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AU0019692

An approximation method for solution of the coupled channels

inverse scattering problem at fixed energy*

M. Eberspacher1'2, K. Amos2, W. Scheid1 and B. Apagyi3

1 Institut filr Theoretische Physik der Justus-Liebig-Universitat, Giessen, Germany

2School of Physics, The University of Melbourne, Parkville, Australia

3 Institute of Physics, Technical University, Budapest, Hungary

(December 6, 1999)

Abstract

We present a method for the quantum mechanical inverse scattering prob-

lem at fixed energy for coupled channels in reactions with particles having

internal degrees of freedom. The scattered particles can be excited by a local

interaction between the relative motion and the internal dynamics which can

be expanded in multipoles. The inverse scattering problem is solved by an

extension of the modified Newton-Sabatier method, assuming a special ansatz

for the integral kernel in the radial wave function. Application has been made

for a hypothetical scattering of two nuclei interacting by a dipole-type inter-

action. Good agreement between the obtained potentials and the input data

is found.

03.65.N, 24.10.E, 24.10.H, 25.70.

* This work is part of the doctoral thesis of M. Eberspacher, Giefien (D26),

1999

Typeset using REVTEX

3 1 / 3 5

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I. INTRODUCTION

Solution of quantum inverse scattering problems (ISP's) usually is by means of a two

step process. In the first step, a set of phase-shifts (or S'-matrix elements) is calculated from

a given differential cross section [1]. The second step is the actual inversion process with

which the underlying interaction is calculated from those phase shifts (or the S'-matrix).

The result takes the form of a local potential matrix [2,3], which is angular momentum or

scattering energy dependent according as one is solving the ISP at fixed angular momentum

or at fixed energy. In the first case one needs as input data the S'-matrix, as a function

of energy at a fixed angular momentum plus the energy values of any bound states. Thus

to solve the ISP at fixed angular momentum, the S'-matrix must be interpolated between

the experimentally known data and extrapolated to higher energies. A first approach to

solve this ISP for coupled channels was given by Cox [4], while applications of his method

to analytic and experimental S-matrices later were made by Kohlhoff et al. [5].

The fixed energy ISP requires knowledge of the S'-matrix as a function of the angular

momentum. Diverse methods of solution of that ISP for elastic scattering exist. They include

the methods of Lipperheide and Fiedeldey [6], the finite difference method of Hooshyar

and Razavy [7], the generalized Darboux transformations of Schnizer and Leeb [8], and the

Newton-Sabatier method which is discussed below. We also note that those quantal ISP have

been solved with numerical inversion methods for which the iterative-perturbative scheme

of Mackintosh et al. [10] can be applied. All those methods, however, can not encompass

scattering which involves a spin-orbit potential save under approximation [9].

For the solution of the ISP at fixed energy allowing a spin-orbit interaction, Sabatier

found an interpolation formula for the representation of the wave function [11], which Hoosh-

yar [12] has used to extract the elastic scattering potential from the known phase-shifts. For

this a non-linear set of equation must be solved. The approach was improved numerically

and applied to analytic input data by Huber and Leeb [13], while Lun et al. found a way to

use Sabatier's formula directly for the inversion by a linearization scheme [14,15].

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Newton-Sabatier methods are among the most successful methods for the solution of the

ISP at fixed energy [16]. First numerical tests of the methods were carried out by Quyen Van

Phu and Coudray [17]. Miinchow and Scheid modified the Newton-Sabatier method under

the assumption that the potential is known in the outer region, i.e. in the infinite interval

(jRint,oo) [18]. That method was developed and tested for neutral particle scattering. Later

it was extended by May et al. [19] to consider charged particles, so that the approach became

applicable for analysis of elastic experimental data from heavy ion scattering [20].

The modified Newton-Sabatier method has been extended to facilitate inversion of in-

elastic scattering under the assumption that the underlying transition interaction does not

depend on the angular momentum of the relative motion [21]. With that assumption, only

monopole transitions could be described. The method was tested for the case of neutral

particles using coupled square wells [22] and using coupled Woods-Saxon potentials [23].

For the case of charged particles, a transformation of the 5-matrix was established by which

the actual ISP is reduced to the same form as for the case of neutral particle scattering [24].

That extended modified Newton-Sabatier method was the first exact attempt to solve the

ISP with coupled channels at fixed energy. In this paper we report on a new approximation

scheme which allows the calculation of angular momentum dependent coupling potential

matrices from an ^-matrix which is given at fixed energy. This inversion scheme also is

based on the modified Newton-Sabatier method.

In the next section we review the coupled channel radial Schrodinger equations for the

case of multipole transitions. In section III a procedure analogous to the Newton-Sabatier

method is presented which is based on a kernel needed for the calculation of the inverse

potential matrix. In section IV an ansatz for this kernel is introduced, while in section

V we discuss the solution of the ISP for multipole transitions and neutral particles. An

application of our method to a model with analytic dipole interactions is given in section

VI and a summary and conclusions follow in section VII.

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II. FORMULATION OF THE COUPLED CHANNEL PROBLEM

As an example let us consider the nuclear reaction of a projectile nucleus with mass

number Ap and charge number Zp with a target nucleus with mass number At and charge

number Zt. These numbers shall stay constant during the whole scattering process thus

excluding transfer reactions. We presume that the scattering of the two nuclei is described

by the Hamiltonian:

H(r, 0 = T(r) + /»(£) + Wft 0. (1)

T(f) is the relative kinetic energy, the Hamiltonian /i(£) describes the intrinsic states of

the nuclei, and W(r,£) is the interaction operator which couples the relative motion to

the intrinsic dynamics of the particles. The vector f connects the centers of masses of the

projectile and the target, and £ designates the set of intrinsic degrees of freedom. The

eigenfunctions Xa (0 and energy-eigenvalues ea of the Hamiltonian are solutions of:

where a denotes a set of quantum numbers which uniquely describe the eigenstates of

The interaction operator W(f,^) can be expanded in multipoles by:

(3)

A v

The multipole moments Q\v describe the coupling between the relative and the intrinsic

motion, and the Y\u are spherical harmonics depending on the angular coordinates of f.

The intrinsic state a has the intrinsic angular momentum Ja, which is coupled with

the angular momentum of the relative motion £a to specify the total angular momentum I:

Joc®^a = I- Thus, the channel wave functions can be written as a tensor product:

The index a = 1,..., N denotes the channel number, where N is the total number of channels.

Each channel a is fixed by the quantum numbers JQ and ia, and the eigenenergy eQ. All

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other quantum numbers, which are necessary to uniquely determine the scattering channel

either are constant in all channels, e.g. the mass and charge numbers, or are determined

by the before mentioned three values, e.g. the channel energy Ea = E — ea. To get the

eigenfunctions for the Hamiltonian of Eq.(l) we need to multiply the channel wave functions

with the radial wave functions R^^r) and sum over all special solutions of the eigenvalue

problem

O- (4)

Thus we get

< = E E ; 4 W ftaYea(Q) ® Xa(0]£ •

an I '

(5)

The index n denotes the number of degenerate solutions in every channel. Every one of

the a — 1, ...,N differential equations, Eq.(4), has as many linearly independent solutions

as there are coupled channels (i.e. N solutions). With the Hamiltonian of Eq.(l) and the

eigenfunctions specified in Eq.(5), the Schrodinger equation, Eq.(4), can be transformed to

[25]:

e°"

E]

RLir) ="

For every total angular momentum / and every channel a there is one such equation. Here,

// denotes the reduced mass of the two nuclei:

ApAt

where mu is the atomic mass unit. The potential matrix V is a sum over products of angular

momentum dependent and independent parts

Ki(r) = EG>i»(O, (7)

with

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(8)

The reduced matrix elements of Q\(r,£) depend on the multipole order A, but not on the

angular momenta /, £a and 1$.

III. THE INVERSE SCATTERING PROBLEM

The Schrodinger equation, Eq.(6), defines the inverse scattering problem:Given the

asymptotic wave function R^if > Rmt) in the outer region, calculate the potential ma-

trix V^g(r) in the region of the nuclear interaction 0 < r < i?jnt. To bring Eq.(6) to a form

formerly used in Ref. [21] we rewrite Eq.(6):

(rl (9)

13=1

with the differential operators

- 6Q] 5Q0 - Vi(r)| + (/(/ + 1) - £*(£* + 1)) V (10)

To solve the ISP given by this equation, we proceed in a way analagous to that used be-

fore [21], by choosing a symmetric reference potential matrix V^{r) — V^{r) with known

solutions R°Jn{r) of the reference Schrodinger equations

E D^^R^r) = 1(1 + l)R°Jn(r), (11)

where the differential operators D^(r) have the same form as those of Eq.(10) up to the

replacement of the potential matrix by the reference potential matrix V^(r). With these

known reference wave functions an ansatz for the wave functions belonging to the sought

potential matrix is made in the form of an extended Povzner-Levitan-representation,

RL(r) = Rl(r) ~ E frK^0I'(ry)R^n(r')%. (12)

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Since the wave functions (12) should solve Eq.(9), one obtains by inserting (12) into (9):

N

2h2 d

i i ->o r

"(r,r')} = 0, (13)

0,7=1'

where the boundary conditions

K.v'v0111

(14)

are used. Assuming that the kernel matrix Kv'v°' 7(r, r') is given or found, the corresponding

potential matrix V7(r) is given by:

r 0it flv

KyIy01 j , ,

h2 N

E

2/Lir

(15)

Here, we need to assume that the determinant of the matrix R0/(r) is different from zero

for any and all values of r except if the integral in (15) vanishes. The formula (15) leads to

the Newton-Sabatier class of potentials if the integrands in (15) are set to zero for all r,r':

N w N

^7/3 (r )Ka0 (r>r )• (16)

Then the potential matrix is simply given by

yi (r\ _ yM(r\ _ ?L — —

VaB\r) - VaB\r) ~n.. J~

(17)

Motivated by the integral kernel formerly used for the solution of the ISP at fixed energy

and with scattering channels coupled by monopole interactions [21], one might introduce

the following ansatz

N

'>'); (is)

n,n'=\

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an ansatz that fulfills the conditions (14) and (16). As the kernel is determined for r > Riat

by the given S-matrix, the spectral coefficients c'nn, can be calculated by using (12) as

described in [21]. The potential for 0 < r < Rint then follows from (17).

However, numerical test calculations using this ansatz and analytic potential matrices

showed that input potential matrices could not be reproduced even if large numbers of

channels N were taken into account. The reason for this is that the functional basis R7R0/

is too limited irrespective of the number of channels used. Thus, a new ansatz for the integral

kernel has to be found, which has a larger functional space than (18).

We note that (15) is also valid for kernels which do not fulfill (16). Therefore, one can

apply a larger class of kernels than the ones of the Newton-Sabatier method for the inverse

coupled channel problem. Then the problem arises to chose a reference potential so that

either the inverse matrix R0/(r) exists, or the integrands of Eq.(15) decrease to zero faster

than the determinant of the reference wave function.

IV. APPROXIMATE CHOICE OF KERNEL

The interaction potential matrix V7 consists of a product of known angular momentum

dependent coefficients C7Aj and the proper potential matrix elements v^(r) which are sought.

The ratio of the number of matrix elements V^p to the number of v*p is proportional to

(/max + l)/(Amax + 1) which is usually a large number (e.g. 7max ~ 20, Amax « 2). Therefore,

the i>-matrix elements contain sufficient information about the coupling potentials v^(r).

To exploit the special form of the potential matrix V7(r), Eq.(7), let us consider Eq.(17).

Integrating this equation gives the diagonal (r = r') part of the kernel matrix:

r'dr'

— 2^ °<*/3 I fc2 I / \vaf3\r ) va/3\r))rar- Viy/

As this diagonal kernel consists of an angular momentum dependent (namely the C^) and

independent part, summed over A, we can assume the kernel matrix elements of the form

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(20)

A=0

where the new kernel matrix elements JC*p should be independent of /, ia and 1$. The

coefficients Q1^ take account for the angular momentum coupling of the channels and are

defined as

I

1 : else

The coefficients C ^ ( / 0) approach their asymptotic values CI

apcoX very rapidly with in-

creasing angular momentum quantum number /. Hence, for larger values of / (small values

of A) all coefficients Q1^ become unity. If the kernel (20) would fulfill Eq.(16), then the

kernels K,^{r, r') with C^ = 0 would have to vanish for r = r' in order that the kernel (20)

is equivalent to (19) for r = r'. Then we can write

r->r — r) - ^a

o

{r ) vr

l

r

That the kernel matrix elements JC^p are independent of 7, £Q and ip is an assumption which

can have the consequence that the kernel Kv//v°//(r,r'), Eq.(20), does not fulfill Eq.(16).

This has the further consequence that the resulting potential matrix calculated according

to Eq.(17) is an approximation only.

Under the assumption of an independence of I, £a and £p we can expand /C^ (r, r') in a

larger basis than used in Eq.(18):

/max N

E <&Kn(r)tfL.(r'),

/=0 n,n'=l

^{ry)=Y. (23)

or inserted into Eq.(20)

CM ;(^')=E«E E c2.RL(r)Rt(r').

A=0 r=0n,n'=l

(24)

The summation over the total angular momentum /' is infinite in general, but we suppose

there exists a cut-off angular momentum, 7max, which is the total angular momentum of the

last partial wave which still contributes significantly to the scattering within the numerical

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accuracy to be considered. The effect of the choice of this cut-off angular momentum on

the quality of the inverted potentials within the Newton-Sabatier inversion method has been

discussed in detail in Ref. [22]. The results given there are applicable to the further method

described in this paper.

V. THE APPROXIMATE INVERSION METHOD FOR NEUTRAL PARTICLES

With Eq. (12) as the ansatz for the wave function and Eq. (24) for the integral kernel

matrix, the inversion procedure can be carried through in the same way as described in

Ref. [21] for monopole transitions. We first consider the scattering of neutral particles, for

which the convenient reference potential matrix Vo/ is chosen as zero. The reference wave

functions in that case are the spherical Ricatti-Bessel functions. We write (£a = £a(I))

R°an (r) = T" (r) = rkajea (kar)5an (25)

with the channel wave number ka = J — (E — ea). The solutions of the Schrodinger equa-

tion, Eq. (9), in the interaction free region (r > i?int) are given by

TLir > Rim) = \iyjKhs [hja(kar)5an - S^ft* (kar)] ,

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(26)

where hfa = ±i(jza±inea) are the incoming (—) and outgoing (+) spherical Hankel functions,

respectively, and S7 is the ^-matrix. The wave functions in channel a are written as the

superposition of all degenerate solutions in this channel,

(27)

n'=l

Together with the kernel matrix, Eq. (24), the set of equations for the radial wave functions,

Eq. (12), then takes the form:

* *

E

n'=l

/ Amax •'max \

TUr)Ailn + £ Q^ £ T£,(r)&M>) = ^(r)

A=0 /'=0

(28)

\

/

with the new matrices therein defined by

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Li!'(r) = kl f nn{I){knr')3in(n{knr')dr\

JO

(29)

C = E AL»<#>«>- (30)

Eqs.(28) areiVxiVx (/max+l) linear equations for the unknown coefficients Al

nn, and 6™, and

for the wave functions T^n(r) in the interaction region r < Rmt. They are solved in a two-

step procedure. In the first step the coefficients A^n, and b1^, are calculated by using these

equations in the outer region r > i?int. In this region the wave functions are known when

the 5-matrix is given. Because the number of coefficients is N x N x (Jmax +1) x (Amax + 2),

we have to solve Eqs.(28) on Amax + 2 radii r^ > Rint or on more radii with a least square fit.

If these coefficients are known, then in a second step, Eqs.(28) are solved at discrete radii

0 < 7-j < jRjnt and the wave functions are obtained in the interaction region. Then the kernel

and potential matrices can be calculated with the equations stated in section III and IV.

In the case of charged particles, the S-matrix can be transformed to an asymptotic

constant potential using the modification technique developed in Ref. [24]. That new S-

matrix can then can be used as input for the solution of the ISP as though the particles

were neutral ones.

VI. APPLICATION TO ANALYTIC DIPOLE-INTERACTIONS

The coupled channels inversion procedure has been applied to the case of a dipole channel

coupling interaction supplementing single channel optical potentials, viz.

V&(r) = MK.(r) + iWa{r)) + C>^(r). (31)

For simplification, we replaced the orbital angular momentum-dependent term in Eq.(6) by

—1(1 + l)/r2 which means a dependence on total angular momentum. In that case the

2fj,

Bessel functions jea in Eq. (25) have to be replaced by jj and the indices ln(I) and ln(I') in

Eq. (29) by I and /', respectively. With this approximation, the kernel matrix (24) fulfills

(14) and (16).

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As we consider a pure dipole interaction, we can drop the part of the kernel matrix (24)

describing other transitions. This means that we limit the sum over A in (24) and (28) to

A = 1 in the expectation of a sufficient reproduction of the potential (31).

As a test example, we consider the scattering of two nuclei with mass and charge numbers

and the charge radius

Ap = 4, Zp = 2, At = 12, Zt = 6, Rch = 3.3 fm (32)

which corresponds to a hypothetical system 477e +12 C. We chose four coupled channels,

namely

a = l, 4 = 7, Ja = 0, ea - OMeV,

a = 2, £a = I, Ja = 1, ea = 3AfeV,

a = 3, 4 = 7 + 1, Ja = l, ea = 3MeV,

a = 4, 4 = 7 - 1 , Ja = \ ea = 3MeV. (33)

There are a ground state with Ja = 0, ea = 0 and an excited state with Ja = 1 and

ea = 3MeV. The channel a = 4 exists only for 7 > 1.

For these channels the quotient matrix Qn is shown in Fig. 1. Only the matrix elements

Q[\ = Q[\ and Ql2\ = QQ (dotted curve) and Q[\ = Q{\ and Qg = Q$ (dashed curve) are

different from 1 for smaller quantum numbers 7 of the total angular momentum. Note that

the quotient matrix is bound in a very small interval around 1.

To test our inversion procedure we first introduce an analytic potential matrix (31) from

which the input S-matrix for the inverse problem then was calculated. We chose

Vj

Va(r) + iWa{r) = --^

^^ : forr>/?Ch,

Z 7,^ ( r^ \ (35)

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Tables I and II contain lists of the parameters Va, Rva, o-va, WDa, RDa and aDa of the optical

potential (34) and of Va/3, RvaB and aVa(i of the coupling matrix (36), respectively. With

the potential matrix (31) the S'-matrix was then calculated for / = 0,1,2,..., 30 at a center

of mass kinetic energy Ecm of 80 MeV by numerical evaluation of the coupled set of the

Schrodinger equations (9). The resultant S'-matrix was transformed to one with asymptotic

constant potential at re = 10.0 fm with the method described in Ref. [24]. This modified

S'-matrix was then used in the inversion procedure.

The radii r\ and ? ~ 2 for the calculation of the unknown coefficient matrices A and b

were taken as 10.1 fm and 10.2 fm, respectively. For four channels and 7max = 30, the set

of equations (Eq. (28) restricted to A = 1) to be solved consists of 978 coupled equations

for A and b. The potential was calculated with a step width of Ar = 0.05 fm in the

interval [0.05 fm, 10.2 fm]. At every of the 204 radii of this interval, 123 coupled equations

for a=l,2,3 and 120 coupled equations for cv=4 had to be solved to get the wave functions

Tln(r) and then the total potential matrix via Eq. (17).

Fig. 2 shows the inverted potential matrix for the quantum number / = 1 of total angular

momentum by dashed and dotted curves in comparison with the analytic input potential

matrix (solid curves). As can be seen the approximate method yields good results. While

the zero potentials (from forbidden transitions for example) are not shown in the figure, they

are also reproduced with a high accuracy. From the potential matrix all other potentials (for

/ = 0,2,3,..., oo) can be calculated simply by multiplying with the appropriate coefficients

The inverted potentials oscillate slightly around the input ones. This is typical for

all Newton-Sabatier-like inversion methods. The reason for and the energy dependence of

these oscillations have been discussed in detail in Ref. [21]. It is also of note that there is

no difference in qualities between the inverted potentials shown here and those obtained in

earlier works [14,19,21] which used exact inversion methods for uncoupled channels problems.

We now investigate the quality of the reproduction of the input S'-matrix by the inverted

potential matrices. To do this we calculated the S'-matrix by solving the coupled Schrodinger

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equations (9) with the numerical potential matrix obtained by the inversion procedure.

These 5-matrix elements are shown in Figs. 3 and 4 in comparison with the input ones.

The reproduction is very good. Note that the scale of the individual diagrams vary, so that

absolute errors of small S-matrix elements (for instance S13 and 531) are emphasized. The

accuracy of results also depends on the radial step size used in the numerical calculations.

To get optimum input data, the original 5-matrix elements were calculated with a step

size of Ar = 0.01/ra. To speed up the inversion calculation, the potential matrices were

calculated therein with a step size of Ar = 0.05/m.

VII. SUMMARY AND CONCLUSIONS

In this paper we have extended the modified Newton-Sabatier method for coupled chan-

nels with monopole transitions to the general case of multipole transitions. The basis of

our method is the ansatz of the wave functions in form of an extended Povzner-Levitan

representation. This ansatz contains a kernel which determines the wave functions and the

potential matrix. If the partial differential equation (16) is fulfilled, the potential matrix can

be obtained in the form (17). The essential point of our method is an ansatz for the kernel

(Eq.(24)) which is suggested by the form of the coupling matrix elements. The functional

basis of this kernel is large enough to determine the full potential matrix V7(r).

As a first application of this method we treated a model case with dipole-transitions. The

approximation of this successful treatment is the use of Eq.(17) in calculating the potential

matrix with the kernel (24) thus presuming, that (24) fulfills (16) and (14). Since the

coupling coefficients C^p already approach their asymptotic values (J — > 00) for small values

of / if the multipolarity of the transition is small (A < 2) and if the angular momentum

quantum numbers Ja of the states are not too high (Ja < 2), the partial differential equations

(16) of the kernel are approximately fulfilled as proven by numerical calculations. This

establishes the quality of the inversion in the model with dipole transitions demonstrated in

this paper and with quadrupole transitions carried through in Ref. [25].

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An exact ansatz for the kernel which is general enough, reaching farther than Eq.(18),

and fulfills Eq.(16) is not known to us for the inverse coupled channels problem at fixed

energy with multipole transitions. Further research on this subject is needed.

The method presented in the paper is now suitable to be applied to experimental cross

sections of reactions with inelastic excitations of the scattering partners. Here, the problem

arises to extract a reliable and complete S-matrix from the experimental data. In most

cases, only a row of the S'-matrix can be extracted from experiment and the rest has to be

added by physical reasoning.

Acknowledgments: One of the authors (ME) is thankful to the support of the DAAD and

the hospitality during his stay at The University of Melbourne. He also received a fellow-

ship of the DFG-Graduiertenkolleg "Theoretische und experimentelle Schwerionenphysik".

This work has also been supported by DFG (Bonn) in a collaboration project between the

universities of Giessen and Budapest. One of the authors (BA) thanks for partial support

of OTKA.

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