Physics of
Nucleon-Nucleus Scattering
[For Postgraduate Students]
Dr Pradip Deb
BSc Honours, MSc (C.U), MAppSc (Medical Physics)
(QUT), PhD (Nuclear Physics) (University of Melbourne)
Discipline of Medical Radiations
School of Medical Sciences
RMIT University, Melbourne, Australia
Dedicated to
My teacher
Professor M. H. A. Pramanik
who taught me Quantum Mechanics and
how to love rationality
Preface
Cross sections from nucleon-nucleus
scattering and reactions are central quantities of import in many and diverse
fields of study. That has been so for almost a century and remains so today. For
example, nucleon reaction cross sections at many energies are important input for
studies of radioactive waste management by transmutation of long lived
radioactive waste into shorter lived products, which together with energy
production, uses accelerator driven systems. Such reaction data are basic in
other studies as well, e.g., in nuclear astrophysics, in nucleon radiation
therapy and protection of patients, in special material science for radiation
safety of astronauts and air crew as well as radiation damage and interference
effects in electronics, and also in basic nuclear physics with the advent of
beams of exotic radioactive nuclear ions for experimentation.
Those many applied and as well as basic
research fields all require a theoretical backdrop from which reliable predictions
of the nucleon-nucleus scattering can be made.
Only with such can credible calculations be made of radiation dosimetry,
of reaction rates important in exotic processes in cosmic and explosive stellar
events that are crucial in nucleogenesis in the universe, and of selective
prescriptions of nucleon structure within nuclei, particularly of neutron
distributions.
Within the last decade or so such very
credible theoretical predictions have been feasible. The central facet of that
success is the formulation of microscopic optical model potentials built upon realistic two nucleon
(projectile-target nucleon) interactions. Success has been found with those
potentials determined either in momentum or coordinate space. In this book, I have
concentrated upon the development and application of a coordinate space
specification of those nucleon-nucleus optical potentials.
A microscopic model specification of the
nucleon-nucleus (NA) optical potential has been obtained in coordinate space by folding complex energy and density
dependent effective nucleon-nucleon (NN) interactions with one-body
density matrix elements (OBDME) and single particle bound states of the target
nucleus generated (often) by large space
shell model calculations. As the approach accounts for the exchange terms in
the scattering process, the resulting
complex optical potential is nonlocal.
This model has been applied successfully to calculate elastic (and,
within the distorted wave approximation or DWA, inelastic) scattering of
protons from many stable and unstable nuclei ranging from 3He to 238U
at different energies between 25 MeV and 300 MeV. Differential cross-section as
well as analyzing power data have been reproduced by this model. The
approach has also been used with some success to explain proton scattering from
12C with energies to 800 MeV and, for the 3,4He
isotopes, at energies of 700 and 800 MeV. As the effective interaction and the
structure details are all preset and no a posteriori adjustment
or simplifying approximation is made to the folded optical potentials, the
observables obtained then are predictions.
The potentials
obtained have strong nonlocality and it is very important that such nonlocality
be treated exactly for quality results. It is also crucial to use effective NN
interactions which are based upon `realistic' free NN
interactions but which allow for modification due to nuclear medium effects of
Pauli blocking and an average mean field.
The differential cross sections and analyzing powers from the elastic
scattering of 25 to 40 MeV protons from many nuclei have been studied. Analyses
have been made using the fully microscopic model of proton-nucleus scattering
seeking to establish a means appropriate for use in analyses of radioactive ion
beam (RIB) scattering from hydrogen targets with ion energies 25A and 40A
MeV, since the procedure, under inverse kinematics, explains observed data from
the radioactive beam experiments in which exotic, halo nuclei, are scattered
from hydrogen targets. By this means it has been shown that 6He has neutron halo character while 8He
does not (although it does have a neutron skin). New results on 6He
scattering using data from GANIL have been used to illustrate that.
With a no parameter DWA, good predictions of inelastic
scattering data can be made also. Results so found further show how RIB
scattering data can be used to identify extended nucleon distributions in
`exotic' nuclei. The case of 6He, with inelastic scattering to
the 2+ (1.8 MeV) first excited state, have been used to demonstrate
this. The sensitivity of the inelastic scattering data to the structure of 6He
and the success of the coordinate space scattering theories based upon effective NN
interactions in analyses of proton scattering from stable
nuclei, open large perspectives for the study of the microscopic structure of
exotic systems.
A measure of the
neutron density of different nuclei has been sought from analyses of
intermediate energy nucleon elastic scattering. The pertinent model for such
analyses again is based on a coordinate space nonlocal optical potentials obtained from model nuclear
ground state densities. Those potentials give predictions of integral
observables and of angular distributions that, when compared
with data, show sensitivity to the neutron density. New results for stable
targets are discussed and reaction cross sections as functions of mass and of energy (at and
above 20 MeV) are shown. They also compare very well with observation and are
further evidence of the effects of the medium upon the interaction between a
bound and a continuum nucleon.
A simple functional form has been found that gives good representations
of the total reaction cross sections for the scattering of nucleons from nuclei. Such
is of value in studies as diverse as radiation therapy and protection and of
the spallation process in the search for treatment of radioactive waste.
Very special thanks are due to Professor
Ken Amos for his continual support, encouragement and enthusiasm. I thank
Professor Steven Karataglidis for his help in the use of the SHELL model code
OXBASH, and for the supply of OBDME. I also thank Dr. Peter Dortmans
for his help with XMGR program. I am thankful to Ajit Podder and A/Prof
Sharaboni Paul for their valuable suggestions and encouragement.
Pradip Deb
pradip.deb@rmit.edu.au
School of
Medical Sciences
RMIT University,
Australia
Chapter One
INTRODUCTION
The need for accurate values of
nucleon-nucleus scattering and reaction cross sections for energies to 300 MeV and above has grown
with time. Far from being a past study field providing just tabulations residing
in a data bank, new developments in old as well as new fields of study and
application require such values to be predicted for circumstances, and with matter, that may be termed exotic. In many such cases,
experimental values either have not been, or cannot be, measured to abrogate
the need for evaluation. Even where such experiments can be made, theoretical
analysis is crucial to ascertain the physics of the target system and of its
reactions.
Nuclear data for neutron and proton
radiation therapy and for radiation protection has long been a concern internationally
with numerous studies made of those topics under the aegis of the International
Commission on Radiation Units and Measurements for example. An excellent report
on such, from which much of the following information has been taken, is the ICRU report by Chadwick [1]. With neutron therapy, neutron
cross sections are needed to determine the production of secondary charged particles,
neutrons and gamma-rays. That information is needed to calculate the absorbed
dose taken by a patient. Further, beam
collimation and shaping as well as neutron transport in a patient is seriously affected by those
secondary particle and photon productions. Likewise estimations of energy
deposition in a patient crucially depend on having accurate values of nucleon
reaction cross sections. With proton therapy, such cross
sections are again vital, for, although protons passing through matter lose
most of their energy via electromagnetic processes, their nuclear
reactions within a patient produce many, and troublesome, neutrons. Those
neutrons are troublesome since they penetrate much further in a patient and so
can produce subsequent (heavy) charged particles which in turn enhance the
biological effect of the input radiation.
Radiation protection, of humans and
equipment, are very important topics for any system that operates especially
above about 10 km above the earth. Cosmic radiation at that height, about 50 % of which dose
equivalent is caused by neutrons, is then obviously an important consideration
for the health and safety of flight crews.
Electronic equipments in aircraft and space vehicles also have been
effected in their operation by the complex radiation fields that are formed by
high energy secondary radiation in the outer layers of the atmosphere. Indeed
the occurrence of errors in integrated circuits caused thereby is a regular
concern in both aircraft and space vehicles. Chadwick [1] notes that it is from fast
neutrons (10 to 150 MeV) that most such effects are caused and values of
reaction cross sections of such energetic neutrons from 28Si especially need be determined.
Another area of study requiring input
knowledge of nucleon-nucleus reaction cross sections is the emerging field of accelerator driven technologies.
High energy protons on a spallation target to produce a flood of lower energy
neutrons are a serious proposition to seek transmutation of radioactive waste into shorter lived (less nasty) radioactive
nuclei. Accurate cross section values need
be predicted so that reliable calculations can be made of the numbers of
spallation neutrons produced per incident proton, of radiation heating and
damage to samples exposed, of shielding design in accelerators as will be used in
new material science studies and for design of proton and neutron radiography
facilities.
All such topics discussed above
constitute a major reason for the establishment of radioactive ion beam (or in
the USA, rare isotope accelerator) facilities. Such are important also for more
esoteric studies. The ability to produce for experiment beams of nuclei that
lie off of the mass stability line and in the proton and neutron rich fields up
to the nucleon drip lines opens a vast new field for study of the way in which
nucleons can amalgamate into (quasi) bound systems. Already we know that some of the exotic nuclei
have (neutron) matter distribution noticeably extended from what one might
expect given the current model prescriptions of nuclear systems. In that
context, and using inverse kinematics, the scattering of radioactive ion beams
from hydrogen targets equates to proton scattering from the radioactive
nucleus. It transpires that proton-nucleus scattering is particularly sensitive
to the neutron matter distribution in the nucleus. Given a means of predicting
angular variables such as the differential scattering cross sections, one thus
has a good means of probing neutron attributes of a nucleus.
But not only is the nature of the
radioactive nuclear systems of interest in their own right, their character
needs be known for estimations to be made of the nucleogenesis in the universe as caused by cosmological
processes in the early universe and also by explosive events
(nova) of more recent times. It is well
known that considerable effect in that nucleogenesis occurs through scattering,
capture and radiation processes that proceed through nuclear chains not
involving the stable nuclei. The r- and rp-processes are examples.
Given the import of predicting
properties of nucleon-nucleus scattering well,
I present next a (brief) history of the developments that have lead to the current, fully
microscopic specification in coordinate space of the nucleon-nucleus optical potential; and with which I believe that
cross sections as needed in the above described fields of import can be given.
Elastic scattering is the predominant event in the interactions
of nucleons with nuclei. This process has been extensively studied over many
decades both experimentally and theoretically, and there now exist considerable
data on the scattering of nucleons from stable nuclei. All formulations of the
nuclear optical model for elastic scattering have in common an
allowance of flux loss from the incident beam to nonelastic channels when
energies are above inelastic and reaction thresholds. These model formulations
range from strong geometric forms to ones based upon complex potential
representations. The geometric approaches [2] remain valid and
appropriate for high energies, typically above 1 GeV, while use of complex
potential models is most appropriate for energies below that. Those complex
potential models broadly fall into two classes, the first being
phenomenologically formed and the other microscopically based.
The concept of a complex optical
potential as a single particle representation of NA
interactions dates at least to the study by Bethe [3] of
neutron-nucleus cross sections. All early optical potentials were
phenomenological. Studies of that phenomenology proceeded apace thereafter,
culminating in attempts to prescribe global forms for all target masses and for
projectile energies typically to 40 MeV. Tabulations of those potential parameters
have been made [4]. Likewise there
have been a number of reviews of the topic of which those of Refs. [5-8] are a selection
that I have found useful. Phenomenological and semiphenomenological optical
potentials are used still to interpret elastic scattering data as well as to
define the distorted waves required in DWA analyses of nonelastic processes. Likewise the semiphenomenological approach
has reached a very sophisticated stage. With it data from many nuclei, for
energies ranging from keV to GeV, have been analyzed successfully [9].
It has been about 60 years since Chew [10] and Watson and collaborators [11, 12] gave theoretical
justification for the NA optical potential built in terms of underlying nucleon--nucleon
(NN) scattering amplitudes. For sufficiently high incident energies it was
supposed that those NA interactions would be ascertained from free NN
scattering. Bethe [13] showed that the
cross section and polarization from the scattering of 310 MeV protons from 12C at forward scattering angles were consistent
with that conjecture. Then Kerman, McManus, and Thaler (KMT) [14] developed the
Watson multiple scattering approach expressing the NA optical potential by a series expansion in terms of the free NN
scattering amplitudes. Those
formulations result in the definition of an effective interaction between
projectile and the target nucleons.
Feshbach [6] and Adhikari and Kowalski [15] give lucid reviews of these
theories. They also give many details of
the NN scattering amplitudes and t matrices. However, adequate numerical implementation of
those theories of NA scattering did not follow for quite some time, in
part due to lack of knowledge of the underlying NN scattering
amplitudes. To some extent this spurred
study of NN phase shift analyses [16, 17] and the
development of NN potentials.
About 30 years ago there was a watershed
in the studies of NN and NA scattering. First, reliable NN scattering amplitude
and phase shift analyses [18] to the pion
threshold were made. Second the Nijmegen [19], Paris [20], and Bonn [21] NN
potentials were developed to fit those NN amplitudes and phase shifts.
Then the status of the microscopic NA optical potential theories was reviewed at the seminal topical
workshop in Hamburg in 1978 [22].
Finally, experimental programs in those years produced many and varied
high quality data sets for energies to 1 GeV [23], adding
incentive to effect implementations of those theories.
The review of Amos et al. [7] encompasses the
developments since that period. The most important of which was an
understanding of how NN scattering is altered in the presence of other
nucleons. Such is designated hereafter as an effective interaction. Those modifications are caused by the two
nucleons interacting within the nuclear medium and are due to Pauli blocking and mean field effects for both projectile and
bound state nucleons. In addition, there
are other effects due to the convolution of the NN scattering amplitudes with target structure that require
off--the--energy--shell scattering amplitudes. Also of importance is the complete
antisymmetrization of the A+1 nucleon scattering system which leads to
direct and knock out exchange amplitudes for NA scattering. The effects of such exchange amplitudes are
not small at most energies to 800 MeV and they are a source of nonlocality. That is also the case with the other medium effects
[24].
The underlying principle of
nonrelativistic multiple scattering theories is that the fundamental dynamics
leading to the NN potential is unaltered although medium effects vary
the NN amplitudes from what the free NN system define them to be. There are other approaches to ascertain
medium effects upon the NN potential.
One of these is to use quark-gluon dynamics directly [25]. That approach involves a
limited number of open parameters whose values, very often, are constrained by
chiral symmetry. With that means, NA
scattering has been analyzed allowing medium variation of coupling constants,
masses, and form factors of boson exchange interactions. However, that approach is relevant with
nuclear densities significantly above nuclear saturation as is the case in some astrophysical
problems.
If one assumes that only the free NN
t matrix on the energy shell is necessary in
calculations of the optical potential, the experimental NN
amplitudes can be used. But if both the on-- and off--shell properties of the t
matrix are important, suitable representations of those properties are
required. One of the first of such representations for the effective
interaction used a local superposition of Gaussians or Yukawas in coordinate space [26-28]. At the same time, Love and Franey
[29] also defined an effective t
matrix comprising a sum of central,
spin--orbit, and tensor components. However, their parameter values were chosen
to match the then existent NN phase shifts [18] in the energy range to 1 GeV. So their force was controlled solely
by the on--shell NN amplitudes. The Yukawa form factors were preferred since such forms were required
with the suite of programs then in general use for NA scattering. Those programs were early versions of DWBA91 [30]. The same structural form remains in use today
with parameters defined to match both on-- and off--shell properties of the NN
t- and g matrices as are required in use the code, DWBA98 [31]. With DWBA98 not only can one calculate
nonlocal microscopic NA optical potentials, elastic
scattering observables, and distorted
waves, but also a consistent evaluation of inelastic and charge exchange
reactions can be made with the same effective interaction acting as the
transition operator. These calculations include direct and knockout amplitudes
for all processes.
To obtain the off--shell properties one
relies on NN scattering theory. One such theory involves one boson exchange
potentials (OBEP); potentials
which are termed realistic when their use reproduce NN scattering phase
shifts. The most commonly used realistic
potentials are those given by groups from Nijmegen [32], Paris [20], Bonn [33], Argonne [34], and Hamburg [35]. These
potentials give very similar results for the phase shifts in all NN
channels and for energies below pion threshold.
This approach can be applied also in the
energy regime above pion threshold to specify t- and g matrices on-- and off--shell. For example, the basic OBEP approaches have been extended seeking to
incorporate other channel information [34, 36, 37], and a coupled
channel method for NN scattering above
threshold has been developed by Ray [38].
The nuclear target and its microscopic
structure is the other essential element in NA scattering theory. Originally,
the structure assumed was that of a simple independent particle model. Today refinements
and extensions to the shell model have made it possible to calculate wave
functions in a large space allowing all, or a large fraction of, nucleons to be
active. Other approaches to structure in
NA scattering analyses include projected Hartree-Fock methods and mean field
models. Irrespective of what model of
structure is chosen, one body density matrix elements (OBDME) must be specified to construct
an optical potential based on the NN interaction. To validate the wave functions obtained from
the assumed structure model, and hence OBDME, it is often necessary to analyze
complementary scattering data. Electron scattering form factors provide just
such data as they reflect the target charge and current densities that are
defined by the same OBDME.
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