Sunday 18 August 2019

Physics of nucleon-nucleus Scattering - Chapter I

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


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
School of Medical Sciences
RMIT University, Australia

Chapter One


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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