Complete Fusion of Heavy Ions

Complete fusion of heavy ions is theoretically treated in the framework of a statistical compound reaction mechanism. In heavy ion collisions, a large number of resonances is excited in the compound system, involving many degrees of freedom. A complete description of such a complex collision process is almost impossible to obtain. However, the mean value of crosssection averaged over several resonances is generally of interest, and can be estimated using the statistical approach. The statistical compound reaction model is founded on the works of Bohr, Bethe, and Weisskopf. Wolfenstein and Hauser and Feshbach extended the model to include the conservation of total angular momentum. The statistical compound model was further refined by Moldauer and Lane and Lynn.

Nuclear reactions may be classified in terms of different parameters, including the reaction time. Fast reactions involving reaction times of the order of the time taken by a nucleon to pass through the nucleus (≈10–21 s) corresponds to direct reactions. Slower processes of reaction times of the order of 10–16 s or so come in the category of compound and pre-compound (or pre-equilibrium, or multistep compound and multistep direct) reactions. The compound reaction mechanism, being the slowest, assumes that the excited compound nucleus formed by the fusion of the target and the projectile lives long enough, without decay, for thorough mixing of the target and projectile nucleons to take place and a thermodynamic equilibrium be established in the compound system. Sometimes, it is convenient to call the fused system formed by the amalgamation of the projectile with the target, before the establishment of thermal equilibrium, as an excited composite system that becomes the compound nucleus (CN) when thermal equilibrium is established. Pre-compound reactions occur during the time taken by the excited composite system to transit to the compound nucleus. In this section, we consider the pure compound reaction mechanism and assume that the composite system becomes a compound nucleus without losing any nucleons or clusters. Almost all nuclear models that aim to determine reaction cross-sections make use of the optical model which enables the separation of the total cross-section into different components and provides transmission coefficients that are used in the compound nucleus model.

This book addresses the single particle dynamics heart of the question; ‘How does an accelerator work?’, for readers who are accelerator users and operators, who are accelerator physicists or who are interested in real-world linear and nonlinear difference systems. The reader might be a synchrotron light source user, a collider experimentalist, a medical accelerator operator, an engineer in a beam instrumentation group or a controls professional in industry.

The level of the discussion is appropriate for graduate students, final-year undergraduates and practicing accelerator professionals. This is not an exhaustively complete reference handbook, at a high technical level. Rather, it is a pedagogical introduction to the subject, telling a self-contained and accurate story about a field of physics that was born and grew rapidly in the second half of the twentieth century, and which continues to mature by leaps and bounds in the twenty first. The treatment is rigorous enough to be accurate and useful, without letting unnecessary detail obscure the central theme. Deeper investigations of the ‘back-stories’ are left to other sources.

The central theme is that repetitive motion through an accelerator is a natural, convenient and well-motivated introduction to the generic linear and nonlinear behaviour of highly iterated difference systems. A circular accelerator – or even a linear accelerator – is one answer to another question: ‘How do difference systems manifest themselves in the real world?’

Even the highest-energy proton linear accelerators, with kinetic energies of order 1 GeV, are not fully relativistic. Their β = v/c values are significantly less than one. This is in stark contrast to electron linacs, in which β ≈ 1 even for energies of only a few MeV. Because of this, the dominant dynamical issues are different in proton and electron linacs. Also, the menagerie of RF structures that accelerate and manipulate protons – each tuned for a different range of β values – is very diverse.

The technology map of Figure 13.1 shows some of the RF structures typically found in a modern large proton linac. Not shown are other common technologies, including Drift Tube Linacs (DTL), side coupled linacs, quarter wave resonators and double and triple spoke resonators. Fortunately an introduction to proton linac dynamics is possible without a detailed understanding of these diverse technologies, which are more fully discussed elsewhere [36, 57]. Instead it is possible to build on the generic discussion of pill-box and elliptical cavities in Chapter 7, adding only one instructive special case, the Radio Frequency Quadrupole (RFQ).

Even low energy proton rings – for example 250 MeV cancer therapy synchrotrons – need an injection linac, although these typically have maximum kinetic energies of less than 10 MeV. More interesting and comprehensive are high power linacs, such as the MW-class linacs listed in Table 13.1, which are used to generate neutrons for material science experimentation (ESS and SNS), to create rare isotopes for nuclear physics (FRIB), or to provide protons for high energy physics experimentation (PIP-II). In all four cases the hadron beam is absorbed in a target with a specialised design serving the need at hand.

This chapter does not discuss collective motion dynamics, or any of the related conceptual tools (such as wakefields and impedances), except to briefly introduce space charge as a phenomenon that is vital to linac design, and which connects to the beam-beam discussion in Chapter 15. Rather, the focus is on single-particle dynamics, especially the longitudinal phase space behaviour that dominates single-pass linacs, in contrast to the transverse dynamics that usually dominate multi-pass circular rings.

The standard map emerges when the differential equation of a pendulum is translated to a pair of difference equations, as shown in the discussion of longitudinal motion and synchrotron oscillations in Section 4.4. Its only control parameter, Q0, is analogous to the synchrotron tune of small oscillations in a storage ring with a single RF cavity – or with a set of cavities that are close enough to act as one. The rich phase space behaviour that emerges as Q0 increases is shown in Figure 4.6. A small chaotic area emerges when Q0 = 0.12, easily identifiable as a scattering of dots. Stable resonance islands emerge from a broad chaotic sea in the extreme case with Q0 = 0.18. Stability and chaos are intermingled at all scales – stability within chaos, chaos within stability – ad infinitum. This is what Hofstadter means, when he refers to ‘ … an eerie type of chaos … just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order’ [20].

The Hénon map is analogous to a storage ring with a single sextupole, as discussed in Section 9.3. This map generates the rich horizontal behaviour shown graphically in Figure 9.3. The taxonomy of 1-D motion discussed in Section 9.4 categorises turn-by-turn phase space trajectories as:

1. 1. Regular non-resonant;

2. 2. Rapidly divergent;

3. 3. Regular resonant; or

4. 4. Chaotic.

This taxonomy is universal. It emerges in phase space motion driven by one sextupole or many, by beam–beam interactions, or by RF cavities.

Much of this book addresses practical and quantitative aspects of linear and nearlinear regular non-resonant (non-chaotic) motion. Rapidly divergent trajectories are discussed semi-quantitatively in Chapter 10, in the practical context of slow extraction from a storage ring. Regular resonant motion is discussed and quantified in Chapter 15, using the convenient and fundamental example of the 1-D beam– beam interaction.

Chaos can also be predicted and discussed semi-quantitatively, beyond the qualitative observation that it appears as dots in phase space, and beyond its formal definition using Lyapunov exponents. The routes to chaos examined in this chapter inevitably require more than just one real-space dimension, especially when more practical and realistic situations are addressed.