*Special Topics in Accelerator Physics*, by Alexander Wu Chao, World Scientific

*Special Topics in Accelerator Physics *by Alexander Wu Chao introduces the global picture of accelerator physics, clearly establishing the scope of the book from the first page. The derivation and solution of concepts and equations is didactic throughout the chapters. Chao takes readers by the hand and guides them through important formulae and their limitations step-by-step, such that the reader does not miss the important parts – an extremely useful tactic for advanced masters or doctoral students when their topic of interest is among the eight special topics described.

In the first chapter, I particularly liked the way the author transitions from the Vlasov equation, a very powerful technique for studying beam–beam effects, towards the Fokker–Planck equation describing the statistical interaction of charged particles inside an accelerator. Chao pedagogically introduces the potential-well distortion, which is complemented by illustrations. The discussion on wakefield acceleration, taking readers deeper into the subject and extending it both for proton and electron beams, is timely. Extending the Fokker–Planck equation to 2D and 3D systems is particularly advanced but at the same time important. The author discusses the practical applications of the transient beam distribution in simple steps and introduces the higher order moments later. The proposed exercises, for some of which solutions are provided, are practical as well.

In chapter two, the concept of symplecticity, the conservation of phase space (a subject that causes much confusion), is discussed with concrete examples. Naming issues are meticulously explained, such as using the term short-magnet rather than thin-lens approximation in formula 2.6. Symplectic models for quadrupole magnets are introduced and the following discussion is extremely useful for students and accelerator physicists who will use symplectic codes such as MAD-X and who would like to understand the mathematical framework of their operation. This nicely conjuncts with the next chapter and the book offers useful insights to how these codes operate. In the discussion about third-order integration, Chao makes occasional mental leaps, which could be mitigated with an additional sentence. Although the discussion on higher order and canonical integrators is rather specialised, it is still very useful.

The author introduces the extremely convenient and broadly used truncated power series algebra (TPSA) technique, used to obtain maps, in chapter three. Chao explains in a simple manner the transition from the pre-TPSA algorithms (such as TRANSPORT or COSY) to symplectic algorithms such as MAD-X or PTC, as well as the reason behind this evolution. The clear “drawbacks” discussion is very useful in this regard.

The transition to Lie algebra in chapter four is masterful and pedagogical. Lie algebras, which can be an advanced topic and come with many formulas, are the main focus in this section of the book. In particular, the non-linearity of the drift space, which is absent of fields, should catch the reader’s attention. This is followed by specialised applications for expert readers only. One of this chapter’s highlights is the derivation of the sextupole pairing, which is complemented by that of Taylor maps up to the second order and its Lie algebra, although it would be better if the “Our plan” section was placed at the beginning of the chapter.

Chapter five covers proton-spin dynamics. Spinor formulas and the Froissart–Stora equation for the polarisation change are developed and explained. The Siberian snake technique remains one of the most well-known to retain beam polarisation, which the author discusses in detail. This links elegantly to chapter six, which introduces the reader to electron-spin dynamics where synchrotron radiation is the dominant effect and therefore constitutes a completely different research area. Chao focuses on the differences between the quantum and classical approach to synchrotron radiation, a phenomenon that cannot be ignored in high-brightness machines. Analogies between protons and electrons are then very well summarised in the recap figure 6.3. Section 6.5 is important for storage rings and leads smoothly to the Derbenev–Kondratenko formula and its applications.

**Echoes**

Chapter seven looks at echoes, a key technique when measuring diffusion in an accelerator, where the author introduces the reader to the generality of the term and the concept of echoes in accelerator physics. Transverse echoes (with and without diffusion) are quite analytical and the figures are didactic.

The book concludes with a very complete, concise and detailed chapter about beam–beam effects, which acts as an introduction to collider–accelerator physics for coherent- and incoherent-effects studies. Although synchro-betatron couplings causing resonant instabilities are advanced topics, they are often seen in practice when operating the machines, and the book offers the theoretical background for a deeper understanding of these effects.

*Special Topics in Accelerator Physics* is well written and develops the advanced subjects in a comprehensive, complete and pedagogical way.