It was a rainy day in December and we were sitting in an office at the Nuclear Physics Center in Lisbon deeply involved in a heated discussion about the opening of this book. Should we follow the standard practice, or should we paint the big picture? True to our main motivation, after hours we finally agreed.

The human fascination for a clear starry sky is timeless. It has been around since the early days of mankind and includes the most diverse cultures. Only in the last century, nuclear physics has started to make a very important contribution to our understanding of these phenomena in the sky. And until the present day, many big questions connected to nuclear reactions remain to be answered. One of the prime examples listed amongst the eleven most important physics questions for our century is this: ‘How and where are the heavy elements produced?’.

Why another book? For decades we have come across colleagues, including experimentalists, who would like to learn more about reactions. Some have become fluent in running reaction codes, but cannot find a book at the right level to learn the theory associated with the calculations they are performing. Probably the largest push toward embarking on the adventure of writing this book came after several years of teaching reaction theory to graduate students. The reference nuclear reaction books have been around for decades, and even though there have been some more recent efforts, nowhere could we find the appropriate level, detail, connection to the present experimental scene, the guiding motivation of astrophysics, and the content consistent with that motivation.

In order to find the cross sections for reactions in terms of the interactions between the reacting nuclei, we have to solve the Schrödinger equation for the quantum mechanical wave function. Scattering theory tells us how to find these wave functions for the positive (scattering) energies that are needed. We start with the simplest case of finite spherical real potentials between two interacting nuclei in Section 3.1, and use a partial-wave analysis to derive expressions for the elastic scattering cross sections. We then progressively generalize the analysis to allow for long-ranged Coulomb potentials, and also complex-valued optical potentials. Section 3.2 presents the quantum mechanical methods to handle multiple kinds of reaction outcomes, each outcome being described by its own set of partial wave channels, and Section 3.3 then describes how multi-channel methods may be reformulated using integral expressions instead of sets of coupled differential equations. We end the chapter by showing in Section 3.4 how the Pauli principle requires us to describe sets of identical particles, and by showing in Section 3.5 how Maxwell's equations for an electromagnetic field may, in the one-photon approximation, be combined with the Schrödinger equation for the nucleons. Then we can describe photo-nuclear reactions, such as photo-capture and disintegration in a uniform framework.

Elastic scattering from spherical potentials

When the potential between two interacting nuclei does not depend on the orientiation of the vector between them, we say that the potential is spherical.

The direct reaction of a two-body projectile with a target constitutes a three-body problem, as discussed in the previous chapter. The next most complicated group of processes involve a three-body projectile, which with a target make a four-body reaction problem. This group includes reactions where the projectile is a two nucleon halo nucleus, and in this chapter we present theories for the structure and reactions of such nuclei. First we introduce the topic of halo nuclei, then describe three-body models for bound and scattering states. Finally we discuss four-body reaction models within DWBA, the adiabatic approximation and the eikonal approximation, and conclude by looking at four-body CDCC.

Definitions of halo and deeply bound states

Stable nuclei are characterized by large binding energies and extremely long lifetimes. Figure 9.1 shows the nuclear chart for light nuclei. As we add protons or neutrons to the system, and move away from the valley of stability (black squares in Fig. 9.1), the binding energy of the valence nucleons becomes smaller and smaller until eventually the system can no longer bind. Around the nuclear dripline, we frequently find exotic structures called halo nuclei [1]. The halo phenomenon comes from a significant decoupling of the valence nucleon (or nucleons) from the remaining nucleons, which form a core.

General structure

Fresco is a general-purpose reaction code, created and frequently updated by Ian Thompson. The code calculates virtually any nuclear reaction which can be expressed in a coupled-channel form. There is a public version of the code which can be downloaded from the website www.fresco.org.uk. Fresco is accompanied by Sfresco, a wrapper code that calls Fresco for data fitting and sumbins and sumxen, two auxiliary codes for integrated cross sections. Although we do not include it here, in the same site you can also find xfresco the front-end program to Fresco for X-window displays.

Its original version was written in Fortran 77 but some important sections were ported to Fortran 90.An important part concerns the input, which now uses namelist format, making it much easier to view the relevant variables. In this section we will discuss the general namelist format of the Fresco input.

There are several different layers of output produced by Fresco. The default output contains the most important information concerning the calculation, repeating the input information, and the resulting observables but most detailed information is contained in the generated fort files, including files ready for plotting purposes. At the end of this section we present the list of output produced by Fresco.

Input file

Input files contain five major namelists regarding different aspects of the calculation: fresco, partition, pot, overlap, coupling.

Approximations in physics are often very useful. In nuclear reactions in particular, depending on the specific regime, some approximations may offer a very large simplification of the problem and still provide great accuracy. Of course, to some extent, all methods in nuclear reactions are approximate, but let us consider that the solution methods discussed in Chapter 6 are the starting point to which approximations can be considered.

One idea appears when there are variables with distinct timescales, as then an adiabatic approximation can be made. Another idea is based on classical arguments. By taking a certain trajectory for the projectile, we can separate out the dynamics of the reaction and treat just what is happening to the projectile within quantum mechanics (semiclassical methods). For instance, at very high energies, the projectile's trajectory is hardly bent and the straight-line approximation is valid (called the eikonal approximation). For cases where the reaction is Coulomb dominated, taking a Rutherford trajectory for the center of mass of the projectile may be adequate, and if so provides useful simplifications of the problem. When the potential is very smooth and for slow reactions (low energies), we may make use of theWKB approximation. In this chapter we summarize the most important approximations used in reactions and discuss their limits of validity.

Given the R-matrix, DWBAor coupled-channels theories of the previous chapters, we now discuss how to vary any unknown parameters to fit experimental measurements. The fitting parameters of a reaction model could be those which specify Woods-Saxon potentials, or they could be structural parameters such as deformations or spectroscopic factors. Sometimes the parameters linearly determine the model predictions (such as spectroscopic factors in one-step DWBA), but more often the predictions depend non-linearly on the model parameters (especially those defining the potentials). When there are multiple non-linear parameters, more systematic methods are needed.

Data fitting is thus the systematic variation of the parameters pj(j = 1, …, P) of a theory in order to find a parameter combination which minimizes the discrepancies with experiment. This is most commonly done by minimizing the χ2 measure of these differences, as defined below, and in Section 15.5 we discuss strategies for this minimization.

χ2measures

Specifications

We want to compare theoretical with experimental cross sections, and use this comparison to improve the theory input parameters {pj} so that observed and predicted cross sections agree better.

Nuclei close to the driplines have large breakup cross sections. The breakup mechanism connects a bound state of the projectile with its continuum states, so the process is rather similar to inelastic excitation, except that the final state is unbound. If the target has a strong Coulomb field, then breakup reactions measure the electromagnetic response that populates scattering states. These can be relevant for astrophysical processes either directly, in photo-disintegration, or in reverse, for capture reactions. The excitation into the continuum also allows the study of resonances, and this is important for structure studies because the understanding of the spectra of unstable nuclei depends strongly on resonant properties. Because of these many interests, it is important to establish reliable models for the description of breakup reactions. For simplicity we shall neglect target spin in this chapter. The connections to astrophysics and nuclear structure will be considered in Chapter 14.

In the simplest case, breakup will leave the target nucleus in its ground state. This is called elastic breakup (also called diffraction dissociation). However, there are experiments that measure the inclusive reaction, and then inelastic breakup (also referred to as stripping) can be an important contribution. Elastic breakup of a two-body projectile can be modeled as a three-body problem, and this is the approach presented in Sections 8.1 and 8.2. Later in subsection 8.3.2 we discuss methods for inelastic breakup.

In order to understand nuclear reactions, we have first in Section 2.1 to name the arrangement of nucleons in a nucleus in terms of the quantum-mechanical state of a nucleus, and then describe the different ways in which these nucleons may be rearranged during nuclear reactions. Reactions which proceed quickly, and thus called direct reactions, are distinguished in Section 2.2 from the comparatively slow reactions that also occur, which are called compound nucleus reactions. Almost all reactions involve the collision of two nuclei, and Section 2.3 shows how the conservations of mass, energy and momentum may be described in either non-relativistic or relativistic kinematics. Section 2.4 describes how the rates of nuclear reactions are measured in terms of cross sections, which have units of area. We show how these cross sections are different in the laboratory and center of- mass coordinate frames of reference, then in the final subsection 2.4.4 how the cross sections may be determined from the wave functions that are solutions of a Schrödinger equation for the pair of reacting nuclei.

Kinds of states and reactions

States of nuclei

Nuclei are aggregations of Z protons and N neutrons in a particular configuration or state described by a wave function φ determined from quantum mechanics, given the strong and electromagnetic potentials V between the A=N + Z constituent nucleons. A state is called bound if energy is needed to remove one or more nucleons to large distances.

So far, we demanded that only one scalar field varies during inflation. Now we allow two or more fields to vary. In the first section we generalize the slow-roll paradigm, and in the second section we give a more general discussion which applies even if the fields are not responsible for inflation.

Multi-field slow-roll inflation

Let us suppose that more than one field varies during inflation, corresponding to an inflationary trajectory φ1(t), φ2(t), …. We continue to assume Einstein gravity, with the energy density dominated by the scalar field potential of canonically normalized fields.

We recover the single-field case if the inflationary trajectory is a straight line in field space, because we can then perform a rotation of the field basis so that just a single field φ has significant variation. Even if the trajectory is not straight, we recover the single-field case provided that the trajectory lies in a steep-sided valley of the potential. The inflaton field then just corresponds to the distance in field space along the valley bottom.

To get something non-trivial, we suppose that the inflationary trajectory is not straight, and that it is one of a continuous family of possible slow-roll trajectories. The family is obtained by displacing the original trajectory sideways in field space. We then say that there is multi-field slow-roll inflation.

The field theory action is constructed to be invariant under Lorentz transformations and spacetime translations. These don't mix different fields. Now we consider internal symmetries, corresponding to transformations which mix different fields but don't involve spacetime.

Internal symmetries come in two kinds, called global symmetry and gauge symmetry. A gauge symmetry must be exact, but a global symmetry may only be approximate.

What is the motivation for considering a symmetry? One might first choose an action, guided by observation and/or aesthetics, and then look to see what is its symmetry group. That was the case when Quantum Electrodynamics was first formulated. Nowadays, the more usual procedure is to first choose a symmetry group (guided again by observation and/or aesthetics) and then to write down the most general action consistent with the symmetry group, which turns out to be quite strongly constrained. Often, the action arrived at in this way turns out to have additional global symmetries which were not imposed from the start, called accidental symmetries.

In this chapter we look at some examples of internal symmetry, as it applies to scalar fields at the classical level. Although we won't generally write down formulas involving spin-1/2 fields, it will be important to remember their existence.

Symmetry groups

To say that the action is invariant under some transformation is to say that it is the same before and after the transformation. It follows that the action is invariant also under the inverse of the transformation.

The beginning of the twenty-first century stands a good chance of being identified in history as the time when humankind first came to grips with the Universe. In a rush of observational progress, the charge led by the 1992 discovery of cosmic microwave background anisotropies by the Cosmic Background Explorer (COBE) satellite, the elements required to build accurate cosmological models were assembled. Complementing this, development of theoretical methods allowed accurate predictions to be made to confront those observations.

The landmark was the 2003 announcement of precision cosmic microwave measurements from the team operating the Wilkinson Microwave Anisotropy Probe (WMAP). Ironically, this success lay in a kind of failure – a failure to uncover anything new and unexpected. In the words of astrophysicist John Bahcall at NASA's announcement press conference, “the biggest surprise is that there are no surprises”. Instead, then, the power of the observations became fully focussed on determining the properties of the cosmological model, and for the first time many of its components were determined to a satisfying degree of accuracy: the percent level for quantities such as the geometry of the Universe, its age, and the density of the baryonic material, and the ten percent level for many other aspects.

In 2000, we published a graduate-level textbook, Cosmological Inflation and Large-Scale Structure, written during the late 1990s and which described many of the ideas underpinning the modern cosmology.

The time dependence of each perturbation is well defined, being determined by laws of physics. Viewed instead as a function of position at fixed time, the perturbations have random distributions. It is the statistical properties of these distribution that we wish to uncover via observation, and relate to fundamental physics models for the origin of perturbations. Those are usually referred to as stochastic properties.

The inherent randomness means that one shouldn't aim to predict things like the precise location of particular galaxies. Questions should refer to stochastic properties only. Don't ask ‘how far is it to the nearest large galaxy?’; instead ask ‘what is the typical separation between large galaxies?’. This randomness echoes simple quantum mechanics, e.g. one shouldn't hope to predict the precise position of a single particle in a closed box, but could compute the typical distance of the particle from its centre averaged over many such boxes. Indeed, we will see that in the inflationary cosmology the randomness of cosmological perturbations does have its origin in quantum uncertainty.

To describe the stochastic properties of the perturbations, one invokes the mathematical concept of a random field. In this chapter we describe the relevant aspects of that concept, without tying ourselves at this stage to any particular perturbation.

Random fields

Consider just one perturbation, evaluated at some instant, which we denote by g(x). We take g(x) to be associated with what is called a random field.

In the previous two chapters we considered the cosmological perturbations before horizon entry. As no causal processes can then operate, the description of these perturbations is simple, but they are not directly observable, except on the very largest scales via the cosmic microwave anisotropy. Over the next few chapters we follow the evolution of the cosmological perturbations after horizon entry, under the influence of causal processes.

The first section of this chapter is devoted to an overview of the evolution. In the rest of the chapter we see how to calculate the evolution in the regime of Newtonian gravity.

Free-streaming, oscillation, and collapse

As summarized in Table 7.1, the cosmic fluid at T < 1MeV has four components: baryons, cold dark matter (CDM), photons, and neutrinos. The CDM and neutrinos have negligible interaction. Until the temperature falls below its mass each neutrino species has relativistic motion, so that it behaves as radiation rather than matter up to that point.

The essence of what happens after horizon entry can be stated very simply. For each component, there is a competition between gravity, which tries to increase the density perturbation by attracting more particles to the overdense regions, and random particle motion which drives particles away from those regions. The relative importance of these effects depends on which component we are considering, and it may depend also on what era we are talking about.

Our picture of the physical world at its most fundamental level, a model that also has a very high degree of experimental support, runs along the following lines. There are only three types of interaction: QCD (Quantum ChromoDynamics), which binds quarks into hadrons, that is, nuclear particles like protons and neutrons, pions and so on; the electroweak interaction, which is the unification of electromagnetism with the weak nuclear force (responsible for beta decay); and gravity. The first two interactions are understood in the context of quantum field theory, more particularly gauge field theory, and the interactions are transmitted by the field quanta, which are gluons (for QCD), and the photon and the W and Z bosons which mediate electroweak interactions. Gravity is described by General Relativity. What is immediately obvious about this statement is that General Relativity is, conceptually, a completely different sort of theory from the other field theories, because of its explicitly geometric nature. The whole enterprise of physics, however, is to reduce the number of fundamental theories and concepts to the absolute minimum, and as a consequence a large number of physicists now work on unification schemes of one sort or another – supergravity, superstring theory, brane worlds, and so on. One guiding principle at work in these endeavours is to unite the three fundamental interactions into one interaction, and another, equally important, aim is to find a quantum theory of gravity; it is clear that General Relativity is a classical theory since it never at any point employs notions involving wave–particle duality or Planck's constant.