SV16O with WS16O with SVFIG. 1: Comparison of the radial shapes fWS(r) in Eq.(2) and fSV (r) in Eq.(3) for the 16O andfor the 208Pb nuclei with parameters listed in Table I.A c1 ρ0 ρ1 ρ¯ R a ∆min16 0.066 5.082 2.707 3.892 3.125 0.63 5.6×10−3208 0.997 10.963 8.328 9.644 7.525 0.7 2.3×10−3TABLE I: Geometrical parameters of the SV and WS potentials and the minimal deviation fornuclei 16O and 208Pb. All distances are in fm units.80 0.2 0.4 0.6 0.8 1Re(k)     [fm-1]-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10Im(k)     [fm-1 ]Rmax = 8.0 fmRmax = 9.4 fmRmax = 11.0 fmRmax = 12.0 fmFIG. 2: Trajectories of the n = 2 resonant poles in the complex k-plane for l = 1 for WS potentialsfor 16O with different values of the cut-off radius Rmax. The Re(k) =3pi2Rmaxvalues are: 0.59 fm,0.5 fm, 0.43 fm and 0.39 fm, for increasing Rmax values, respectively.90 0.5 1 1.5 2 2.5 3Re(k)     [fm-1]-1-0.8-0.6-0.4-0.200.20.4Im(k)     [fm-1 ]n = 1n = 2n = 3n = 4pi/(2ρ−) 3pi/(2ρ−) 5pi/(2ρ−) 7pi/(2ρ−)FIG. 3: Trajectories of the resonant poles in the complex k-plane for l = 1 for SV potentials fordifferent n values for 16O.100 0.5 1 1.5 2 2.5 3 3.5Re(k)     [fm-1]-1-0.8-0.6-0.4-0.200.20.4Im(k)     [fm-1 ]n = 1n = 2n = 3n = 4pi/ρ− 2pi/ρ− 3pi/ρ− 4pi/ρ−FIG. 4: Trajectories of the resonant poles in the complex k-plane for l = 2 for SV potentials fordifferent n values for 16O.110 0.5 1 1.5Re(k)     [fm-1]-0.8-0.6-0.4-0.20Im(k)  [fm -1 ]n = 1n = 2n = 3n = 4pi/ρ− 2pi/ρ− 3pi/ρ− 4pi/ρ−FIG. 5: Trajectories of the resonant poles in the complex k-plane in SV potential for l = 2 fordifferent n values for 208Pb.12

Trajectories of S-matrix poles in a new finite-range potential

Trajectories of the S-matrix poles in Salamon-Vertse potential

Trajectories of the S-matrix poles in Salamon-Vertse potential

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