Physics of nucleon-nucleus Scattering - Chapter IIA

Chapter Two

MICROSCOPIC OPTICAL MODEL

All microscopic optical models of NA scattering that have been developed so far, depend on a (model) prescription of basic interactions between nucleons. Consequently, to date there is no theory of such scattering as exact or reliable as exists for electromagnetic scattering. Nevertheless the full-folding microscopic optical model has reached a very successful stage. In this review chapter some of the essential details of the developing process of the full folding coordinate space optical potentials, and of the input to the codes that have been used in the calculations, are reported briefly. All the relevant details have been given in much fuller form  in a review [7].

Formal theory of the nucleon--nucleus optical potential

In the optical model approach, the many-nucleon-problem is reduced to one for a single particle with the kinetic energy being that of the projectile. The optical potential accounts effectively for all of the complexity of scattering in the many body system. Note that as the term generalized optical model includes more than solely entrance channel phenomena, my use of the term optical potential is to be taken to mean the single elastic channel reduction.

The Feshbach formalism [39, 40] is a convenient device to use. One divides the Hilbert space of the A+1 scattering system using two projection operators P and Q. P projects onto the elastic channel and Q on to all others.  Thus P + Q = 1, PQ = QP = 0 and

With these projection operators, and with H_PQ = H_QP etc., the Schrodinger equation segments as

                

Using the second equation to eliminate from the first gives

 ........ ..  (2.1)

where outgoing wave boundary conditions are assumed. The Feshbach formalism reduces the many nucleon problem to an effective one body one by invoking explicitly the ground state and the influence of transitions to other channels from the ground state, with

....(2.2)

The single nucleon distorted wave  for elastic scattering is thus defined.  This optical potential is complex as the intermediate state propagator is complex (due to pole contributions).  It contains the whole complex spectrum of many body excitations of the projectile and target nucleons in bound as well as in continuum states, with

......................................................(2.3)

Thus the optical potential is identified by 

..........(2.4)

In general it is nonlocal, and energy dependent as well as being complex due to the second term.  With a local V, the leading term is real, local, and energy independent. That is so provided antisymmetrization is ignored. Multiple scattering effects are contained within the second term.

           

But this concentration of all multiple scattering effects in the second term is largely ineffective since realistic NN potentials are short ranged and multiple scattering may be significant. Treating such  explicitly is the leading idea behind the Chew [10] and Watson [11] multiple scattering theory.  They replace the potential in Eq. (2.4) by the complex NN t matrix. As a result, the leading term is now complex, nonlocal, and energy dependent.  In the simplest versions it is assumed that only pairwise interactions between the projectile and target nucleons are important. With the projectile tagged 0, and target nucleons tagged i,

..................................................................(2.5)

Furthermore, the NN t matrices are associated with the Lippmann-Schwinger (LS) equation

..........................................................(2.6)

while the NN  g matrices which will be used in the current approach  are associated with the Brueckner--Bethe--Goldstone (BBG) equation

.......................................................(2.7)

Here differs from the free NN propagator by including medium effects such as Pauli blocking and a mean field.  When such t  and g  matrices replace V  in Eq. (2.4)  the  formulations of the microscopic optical models  are  in the spirit of KMT [14]. Adhikari and Kowalski [15] define auxiliary operators that are useful in this development, such as

....................................................................(2.8)

and

.................................................(2.9)

with which a modified pA T matrix is defined by

............................(2.10)

The Feshbach subtraction technique then yields

..........................................(2.11)

and

............................................(2.12)

where the resolvent is that of KMT [14]

..........................................(2.13)

The optical potential itself then is the expectation value taken with the wave function attributes of the i^th nucleon in the target.  It is still a many-nucleon problem as evaluation requires knowledge of the excitation spectrum of the target.

           

In a lowest order model, the complexity of the problem is reduced to a weighted sum of one nucleon expectation values.  The weights, to first order, are the shell occupancy numbers.  The resultant optical potential is still complex but that is now due to the complex nature of the g matrix rather than because of higher order terms in the expansion of Eq. (2.12).  Neglecting medium effects in the g matrix defines the t matrix approximation.  If in addition, the antisymmetrization of the projectile and target nucleon wave function is ignored one has a very simplified fully off-shell tr approximation.

           

As will be developed subsequently, the zero angular momentum transfer I=0 component of the optical potential will be used in specifying radial Schrodinger equations whose solutions are the distorted waves.  There are also

 amplitudes contributing to elastic scattering from non--zero spin targets. They may yield re--orientation amplitudes even though, generally, they are small in comparison to the I=0 ones.  A practical means of evaluation of the relevant  

 scattering amplitudes uses the distorted wave approximation. 

Chapter IIB