Preamble

Walk along an island beach on a clear, breezy, cloudless night, or stand on the spine of a barren mountain ridge after sunset, and behold the firmament of stars glittering against the coal black sky above. They fill the sky with their timeless, brilliant flickering. With binoculars or even a small telescope one finds that even the lacey dark matrix between the vast sea of stars is populated with still more stars that are simply too faint to be seen with the naked eye. Within the Milky Way Galaxy, which stretches from horizon to horizon, the density of stars against the background sky is even greater.

Each twinkling point of light is a star not too unlike our own Sun. The Sun is just an ordinary star but it features prominently on the pages of this book because of its proximity. The next closest star, α-Centauri (which is a triple system in which Proxima Centauri is currently the closest to Earth), is almost a million times farther away (at 4.22 light years), and the remainder are farther still. We may now say with some confidence that many stars are surrounded by planets of various sizes. Some of these orbital companions are so immense that they are stars in their own right: double-star systems are quite common.

With the same measure of confidence we may assert that most of these stars possess magnetic fields; that these magnetic fields create hot outer atmospheres, or coronae, that drivemagnetized winds from their stars; and that these variable plasma winds blow past the orbiting planets, distorting their individual magnetospheres, and push outward against the surrounding interstellar medium.

Preamble

The widespread interest in reconnection results from the fact that it is a fundamental process that occurs in magnetized plasmas whenever the connectivity of the field lines changes in time. Reconnection is most commonly associated with geomagnetic and solar activity because such changes in field line connectivity can be directly observed, but there are many other, less well-known, applications ranging from meteorites and comet tails to accretion disks and galactic jets. Reconnection is also found in laboratory devices that have been built to study the feasibility of controlled thermonuclear reactors, as well as in several experiments that have been specifically designed to study reconnection as a basic plasma process. Those aspects of magnetic reconnection that depend primarily on the topology of the magnetic field tend to be of universal application. However, aspects that depend on the detailed characteristics of the plasma itself, such as its temperature and density, tend to be restricted to the specific application where such characteristics occur. Thus, there is no universal theory that can be applied to all situations.

Basic concepts

The term magnetic reconnection was introduced by Dungey (1953a), who was interested in the problem of particle acceleration in the Earth's magnetosphere. Earlier studies (Giovanelli, 1946; Hoyle, 1949) had considered the acceleration of particles at magnetic neutral points in the presence of an electric field produced by plasma convection, but these studies did not include the magnetic field produced by the current associated with the motion of the particle.

Preamble

One goal this book pursues is to identify phenomena throughout the heliosphere that can be said to be universal. An excellent example of a universal subject is the morphology of magnetically defined structures. The heliosphere is full of distinctive magnetic forms that recur in widely different places. What they are and why this is so are questions we take up here. As a first stab at distinguishing magnetically defined structures, we put them into three groups: current sheets, of which the heliospheric current sheet (HCS) is the largest example; flux tubes, with sunspots as a prototype; and cells, in which we include cavities such as magnetospheres. These three classes make up the common forms of heliophysical magnetic structure that exist on MHD time and distance scales (we are not concerned here with kineticscale structures that inhabit the dissipation range of turbulence). Our tasks are to explain why these magnetic structures arise naturally and to describe examples of each. It is important to note that these structures are idealized mental constructs to approximate a real world; we are trying to describe a continuum in black-and-white terms. We should realize that current sheets are not mathematical planes, flux tubes are surrounded by other fields, and cells are leaky in the real world. Nevertheless, these abstractions should help us to think and communicate.

That magnetically defined structures in the heliosphere have been seen to take common forms has led people to recognize over time that there actually exists such a thing as the magnetic organization of cosmic matter (as described, for example, in the NRC report “Plasma physics of the local cosmos”, 2001, after which this volume is named).

Making predictions for an experiment is much better achieved if the person performing the calculation is aware of the experimental details. In this chapter we therefore address some of the important issues to take into consideration when applying reaction theories. There is a wide variety of experiments related to astrophysics: the direct measurements (e.g. (p, γ), (n, γ), (α, γ) performed at low energy are obviously important, but there are also many indirect measurements (e.g. Coulomb dissociation method), a large fraction of which make use of higherenergy accelerators. Many of the forefront research experiments involve radioactive beams, while there are still important rates to determine using stable beams.

Since a large part of the research activity is taking place in rare-isotope facilities, we will first focus (Section 13.1) on some specifics of these facilities, where the exotic nuclei are produced, including both low- and high-energy laboratories. Next, Section 13.2 focuses on different aspects of present-day detectors that need to be considered when comparing calculations with data. Finally, in Section 13.3,we briefly mention some of the direct measurements involving less-exotic nuclei which are stable enough to be made into targets. Included are reactions with light chargedparticle beams (protons and alphas), neutron beams and photon beams.

New accelerators and their methods

Many of the recent leading studies in nuclear physics involve unstable nuclei. The first step in an experiment with radioactive beams is the production of the radioisotope of interest.

In order to learn about the internal properties of nuclei, some kind of reaction is necessary. The most interesting questions concern the arrangements of nucleons inside the nucleus, and, for this purpose, transfer reactions have been the standard procedure for examining the single-particle structure of nuclei and extracting spectroscopic factors. Transfer experiments in the sixties and seventies were abundant, but they suffered a decrease of popularity following the shutdown of some low-energy laboratories. They are now becoming popular again to study exotic nuclei, as more intense beams are produced in ISOL facilities. In this chapter we primarily discuss the various features of transfer reactions.We present the standard theory used to analyze these reactions, namely the DWBA, look at its advantages and limitations, and then consider other approaches that handle higher-order effects. We compare transfer probes with electron knockout and nuclear knockout reactions. At the end of the chapter we briefly discuss chargeexchange reactions, which are used to extract Fermi or Gamow-Teller transition strengths.

Transfer spectroscopy

We begin by discussing standard DWBA theory for describing transfer reactions. This theory is most useful for reactions that probe the surface regions of the nuclei, and try to measure the spectroscopic factors of the single-particle states.

Compound-nucleus phenomena

In Chapter 2 we saw how nuclear reactions are broadly dominated by two kinds of timescales: the fast direct reactions and the slower compound-nucleus (CN) reactions. The direct reactions are typically described in R-matrix theory by a few poles with large widths, whereas there are usually very many compound-nucleus resonances, each of which has a narrow width.

Direct reactions are generally foward-peaked with respect to the incident direction, whereas the CN process has less ‘memory’ about that direction and gives products which are typically symmetric about 90°. Usually it is possible to experimentally separate the symmetric contributions to a given outgoing channel, and theoretically the direct and CN cross sections are calculated by quite different methods.We will sometimes try to model the detailed resonance structure of theCN process, but usually calculate the reaction rates averaging over manyCNlevels, and use only statistical features of these levels, such as their average spacings and widths.

Thermal averaging

Reaction rates 〈σv〉 and lifetimes

The reactions that we have discussed so far all have cross sections σ(E) that depend strongly on center-of-mass energy E. This dependence may be because of a repulsive Coulomb barrier, so that σ(E) = E-1 exp(-2πη)S(E) for an ‘astrophysical S-factor’ S(E) that is relatively less variable with energy. It may be because neutrons have a σ(E)∝ 1/v behavior for projectile-target relative velocity v near zero. Or itmaybe because of resonances giving sharply peaked cross sections, like Г/[(E - ER)2 + Г2/4] for a single resonance centered at ER with a full width at half maximum of Г.

In a stellar plasma of a mixture of two or more nuclear species, there will be a considerable range of relative energies E (or, velocities v) because of the statistical distribution of thermal energy among all the particles in the plasma. The actual rate of nuclear reactions will therefore require an averaging of the cross sections σ(E) over the thermal distribution of relative energies. We will therefore define an average reaction rate as the number of reactions per second per unit volume, and find an expression for this in terms of σ(E) and the distribution φ(v) of the relative velocities of the interacting particles.

In the previous chapter we presented non-elastic mechanisms based on rotational or vibrational models, and on transfer, capture and knockout reactions based on the separation of a nucleus into a nucleon and a core cluster. In this chapter we see how the necessary properties of these structure models may be related to nuclear structure theories that should be more exact because they are microscopic and take all the many nucleons of the nuclei into account. It is beyond the scope of this book on reactions to discuss detailed methods and numerical examples using microscopic models, so we adopt the aim of establishing a ‘common language’ between structure and direct-reaction theories. In particular, we show how masses, sizes, folding potentials, overlaps and other matrix elements may be defined in terms of structure models and then used for reaction calculations.

Summary of structure models

A nucleus contains a number A of nucleons, all pairs ij of which interact with each other by a nucleon-nucleon potentialV(2) (ri-rj) which is strongly repulsive at short distances (rij ≡ |ri-rj|≾ 1 fm), attractive at medium distances (1fm ≾ rij ≾ 4fm), while protons repel even at large distances. There may also be three-body forces between bodies ijk according to some form V(3) (ri-rj, ri-rk). In general, V(2) and V(3) depend also on the spin states of the interacting nucleons, and should therefore be expanded on a sufficiently complete set of vector and tensor operators.

This chapter shows how the interactions in and between channels may be calculated on the basis of some potential model for a few interacting bodies. That is, a Hamiltonian is defined whose matrix elements give rise to channel couplings, also known as transition potentials. The parameters in this Hamiltonian may be found either from structure models (Chapter 5), or from fitting data (Chapter 15). It is also possible to directly fit to experiment the effects of these couplings on the asymptotic amplitudes of the wave functions, and this is the basis of the R-matrix phenomenology discussed in Chapter 10.

Optical potentials

Beforewecan discuss more detailed reaction mechanisms,weneed to see the typical kinds of potentials used for elastic scattering, and also the binding potentials needed to reproduce the usual single-particle structures of nucleons within a nucleus. We will start by describing the most commonly used optical potentials for elastic scattering, expanding on the introduction in Box 3.4.

Typical forms

The interaction potential between a nucleon and a spherical nucleus is usually described by an attractive nuclear well of depth Vr with a radius Rr slightly larger than the nuclear matter radius, and a diffuse nuclear surface.