Wednesday 21 August 2019

Physics of Nucleon-Nucleus Scattering - Chapter IIB


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,

..(2.14)

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

....................(2.15)

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.

..............(2.16)

Ignoring spin attributes temporarily, with this form of an optical potential, the Schrodinger equation for elastic scattering is


,.................(2.17)

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,

.(2.18)

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

............(2.19)

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 contributions evidenced in Eq. (2.18) are also evaluated in those programs using a distorted wave approximation.


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


and
,

the total energy of the two particles becomes

. .............................................(2.20)

After the scattering there is an equivalent set of momenta  

the scattering g matrix is a solution of the BBG equationq and q' being momentum variables,


, ..............................................................................................(2.21)

where   
 Pauli blocking operator. A simple realization of that operator is

  = 0 otherwise.  ..........(2.22)

Other medium effects are subsumed in an auxiliary potential 
   in terms of which the energy denominator is defined [60, 61] by



. .........................(2.23)

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, 


..............................................................................................(2.24)

with  
 and bars designating angle averaged values. Specifically, the angle averaged Pauli operator is

.............................................(2.25)

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

........(2.26)

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

........................................................(2.27)

where m* and Ui are constants. Since the total energy is the sum of the incoming and struck nucleon energies, one then gets

, ...............................(2.28)

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],

, ..................(2.29)

and an average value for K→K when determining the single particle potential, 



 if 



 if        
 ...........................(2.30)


The interplay between U(p) and the g matrices is given by the sum over two-body channels,

...............................................................................................(2.31)

The integration weight X(k) is

..........................................(2.32)

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 
   is that at which one evaluates the g matrices.  To facilitate calculation, the principal value of the integral is taken [60, 61, 66]

,   .............(2.34)

defining the reactance matrix K(a) which is related to the g matrix by

,.(2.35)

where [60]

 

...............................................(2.36)


involves

...............................(2.37)

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

...........................................(2.38)

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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