Chapter Five
SPIN OBSERVABLES
IN POLARISED
PROTON-HELIUM ELASTIC SCATTERING
Introduction
Elastic scattering transitions have been extensively studied over
the years using the proton as a probe not only to extract information about the
spin dependence of the nuclear force but also to test the validity of various
theoretical models of nuclear structure and reactions. As part of such general investigation, the
few-nucleon systems, 3He and 4He, are important nuclei
for theoretical and experimental studies. The prediction of the properties for
the few-body nuclear system represents a
challenging test for existing theoretical models, many of which are based on
various types of few-body nuclear forces, that have been developed over the
years [167]. However, recently, the shell model approach [168] has been
considered with such light mass systems and very good results for the
structures have been obtained therefrom when large basis spaces and NN g
matrix elements have been used. That has been so with proton scattering data
analyses as well [74].
Thus we have OBDME and SP wave functions which enable us to make
predictions of p-3,4He scattering by defining g-folding optical potentials. Note that with p-3He
elastic scattering, there are contributions from angular momentum transfer (I)
values of 0 and 1. The set of OBDME with I = 0 define the nonlocal optical potential. The I=1 contributions are small
[74] but are included in all
results shown here. They have been
evaluated as in Ref. [74] by using a DWA.
We consider herein just what may be
achieved with that approach in analyses of 800 MeV
scattering allowing
minimal relativity. The process has been used [74] with some success to analyze
scattering data taken
with 200 and 300 MeV incident proton energies. Thus the data [169] have been analyzed
using the structure for 3He
given in that study [74]. The folding to give the optical
potentials have been made using effective interactions built from the g
matrices of the complex NN interactions [17] based upon the
BCC3 base interaction, as was used in my
calculations of p-12C at 800 MeV.
Shell
model structures of 3,4He
The structures of 3H and 3He
have been one of the successes of few-body physics and of the Faddeev approach
in particular. However, it is instructive to consider a shell model description
of such light mass nuclei as this approach offers an alternate means to
investigate the correlations in the wave functions that are naturally
contained, but can be difficult to identify, in the few-body schemes. To be
relevant, this description must give the basic static and reaction properties
of the mass 3 nuclei in reasonable if not as good agreement with observed
results. The structure studies of Navratil and Barrett [170] show
satisfactory agreement.
The simplest shell model that may be
constructed for these mass 3 nuclei is to take 3 nucleons in the 0s shell. That
model however does not involve correlations in the ground state wave function
that implicitly are included in the solutions of the Faddeev equations. Such
correlations can be inherent in the shell model wave functions but only from
shell model calculations made with much larger model spaces. Even so,
convergence with basis size on some properties is slow. For example, from
calculations of the ground state of the 3He performed in a shell
model including up to 32
excitations [170], the binding
energy is still a few percent away from the exact value given by the Faddeev
solution for the specified NN interaction.
excitations [170], the binding
energy is still a few percent away from the exact value given by the Faddeev
solution for the specified NN interaction.
However, the binding energy reflects the
large distance properties of the ground state wave function. On the other hand,
and for the momentum transfer values usually involved, most scattering
processes are sensitive to details of the wave functions within the body of the
nucleus. Hence the interests in using a large space shell model wave function
in calculations of the elastic scattering of electrons and protons from 3He.
shell model using the G-matrix interaction of Zheng et al.
[171]. The shell
model code OXBASH [172]
was used to obtain those wave functions. From that study the wave
function of the ground state of 3He segments as 
,
and
that of the ground state of 4He
segments as
.
Although both nuclei are dominantly 
in nature, the
significant mixing of higher 
components. These
binding energy calculations are of a similar accuracy to those made using the
few body techniques [173-175]. The lack of
convergence in the binding energies with this wave function is demonstrated in
the strength of the 
one. But the essential aspect is that this wave function well
describes the matter properties of 3He to a radius of import for
electron and, at lower energies, proton elastic scattering [74].
in nature, the
significant mixing of higher
The Zheng interaction has a unique
property not associated with the usual fitted (phenomenological) interactions.
It is defined by the G-matrix elements of a realistic NN potential which
required specification of HO SP wave functions at the outset. Thus, in
principle, there are no parameters left for adjustment in making analyses of
electron and proton scattering observables. The longitudinal form factor
for elastic scattering of electrons found using the specified SP wave functions
[74] is almost an exact reproduction
of the data. But the transverse (magnetic) form factor for electron scattering
from 3He was underpredicted by an order of magnitude. However, this
magnetic form factor is a small magnitude quantity.
Analyses of proton scattering data from 3He
at 200 and 300 MeV was also made in that study [74]. Such analyses required the
additional specification of the (complex and nonlocal) optical potential, which was derived from the self
same g-folding approach that has been adopted herein. That
description then was also parameter free, and the predictions for the 200 and
300 MeV proton scattering observables agreed well with data [74]. Those results encourage
credence in use of many-body methods to give a satisfactory description of 3He
within at least a sphere of a few fermi radius,
so long as the structure model is defined by a large basis space
model.
Results
and discussion
Results
from 3He
The differential
cross section, analyzing power, and the spin transfer
observables for 
elastic scattering at
800 MeV proton incident energy are compared with experimental results in Figs. 5.1, 5.2, and
5.3.
elastic scattering at
800 MeV proton incident energy are compared with experimental results in Figs. 5.1, 5.2, and
5.3.
Figure 5.1: The differential cross
sections (top), and analyzing powers (bottom) from the
elastic scattering of 800 MeV protons from 3He are compared with the
results found using the g-folding optical potential (solid curves). Data were taken from Ref. [169].
Figure 5.2: The spin observables, DNN (top)
and DLL (bottom)
from the elastic scattering of 800 MeV protons from 3He are
compared with the results found using the g-folding optical potential (solid curves). Data were taken from [169].
In Fig. 5.1, the differential cross
section and analyzing power are compared with the results found using the g-folding optical potential. Those predictions are displayed
by the solid curves. The cross section and analyzing power data span a
scattering angle range to 50o in the center of mass, with the cross
section values from 30o onwards being of the order of 0.1 mb/sr.
Dashed vertical lines are given in each
plot to note where the cross section has become that small. This indicates
where the model calculation may break down. In comparison to measured values of
the cross section it does.
Small changes in calculated phase shifts
as would be wrought by the uncertainties in specification of the nonlocal optical potential as well as by what one could expect (or hope)
from higher order contributions, will affect the results at the larger
scattering angles where the scattering theory and/or the approximation inherent
with its implementation are not well defined. This may be defined as the
``credibility limit" of the theoretical analysis method used. To that angle the predicted cross section
agrees very well with the data. The calculated analyzing power agrees well with the data to 25o
and trends towards the minimum at 30o without reaching a null value.
Figure 5.3: The spin observables, DSL (top), DLS (middle)
and DLL (bottom)
from the elastic scattering of 800 MeV protons from 3He are
compared with the results found using the g-folding optical potential (solid curves).
The spin transfer observables, DNN, DLL, DSL, DLS and DSS are compared with the results of calculations
in Figs. 5.2 and 5.3. Over the range to 30o the experimental data
for the spin transfer observable, Dnn, is practically 1.0 and
the theoretical calculations reproduce this behavior. Again the theory
credibility limit (at 30o) is shown in each panel by the vertical
dashed lines. To that limit, the variations in measured spin observable values
are quite well reproduced by the results of calculation. In particular DLL
is very well defined; the calculated values of both DLS and DSL
are a little too large in magnitude, while that of DSS is a
bit too small.
Results
from 4He
Predictions of the
differential cross sections and analyzing powers from the scattering of
800 MeV polarized protons from 4He
are given in Fig. 5.4 where they are compared with the experimental data of
Courant et al. [176].
The differential cross sections are well reproduced to 25o
scattering angle. At the larger scattering angles the cross section values are
the order of 0.1 mb/sr. Analyzing powers are not well reproduced. The trend up
to 40o scattering angle is similar to the data but calculation
failed to reproduce the minima.
The spin transfer observables DLL, DLS, DSL and DSS are compared with the results of calculations
in Fig. 5.5 and Fig. 5.6. Data were taken from Ref. [177].
Clearly the general shapes of all the
spin observables are described by the results of those
calculations but there are some problems with the magnitudes. Since the spin
observables are dependent on the differential cross sections, and for 4He
the elastic 800 MeV proton scattering
cross sections are not as well reproduced by calculation as one would like, the
predicted spin observables are affected and it is not surprising that fits are
not as good as expected.
Figure 5.4: The differential cross
sections (top), and analyzing powers (bottom) from the
elastic scattering of 800 MeV protons from 4He are compared with the
results found using the g-folding optical potential (solid curves). Data were taken from Refs.[176].
Figure 5.5: The spin observables DLL (top)
and DLS (bottom)
from the elastic scattering of 800 MeV protons from 4He are
compared with the results found using the g-folding optical potential (solid curves).
Figure 5.6: The spin observables DSL (top)
and DSS (bottom)
from the elastic scattering of 800 MeV protons from 4He are
compared with the results found using the g-folding optical potential (solid curves).
Conclusions
In summary, the BCC3 boson
exchange model NN interaction modulated have been used here by NN
optical potentials with which the SM97 NN scattering phase shifts to 2.5
GeV were reproduced to specify NN t- and g-matrices at 800 MeV.
Coordinate space effective interaction
forms that map those t- and g-matrices have been determined and
then used in a g-folding process to specify a complex and nonlocal optical potential for 800 MeV protons incident on 3,4He.
The structure of the target used in that
folding was determined from a large space shell model calculation; the ground
state wave function of which leads to an electron scattering longitudinal form
factor in good agreement with measured values. Thereby all quantities required
in the folding process have been preset to make solution of the associated
nonlocal p-3,4He Schrodinger equations
predictive of the scattering phase shifts, and so of the differential cross
sections and spin transfer observables. Most predicted results agree quite
well with the observation for momentum transfer values to that at which the
cross section is of the order of 0.1 mb/sr.
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