Coordinate
space optical model
The conventional coordinate space approach to study elastic NA scattering
is a phenomenological one in which it is
assumed that the scattering wave function satisfies a Schrodinger equation containing a local and energy dependent
potential,
But one can anticipate that the
potential U(r; E) should be nonlocal as well as energy dependent. Indeed, nonlocal
potentials do result from microscopic treatments based upon NN interactions.
The
folding model of the optical potential
In practical applications, one uses the
real and imaginary NA optical potentials calculated to first order in NN
t or g matrices as effective interactions. An effective NN
interaction in coordinate space is parameterized such that its plane wave
matrix elements reproduce as best possible the on- and off-shell properties of
those t and g matrices.
Historically, there was concern that no
such parameterization scheme could be designed to guarantee sufficient accuracy
in reproduction of all the important on- and off-shell properties of the t
and g matrices. The first of such parameterizations did not
have that specifically in mind as it was designed to be a significant
improvement on the then existing transition operators for inelastic scattering processes. Thus, at the time, it was a surprise to find
that parameterization led to folding models of the optical potential that gave very good results for elastic scattering [22, 41].
The original parameterization [22, 41] used a potential ersatz of superpositions of
Gaussian or Yukawa form factors. With that ersatz, the strengths
of those form factor terms had to be energy and density dependent. The
parameterization scheme was simple and based directly upon radial wave
functions of the LS and BBG equations. Later the fitting procedure was
altered to directly map large sets of values of on- and off-shell t and g
matrix elements [42].
A key feature of this scheme is that the form factors remain local and
of the form of superpositions of Yukawas as such is required with the programs
of import for analysis of elastic, inelastic, and charge exchange reactions,
DWBA91 and DWBA98 [30, 31].
With ni being shell
occupancies of the target, the optical potential in coordinate space can be defined by
where the subscripts D, E designate the direct and exchange contributions,
respectively. The coordinates r and
r' are projectile coordinates and the summations are taken over
the occupied bound single particle states for which the shell occupancies
(actually OBDME) are ni. The basic ingredient in this approach is the
choice of the direct and exchange g functions which are mixtures of the NN
channel terms of the effective NN interaction. In principle they should be evaluated in the
finite system studied using the most detailed structure information of the
target available. To date that is
impractical. It suffices to assume that
they can be defined locally in terms of the parameterized NN interaction
for the appropriate energies and at densities related to those from infinite
nuclear matter systems, i.e.
Ignoring spin attributes temporarily,
with this form of an optical potential, the Schrodinger equation for elastic scattering is
where the Coulomb potential, usually that of a uniformly charged sphere, is
included in the direct term. Using standard partial wave expansions, the radial
Schrodinger equations have second order integro-differential form,
This reduces to an uncoupled system of
equations for a target with angular momentum J = 0, for which I = 0
only. Likewise for nuclei with J=/0 by considering only
the I = 0 contributions, one obtains a similar uncoupled set of
equations. Collectively they are
The program suites, DWBA91 and DWBA98 [30, 31], evaluate the S
matrix elements from solutions of those uncoupled integro-differential equations. For nuclei with J=/0, I=/0
Nuclear
medium effects on NN interactions
It has been many years since Brueckner and his collaborators [43, 44] established a
particularly useful perturbation theory of nuclear many-body systems based upon
the NN interaction. With the
apparent success of the shell model and thus the implication that nucleons
within a nucleus move independently, short ranged particle correlations [43] did not complicate that
theory. Essentially, their effect was to
create an average field [44]
in which the nucleons move. In
this section, the Brueckner theory by which that average field can be obtained
from NN g matrices [45] is given. In the context of infinite nuclear matter,
the average field modifies the average kinetic energies in the intermediate
propagators [46], which, when antisymmetrization
is included, define the BBG integral equations.
Solutions of those BBG equations are called the g matrices. The Pauli
principle affects those integral equations by restricting intermediate
scattering states to be excited states of a fermionic many-body system. That is
defined as Pauli blocking.
In the development of a Thomas-Fermi
theory of large nuclei, Bethe [47] established
that the local density approximation (LDA) is a convenient connection between the infinite
matter and finite nuclear systems. He found
the LDA to be valid for densities in excess of ~17 % of the central value. In that paper, Bethe also noted the
importance of damping effects due to the auxiliary potentials in the propagators
of the Brueckner theory.
This importance was revealed also by the
study of Yuan et al. [48] on the mean
free path of nucleons in infinite nuclear matter. Allowing for Pauli blocking effects at normal central densities (kf
= 1.4 fm-1) they found the nucleon mean free path to be almost
double that specified by using the on-shell tr approximation. Even so the calculated mean free paths fell
short of the empirical value, due in part to the neglect of the auxiliary
potentials in the calculations. Cheon [49] allowed for
that by using the effective mass approximation to obtain good estimates of the nucleon mean
free path. Other studies [50] confirmed that
effects of the medium can vary predictions of nuclear matter absorption
potential strengths by as much as a factor of five from values obtained using
free NN t matrices. In any event, a primary requirement of
microscopic theories of nucleon scattering from nuclei is to know how free NN
interactions are influenced by the presence of a nuclear medium.
All nuclear medium calculations require
evaluation of the BBG g matrices off-shell.
For example, in an analysis of knock-out reactions [51, 52], it was found
that extraction of nuclear properties becomes tenuous if conventional on-shell
approximations are made. Intriguingly, realistic one boson exchange potentials,
as well as those obtained by inversion of the NN phase shifts [53, 54], have very
similar off-shell characteristics.
Problems therefore lie with the specific medium correction effects to
the on-shell values of the free t matrices to be used in defining optical
potentials with the tr and gr approximations [55-59].
Nuclear
matter g matrices
Consider a fast nucleon of momentum p0
in collision with another of momentum p1 which
is embedded in infinite nuclear matter. The
Fermi-Sea is defined by a momentum kf. This collision involves a relative momentum
and
a center of momentum
In these coordinates, and with separate
particle energies
the total energy of the two particles
becomes
After the scattering there is an
equivalent set of momenta
the scattering g matrix is a solution of the BBG equation, q and q' being momentum variables,
where
Other medium effects are subsumed in an auxiliary
potential
Both Pauli blocking and the energy denominator are functions of
the integrating momentum k'.
It is customary to approximate these by their monopole, or angle
averaged, values. This is justified since the g matrices are weakly dependent upon the momentum K
[61]. Standard partial wave expansions yield well
behaved Fredholm integral equations of the second kind,
with
In the context of nuclear matter, Legindgaard
[62] has shown this angle averaging
to be a good approximation. Subsequently
Cheon and Redish [63] demonstrated
that this remains valid when positive energies of 300 MeV and normal nuclear matter
densities are considered. No such detailed calculation of the accuracy of angle
averaging of the energy denominator exists but a sensible approximation to use
is
U(p) are auxiliary
potentials.
Another simplification of the energy
denominator is the effective mass approximation in which the individual single particle
energies are approximated by
where m* and Ui
are constants. Since the total energy is the sum of the incoming and struck
nucleon energies, one then gets
It is natural to seek a better
prescription of the auxiliary potentials.
One way is to consider the arguments of the potential in Eq. (2.23) with
the Brueckner angle averaged prescription [64, 65],
and an average value for K→K when
determining the single particle potential,
...........................(2.30)
The interplay between U(p) and
the g matrices is given by the sum over two-body channels,
The
integration weight X(k) is
and
..............................................................................................(2.33)
The momentum k is an on-shell
relative momentum value which, with the chosen values of kf
and p, specify K. The energy
defining
the reactance matrix K(a) which is related
to the g matrix by
...............................................(2.36)
involves
Since the average momentum
is dependent upon the energy of the incoming nucleon and upon
the Fermi momentum, both the g and K(a) matrices are
functions of these. Note that if the
Fermi momentum is zero, Eq. (2.35) simplifies to the Heitler equation [66]. It is customary [60, 67] to use the real parts of the on-shell g
matrices in the summations to define a real auxiliary
potential for the BBG equation.
The range of momentum for which Eq.
(2.31) is used to define the auxiliary potentials has been a point of
discussion. A standard choice has been to set U(p0) to zero
if p0 is greater than kf. This is not truly
self-consistent as outside the Fermi surface a zero set potential on iteration
does not give convergent g matrices. It has also
been argued that a continuous choice of the
auxiliary potential is more realistic as that allows for appropriate
cancellation of some higher order terms [68]. Also, with the
continuous choice, the appropriate behavior of the imaginary component of the
optical model potential in the vicinity of the Fermi surface
can be estimated [68, 69].
The utility of the K(a) matrices at
positive energies is that they reflect the off-shell character of the complete g
matrices and that character can be displayed for any
channel by Noyes-Kowalski f-ratios [70-72], which now are
medium dependent, as
with K(k, kf, p)
. Note again that this
ratio emphasizes off-shell behavior by scaling against the on-shell value so
that one must always bear in mind the actual size of the denominator when considering
any significance of f-ratios.
The effective interactions used
throughout have been deduced from g-matrices
found as described above. The influence of the Pauli blocking and of the mean field medium effects both on- and off- shell have been illustrated and
discussed in detail in the review [7].
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