Revista Mexicana de Física 43, Suplemento 1 (1997) 14-23

The modified Newton-Sabatier method for thecoupled channel inverse scattering problem at

fixed energy

MATTHlAS EBERSPACHER,a BARNABAS APAGYI," AND WERNER SCHEIDb

"¡nstitut für Theoretische Physik del' Justus-Liebig- Universitiit, Giessen, Germanyb Department of Theoretical Physics, Technical University, Budapest, llungary

AUSTRACT. The modified Newton-Sabatier method is extended to the inverse scattering problemfor coupled channels at fixed energy. The coupled Schrodinger equations are assumed to dependon a potential matrix which does Bot couple stat.es with different orbital angular momenta. 'Vepresent applications of our method to S-matrices calculated with potential matrices of \Voods-Saxon type. In the case of charged particles the corresponding S-matrix with the asymptoticCoulomb potential is transformed to the S-matrix with an asymptotic constant potentia1. Theresulting S-matrix then serves as input for the inverse scattering problem of neutral particles.

RESUMEN. El método Newton-Sabat.ier modificado es extendido para el problema de dispersión in-verso para canales acoplados a energía fija. Se a':iumeque las ecuaciones de Schr6dinger acopladasdependen de una matriz de potencial que no acopla estados con diferente momento angular orbital.Presentamos aplicaciones de nuestro método a matrices S calculadas con matrices de potencialde tipo \Vooels-Saxon. En el caso de partículas cargadas la matriz S correspondiente al potencialasintótico de Coulomb es transformada a una matriz S con un potencial asintótico constante. Lamatriz S resultante sirve así como entrada para el problema de dispersión inverso para partículasneutras.

PAes: 03.65.Nk; 24.10-i; 25.70-z

1. INTRODUCTION

The inverse scattering problem (ISr) in quantum mechanics consists oI the determinationoI the potentiai in the Schriidinger equation Irom the S-matrix. For e!astic scatteringthe ISr at fixed energy was first solved by Newton [Ij. Sabatier [2,3] investigated amIextended this method which is thercfore named Newton-Sabatier method today. A Iur-ther extension oI their work was given by Miinchow ami Scheid [4] by introducing themodified Newton-Sabatier method. This method assumes that the potential is knownfraIn a certain radius up to infinity, so that thc potential is unknown only in a finitcintcrval. If aue considcrs ncutron scattcring fol' examplc, the asymptotic potential willtend to zero, Ior ion-ion scattering it will be t-he Coulomb potentia1. This modificationtuade the method applicable to mcasurcd data from hcavy ion experiInents [5J. ThcNewtoIl-Sabaticr Illcthod and other inversion methods at fixcd encrgy are rcviewed inmonographies [3,6,7].

TIIE MO[)IFIED NEWTON-SABATIER METIIOD FOR .. . 15

In this paper we extend the modified Newton-Sabatier method from the one-channelcase (i.e. elastic scattering) to the case of N conpled channels (i.e. inelastic scattering).To solve this problem we assume that the channels are coupled by a potential matrixwhich does not depend on the angular coordinates of the relative motion. In Sect. 2 wederive the basic equations of the ISr for couplcd charmels. In Sect. 3 we solve the ISrwith the modified Newton-Sabatier method by using the full S-matrix at a fixed energy.The special case of clrarged partides is reduced to tire case of neutral partides via atransformation wlriclr is given for elastic scattering by May et al. [8]. Finally, in Sect. 4the new metlrod is applied to potential matrices consisting of \Voods-Saxon potentialsfor neutral and clrarged partides.

2. THE ISP FOR COUPLED CHANNELS

We solve the ISr for a particular coupled channel problem, in wlriclr tire potential matrixmay not depend on tire angular momentum of t.he rclative mot.ion [9]. We "-~sume thattire Hamilt.onian describing t.lre scattering system of t.lre t.wo colliding partides is givenby

H = T(r) + h(O + W(r, O. (1)

Here, T denotes tire relative kinetic energy operator ami h the Hamiltonian of tire int.ernalstates of tire two scattering part.ners. The operator W gives t.he int.eraction between therelative and tire int.rinsic motion. It depends on the relative radial coordinate r andthe intrinsic coordinates ( With tlris special choice, only 1II0nopole transitions can bedescribed.

Due t.o t.lris ansatz, the wavefunct.ion of the syst.em factorizes "-~

(2)

wit.1rR~(r) being the radial wavefunction, Yfm describing tire orbital motion amI X,,(Osolving tire eigenvalue equation for t.hc int.rinsic mot.ion with eigenvalues E"

(3)

Using the Schriidinger equat.ion HifJ = EifJ and introdueing t.lre Jlot.ential mat.rix V,,~ =(x"IWlx~), we get tire coupled equations for t.he radial wavefunctions

[,,2 1 d2 ,,2 f(€ + 1) ] f N f

--" --12'" + 2 2 + E" - E R",,(r.) +L V"B(r) R~n(r.) = o.•...¡17.(T 11 r

~=l

(4)

The t.otal number of channels is denoted by N. The reduced mass J1. is assumed t.o beindependent of the chaullel nurnuer a. Becallse af our special choice uf thc intcractiollIV, the potential matrix V does not. depend on the qnantum numher f of the orbital

16 MATTHIAS EBERSPACHER ET AL.

angular momentum of relative motion which means that it does not couple the orbitaland intrinsic motions ..Equation (4) is a set of N linear homogeneous equations for the radial functions R~n"

Thus, we can write the solutions of this system as vectors with N components

(5)

The N linearly independent solutions are enumerated by the additional index n. Theyare degenerate in the total energy E.The ISP at fixed energy now means the calculation of the potential matrix V,,¡J(r) by

starting with the S-matrix S~¡J (a, {3= 1,2, ... ,N) as function of the quantum numbere of angular momentum.

3. THE MODIFIED NEWTON-SABATIER METHOD FOR COUPLED CHANNELS

By introducing dimensionless coordinates and sorne abbrevations

J2J1.EP = kr = fi'2r,

we can rewrite Eq. (4)

u () _ V,,¡J(r),,¡JP- E' (6)

N

LD~¡J(p) 'P~n(p) = e(£+ 1) 'P~n(P)¡J=\

with the differential operator matrix given by

D~¡J(p) = p2 { [:;2 + ~ ] J,,¡J - U,,¡J(p) } ,

(7)

(8)

(9)

where E" = E - E" is the energy in channel a.The first step in solving the ISP with the Newton-Sabatier method is to choose a

reference potential matrix with U~¡J(p) = U3,,(p). The regular solutions 'P~~(p) of thecoupled channel problem with this reference potential matrix must be known

N

LD~3(p) 'P~~(p) = ele + 1) 'P~~(p)¡J=1

with the reference differential operator

Uo ( ) _ 2 { [ d2 E,,] o ( )}D,,¡J P - P dp2 +E Jn¡J - U,,¡J P .

Next, an ansatz for the yet unknown wavefunctiollS is made

(10)

(11)

(12)

THE MODIFIED NEWTON-SABATlER METHOD FOR . . . 17

This is a generalization of the Povzner-Levitan representation to coupled channels [3].The potential matrix can be calculated if the kernel matrix K~!!O is a solution of thepartial differential equations

N N

L. D~[3(p) K%:O(p,P') = L.D~3(P') K~!!O(p,P'),[3=1 [3=1

with the boundary conditions

K~!!O (p = O, pi) = O, K::!!O (p, p' = O) = O.

Under these conditions it can be shown [10] that the potential matrix is given by

2 d KUUO( )Uo[3(p) = u2[3(p) - - d 0[3 p, P

P P P

(13)

(14)

A solution of the kernel in terms of the wavefunctions <pand <pocan be written with yetunknown cocfficients c~nl

00 N

K~!!O(p,p') =L. L. c~n'<P~n(P)<P~~'(P').£:;::0 n,n':;::l

(15)

The kernel fulfills the aboye conditions (12) and (13). Inserting the kernel into Eq. (11)it is possible to calculate the coefficients c~n' and tbe wavefunctions <P~nand then thepotential matrix (14). In this paper we apply the modified Newton-Sabatier method,assuming that the potential matrix Uo[3 is known from sorne radius Po on. In this casethe wavefunctions <P~nare known for p > Po by their dependence on the S-matrix andthe coefficients c~n' can be calculated by solving Eq. (11) for p > Po. The modifiedNewton-Sabatier method yields an unique solution of the ISP in the dass of potentialsgenerated by the kernel (15).

3.1. SCATTERING OF NEUTRAL PARTICLES

Neutral partides like neutrons with a short range interaction have a potential matrixwhich vanishes from a certain radial distance: Uo[3(p > po) = o. In this case the simplestchoice of the reference potential matrix UO is zero: u2[3(p) = O. Then, the regularreference solutions are

(16)

(17)

with k~ = Eoj E.The wavefunction <P in channel a is composed of a superposition of the degenerate

solutions T~n (p) in channel a with yet unknown coefficients A~'n

N

<P~n(P)= L. pT¿n'(P) A~'n.n'::;]

18 MATTHIAS EBERSPACHER ET AL.

The solutions T~n(p) are asymptotieally detennined by the given S-matrix

( 18)

where h¡ are the spherieal Hankel funetions. By introdueing the wavefunetions (16) amI(17) together with the kernel (15) in the Povzner-Levitan representation (11) we obtainthe set of eoupled inhomogeneous equations

with

( 19)

L~I' (p) = 1"jl(knP')je(knp') dp',N

bl - "\" Al Inn' - ¿ nnl/ Cnl/n'.

nll=l

(20)

In order to get a set of finite eoupled equations, we limit the summation over f' to fmax.The effeet of this limitation is diseussed in Ref. 10.The known quantities in Eq. (19) are the referenee solutions TOI and the matrix L.

The solutions TI are only known in the outer region, where they depend on the 5-matrix as given in Eq. (18). Further unknown quantities are the eoeffieients A and b.Equation (19) is used two times to ealculate these quantities.In the first step, we solve Eq. (19) at two points PI, P2 > Po in the onter region. We

then have 2 X N X N X (fmax + 1) equations for the ealculation of the eoeffieients A~n'and b~n' with the solutions TI depending on the S-matrix as given by Eq. (18). In theseeond step we use these eoeffieients and solve Eq. (19) at diserete points Pi in the innerregion O < Pi < Po to get the solutions T~n(Pi). We need to solve N X N X (fmax + 1)eqnations at eaeh radius Pi.Now, knowing the eoeffieients ami the wavefunetions, we can ealculate the kernel (15)

and the potential matrix (14). Thus, the ISr at fixed energy for the inelastie seatteringof neutral partides is solved.

3.2. SCATTERING OF CHARGED PARTICLES

The asymptotie solutions for the seattering of eharged partides like heavy ions are givenin terms of the Coulomb fnnetions

with

ei"~ [ ]JIf(k"p) = ~"P GI(k"p):l: iFI(k"p) ,

(21 )

(22)

'filE MODIFIED NEWTON-SABATIER METIIOD FOR .. . 19

o P. o

op

op

FIGURE 1. The left diagram shows a diagonal elem€nt of the potelltialmatrices Uno. (Real part:-, Imaginary part: ... ) and U!!o. (Real part: - - -, Imaginar,)' part: ... ), the right diagram givesan off-diagonal element. Pn is the transformation radius.

where GI and FI are the irregular and regular Coulomb functions, respectivcly, and a~are the Coulomb phases

(23)a~,= arg f(f + 1+ i 'la), Zp Z, ellt'lo = lil k k"

Zp e and Z, e are the projectile and target charges, respectivcly.In principIe, we could choose the rcference pot.ential, which is a Coulomb pot.ent.ial for

largc valtlcs 01'1', alld salve lhe ISP again in the salllc way as described ahoye. Howevcr,the JlllIllcrical ea.lcula.tion with Coulomb functiolls cOllsumes lIl11ch JIlore time than lheealclllatioll with sphcrical Hankcl functions. Thus, wc cOllsi<ier a transformation 01' theS-matrix frolll an asymptotic Coulmub potcntial lo aH asymplotic COIlstallt potcntia1.The proccdnre is an ext.ension of t.he met.hou describcd by May d ,,1. [8] for the elasticscatt.ering of charged partidcs 1.0 inelast.ic scatt.ering.

'fhe idea of t.he t.ransformat.ion can be seen in Fig. 1. The want.ed pot.ential matrix(full line) wit.h an asymptot.ic Coulomb behavior has the fonn

1':::: Po,

l' > Po.(24)

\Ve transform t.he S-matrix belonging 1.0 this potentiai matrix to a S-mat.rix of a pot.ent.ialmat.rix which l,,~,¡~'ympt.otically constant. clements (PTI > po)

U/J ( ) _, Zp Z, ,,2 k0/3 l' - "a/3 E

1'13

amI is identical to Uo/3(p) in the inner region

for l' > P/J, (25)

for 1) < p/J. (26)

20 MATTIIIAS EBERSPACIIER ET AL.

The new S-matrix SB describes the same potential matrix for P < PB as the originalS-matrix. The solutions of the Schriidinger equations for the outer region (p > PB) ofthe constant potential (25) can be written in a form similar to Eq. (18)

with the new notations

(27)

(k:;)2 = Ea - EnE

EB= E _ Zp Z, e2 k

PB(28)

For the transformation of the S-matrix S~n --7 S~~,we use the eontinuity of thewavefunctions and their derivatives at P = PB. The new wavefunctions (27) can thenbe used as input for the solution of the ISr for neutral partides. By transforming theS-matrix we have reduced the ISr for charged partides to the ISr for neutral partides.This transformation works for all potentials, to which the asymptotie solutions are known.

4. ApPLICATION TO WOODS-SAXON POTENTIALS

In this paper we first consider a potential matrix eonsisting of Woods-Saxon potentials forneutral partides. For simplicity we limit the calculations to two channels. The input S-matrix is calculated with a numerical coupled channel program for the analytie potentialmatrix (in MeV)

-6 12e2(r-3)VlI(r) - ---- -1----

- e2(r-3) + 1 [e2(r-3) + 1f'-5

VI2(r) = V21(r) = e3(r-2.5) + l'-4 . 8e2.5(r-3.5)

V22(r) = e2.5(r-3.5) + 1-1[e2.5(r-35) + Ir

(29)

The imaginary part of the potential describes the absorption from the eonsideredehannels and is taken to be proportional to the derivative of the real part which isattractive. The excitation energy in the first channcl is set to zero: £1 = OMeV, in theseeond ehannel £2 = 1 MeV. The reduced mass is ehosen to model the scattering of aneutral partide with the mass ofan alpha partide on a 12C-nudeus: l' = 2794.50 MeV je2.

For the ealculation of the speetral eoefficients A and b in equation (19) we take the points7"1 = 9.01 fm and T2 = 9.02 fm. The cut-off angular momentum is fmax = 24.Figure 2 shows the inverted potential matrix at the energy E = 50 MeV. The diagrams

at the top and bottom of Fig. 2 are the diagonal elements VII and V22, respeetively. Thetwo diagrams in the middle of Fig. 2 eorrespond to the eoupling potentials V12 and V21.The left hand side shows the real part, the right hand one the imaginary part of thepotentia1.

TIIE MODIFIED NEWTON-SAIlATIER METIIOD FOR . . . 21

o~-I~ .26-3=-4> -5

-6

o~-I~ -26-3~ .4> -5

-6

o~-I~ -26 -3;::¡ .4> -5

-6

Real pan lmaginary part

o_-1 /"~4 _/6 -3~ .4> -5

,6

123456 123456r (fro)

FIGURE2. Inverted potential matrix (- - -) and original potentialmatrix (-) for neutral partides(for details see text).

Thc original potential matrix is given by the full lincs, thc invcrtcd potcntial matrixby the dashcd lincs_ Onc finds an excellent agrccment bctwecn thc original ami invertedpotentials with cxcmption of the neighbourhood of,' = D. For small valucs of r dcviationsarise duc to a singularity of the method at thc origin " = D, which can bc scen fromEq, (14), Although the condition UaIJ = UIJa is not cxplicitly uscd in the method, thenumerical procedurc automatically providcs this symmctry with high accuracy_

In thc second casc wc assnme a potential matrix for chargcd particlcs (in MeV)

-10V22(,-) = c2.5(r-3.5) + 1

-15 12e2 30e2(r-3)Vl1(r) = ----+ - -i-----

e2(r-3) + 1 r [e2{r-3) + 1]2'

-5VI2(,') = V21(r) = 3( -2 ") ,e T .d + 1

12c2 2Dc2.',(r-:1.5)+---i 2"( '1") 2'" [e .,' r-, ..' + 1)

(3D)

22 MATTIIIAS EBERSPACIIER ET AL.

Realpart lmaginary part

4_2:;; O

~-2;;:-4-6-8

4

>2" O6.2

'"';;:-4-6-8

4_ 2

>~ O~-2

~-4-6-8

4_ 2>~-~'"'~-4-6-8

23456 23456r (Cm)

FI(~UHE3. Invertcd potcntiai matrix (- - -) and original potclltialmatrix (-) for charged particles(fol"<1etailssee text).

The excitation energies are él = OMeV and 102 = 1 MeV. The charge numbers and thereduced mass l' = 2794.50 MeV /e2 are chosen to modcl the scattering system ~He + ~2C.

The S-matrix corresJlonding to the asymJltotic Coulomb Jlotential is first transformedto the S-matrix of a constant asymptotic Jlotential with ,.¡¡ = 9 fm. The radii for thecalculation of the sJlectral coefficients A and b are TI = 9.01 fm and '.2 = 9.02 fm. Fig.3shows the inverted potentialmatrix at the same incident energy a., in the first application.The cut-off angular momentum is now fmax = 26.

Let us discuss the consistency of our method. Ir we consider the application 1,0 re-alistic data, we can only compare the original S-matrix with tite S-lIlatrix ealculatedwith tite invcrted potential matrix. Thereforc, \Ve ealculated tlle S-matrices shown byda.,hed lines in Figs. 2 and 3 and took the absolnte differences with respect 1,0 the ma-trix elements or the original S-matrices: IriS~¡11= 15;,¡1 (inverted potentialmatrix) -5~~{orig inal potent.ialmat.rix)l. We obtained values smaller t.han lO-1 in both exampleswhich 1l1l<1(lrlillcS tite quality of the inversion procedllrc.

TIIE MODIFIED NEWTON-SABATIEH METIIOD FOlt ... 23

5. SUMMARY AND CONCLUSIONS

We extended the modified Newton-Sabatier method from the one-channel case to thecase of N coupled channels. Due to the spedal choice of the Hamiltonian, t.he methodis restricted to monopole transitions induced by the radial motion. We need the fullS-matrix as input data. This confronts us with the fundamental problem that the fullS-matrix is usually not known from experimental data. In the experiment. t.he partidesare scattered in their ground 8tate. ThllS, onl)' one COIUIIlIl of the S-matrix can betueasurcd, dcscribing the cxcitation frolll this statc. In the ca.."lC of onIy two challnelsand no absorption OIle luight use the unitary conc1ition for the S-matrix lo ealculate themissing matrix clmnents.

Our method is related t.o the one of Cox [11] who gave a solution of the ISP forcoupled channels al. fixed total angular momentum. In t.he case of Cox the S-matrixmust be known as a function of the energy, whereas in the present method the S-matrixmust be known as a function of the angular mOlnellttlIll of the relative motion. Therefore,in our case t.he angular moment.um of relative motion al. fixed t.ot.al energy plays the samerole as the energy al. fixed total angular momentum in the method of Cox, and spansthe space of functions in whieh the potential mat.rix is expanded. Hence, the pot.entialmatrix in our method can not depend on the angular momentum of t.he relative motiona]](1 is not allowed 1.0 couple t.he orbital motion and the int.rinsic dynamics of t.he scatteredpartides. In arder 1.0 salve t.he ISP al. fixed energy with a coupling between t.he orbitalrclativc alld the intrinsic lI1otions, new ideas are Ilceded.

ACKNOWLEDGMENTS

We thank Professor H.V. van Geramb for providing us his coupled channel codeo One ofthe aut.hors (W.S.) thanks CONACyT for support. of this work.

REFERENCES

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(Springer- Verlag, New York, 1989).4. M. Münchow and W. Scheid, Phys. Rev. Let!. 44 (1980) 1299.5. D. Apagyi, J. Schmidt, \V. Schcid, and H. Voit, Pltys. Rev. e 49 (1994) 2608.6. R.G. Nc\\.ton, Scattering TheOT1J o/ lVaves uwl Particles, 2nd cditioll. (Springer \'erlag. :\'cw

York, 1982).7. D.N. Zakhariev and A.A. Suzko, Dicee! aTlrl 17¡ve7..,e Proble11l", (Sprillger \'erlag. Ikrlill 1990).8. ICE. May, M. MÜIlChow,ami \V. Schcid, I'hy .•. Let!. B 141 (1984) 1.9. 1\1. Eherspiichcr, D. Apagyi, ami W. Schei<l, I'lty.<. Ilev. Let!. 77 (1996) 1921.10. M. Ebcrspacher, Das A40dijizierte Newton-S(jbatier- Verfal17"cn jiir das InveniC Stn~u]Jroblem

bei gekoPIlelteTl KaTliileTl, Diploma thesis. Giessell (1996).11. J.R. Cox, ./. Math. Phy.<. 12 (1967) 2327.