An Approximation Method for Solution of the Coupled Channels Inverse Scattering Problem At Fixed Energy
ABSTRACT We present a method for the quantum mechanical inverse scattering problem 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 interaction. Good agreement between the obtained potentials and the input data is found. 03.65.N, 24.10.E, 24.10.H, 25.70.
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ABSTRACT: Scattering amplitudes are extracted from (elastic scattering) differential cross sections under the constraint that the scattering function is unitary. A modification to the Newton iteration method has been used to solve the nonlinear equation that specifies the phase of the scattering amplitude in terms of the complete (0[degree] to 180[degree]) cross section. The approach is tested by using it to specify the scattering amplitude from simulated data and comparing the result with the amplitude found by using an exact iterated fixed point method of solution. The (modified) Newton method was then used to analyze the cross sections from neutron--[alpha]-particle scattering at low nuclear energies ([lt]24 MeV) and from 1000-eV electron-water molecule scattering.Physical Review A 12/1994; 50(5):4000-4006. · 3.04 Impact Factor
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)
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
03.65.N, 24.10.E, 24.10.H, 25.70.
* This work is part of the doctoral thesis of M. Eberspacher, Giefien (D26),
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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 . 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 , while applications of his method
to analytic and experimental S-matrices later were made by Kohlhoff et al. .
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 , the finite difference method of Hooshyar
and Razavy , the generalized Darboux transformations of Schnizer and Leeb , 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.  can be applied. All those methods, however, can not encompass
scattering which involves a spin-orbit potential save under approximation .
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 , which Hoosh-
yar  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 , while Lun et al. found a way to
use Sabatier's formula directly for the inversion by a linearization scheme [14,15].
Newton-Sabatier methods are among the most successful methods for the solution of the
ISP at fixed energy . First numerical tests of the methods were carried out by Quyen Van
Phu and Coudray . 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) . That method was developed and tested for neutral particle scattering. Later
it was extended by May et al.  to consider charged particles, so that the approach became
applicable for analysis of elastic experimental data from heavy ion scattering .
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 . With that assumption, only
monopole transitions could be described. The method was tested for the case of neutral
particles using coupled square wells  and using coupled Woods-Saxon potentials .
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 .
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.
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:
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
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
Thus we get
< = E E ; 4 W ftaYea(Q) ® Xa(0]£ •
an I '
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
For every total angular momentum / and every channel a there is one such equation. Here,
// denotes the reduced mass of the two nuclei:
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)