# Stephen Myers Herwig Schopper *Editors*

# Particle Physics Reference Library Volume 3: Accelerators and Colliders

Particle Physics Reference Library

Stephen Myers • Herwig Schopper Editors

# Particle Physics Reference Library

Volume 3: Accelerators and Colliders

*Editors* Stephen Myers ADAM SA Geneva, Switzerland

Herwig Schopper CERN Geneva, Switzerland

ISBN 978-3-030-34244-9 ISBN 978-3-030-34245-6 (eBook) https://doi.org/10.1007/978-3-030-34245-6

This book is an open access publication.

© The Editor(s) (if applicable) and The Author(s) 2013, 2020

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# **Preface**

For many years the series Landolt-Börnstein—Group I Elementary Particles, Nuclei and Atoms, Volume 21A (Physics and Methods Theory and Experiments, 2008), Vol. 21B1 (Elementary Particles Detectors for Particles and Radiation. Part 1: Principles and Methods, 2011), Vol. 21B2 (Elementary Particles Detectors for Particles and Radiation. Part 2: Systems and Applications), and Vol. 21C (Elementary Particles Accelerators and Colliders, 2013), has served as a major reference work in the field of high-energy physics.

When, not long after the publication of the last volume, open access became a reality for HEP journals in 2014, discussions between Springer and CERN intensified to find a solution for the "Labö" which would make the content available in the same spirit to readers worldwide. This was helped by the fact that many researchers in the field expressed similar views and their readiness to contribute.

Eventually, in 2016, at the initiative of Springer, CERN and the original Labö volume editors agreed in tackling the issue by proposing to the contributing authors a new OA edition of their work. From these discussions a compromise emerged along the following lines: transfer as much as possible of the original material into open access; add some new material reflecting new developments and important discoveries, such as the Higgs boson; and adapt to the conditions due to the change from copyright to a CC BY 4.0 license.

Some authors were no longer available for making such changes, having either retired or, in some cases, deceased. In most such cases, it was possible to find colleagues willing to take care of the necessary revisions. A few manuscripts could not be updated and are therefore not included in the present edition.

We consider that this new edition essentially fulfills the main goal that motivated us in the first place—there are some gaps compared to the original edition, as explained, as there are some entirely new contributions. Many contributions have been only minimally revised in order to make the original status of the field available as historical testimony. Others are in the form of the original contribution being supplemented with a detailed appendix relating recent developments in the field. However, a substantial fraction of contributions has been thoroughly revisited by their authors resulting in true new editions of their original material.

We would like to express our appreciation and gratitude to the contributing authors, to the colleagues at CERN involved in the project, and to the publisher, who has helped making this very special endeavor possible.

Vienna, Austria Christian Fabjan Geneva, Switzerland Stephen Myers Geneva, Switzerland Herwig Schopper July 2019

# **Contents**

# **About the Editors**

**Stephan Myers** was born in Belfast, Northern Ireland, and worked at CERN, Geneva, from 1972 until 2015. He was responsible for the commissioning and the energy upgrade of the CERN Large Electron-Positron Collider (LEP). In 2008, he was nominated CERN Director of Accelerators and Technology until January 2014; during this time, he directed the repair of the CERN Large Hadron Collider (LHC) after the serious accident and steered the operation of the collider in 2010, 2011, and 2012. On July 4, 2012, the data from the collider allowed the discovery of a "Higgs" boson for which Peter Higgs and Francois Engelert received the Nobel Physics Prize in 2013.

He is currently executive chair of a Geneva-based company (ADAM SA), which is developing a linear accelerator for proton therapy of cancer, and nonexecutive Director of the parent company Advanced Oncotherapy (AVO).

**Herwig Franz Schopper** joined as a research associate at CERN since 1966 and returned in 1970 as leader of the Nuclear Physics Division and went on to become a member of the directorate responsible for the coordination of CERN's experimental program. He was chairman of the ISR Committee at CERN from 1973 to 1976 and was elected as member of the Scientific Policy Committee in 1979. Following Léon Van Hove and John Adams' years as Director-General for research and executive Director-General, Schopper became the sole Director-General of CERN in 1981.

Schopper's years as CERN's Director-General saw the construction and installation of the Large Electron-Positron Collider (LEP) and the first tests of four detectors for the LEP experiments. Several facilities (including ISR, BEBC, and EHS) had to be closed to free up resources for LEP.

# **Chapter 1 Accelerators, Colliders and Their Application**

**E. Wilson and B. J. Holzer**

# **1.1 Why Build Accelerators?**

Accelerators are modern, high precision tools with applications in a broad spectrum that ranges from material treatment, isotope production for nuclear physics and medicine, probe analysis in industry and research, to the production of high energy particle beams in physics and astronomy. At present about 35,000 accelerators exist world-wide, the majority of them being used for industrial and medical applications. Originally however the design of accelerators arose from the request in basic physics research, namely to study the basic constituents of matter.

The first accelerators were inspired by the early experiments in nuclear physics. In the early years of the twentieth century Rutherford discovered that by using alpha particles from radioactive disintegration and detecting the pattern of particles scattered by atoms one might deduce that the nucleus was a tiny but massive central element in the atom. Alpha particles from disintegration can only be of energies of 10 MeV; comparable with the nuclear binding forces. Higher energies were needed and a more reliable and steady supply to ease the tedium of counting occasional flashes of light on the scintillation screen that was Rutherford's detector. De Broglie had shown that there was an inverse relationship between the momentum, and hence the energy of a particle and the wavelength of its representation in quantum mechanics.

$$
\lambda = \frac{h}{p}
$$

E. Wilson · B. J. Holzer (-)

CERN (European Organization for Nuclear Research), Meyrin, Genève, Switzerland e-mail<EMAIL_ADDRESS>
where *h* represents Planck's constant and *p* the particles' momentum which relates to its energy via the well-known equation of special relativity, *E*2 = *p*2*c*2 + *m*2*c*4. The limit to the scale of detail that experiments can reveal is set by the length of the wave which is scattered: rather as the wave breaking in the beach can only be deflected by islands larger than itself. It was argued correctly that higher energy particle, having the property of shorter wavelengths could better reveal the structure of the nuclei that Rutherford has detected.

Such arguments led to the invention of the first accelerators and have sustained the development of particle accelerators of higher and higher energy over the best part of the last 100 years. At first, physicists used accelerators to probe the structure of the nucleus, but went on to use higher energy accelerators to search for structure in the "fundamental" particles—protons, neutrons and electrons they discovered. Inevitably higher energies implied larger accelerators, for it was quickly discovered that the best way to accelerate repetitively was to keep particle in a circular path whose radius was itself proportional to energy, limited by the strength of the magnetic field one might use to do the bending.

As energies were raised physicists found new and interesting particles to fit into the pattern of those that their theories might predict. Einstein's

$$E = mc^2$$

tells us that only high energies will create the more massive particles. The latest and largest accelerator, LHC, flagship of the whole community, was designed to search for the Higgs Boson and the successful discovery of this missing puzzle piece in 2013 allowed us to complete the Standard Model of Particle Physics.

As we write, this machine is carrying on the search for physics beyond the standard model, seeking to disclose the nature of dark matter and dark energy.

As more powerful accelerators have been developed for high energy particle physics, advances in the field have been exploited in a whole range of smaller accelerators for other applications. From the time of the first cyclotrons they have been used for producing isotopes and for treating cancer. The development of compact high-frequency linac structures triggered the manufacture of hundreds of small electron linacs producing X-rays for cancer treatment in hospitals around the developed and, latterly, the developing world. Electron rings of a few GeV, specially designed to produce beams of synchrotron radiation have become popular. Each facility serves scores of experiments to investigate the structure of complex molecules—particularly the proteins of today's biomedical studies. Proton accelerators of about 1 GeV produce pulsed beams of neutrons by spallation which are used principally to study the structure of materials. In addition thousands of lower energy accelerators are used in industry for sterilisation and ion implantation in the fabrication of sophisticated CPU chips for computers.

# **1.2 Types and Evolution of Accelerators**

The development of accelerators to ever higher energy is marked by a number of milestones. Each of these marks the invention of a new type of accelerator or the invention of a new principle of transverse or longitudinal focusing which enables a higher energy to be reached for a lower unit cost. The best way to describe this evolution and introduce the different types of accelerator is to follow the road charted by these milestones. Each is described in one of the sections which follow.

# *1.2.1 Early Accelerators*

The nineteenth century had produced a number of electrostatic high-voltage generators. They were unpredictable in performance and electrical breakdown became a serious problem above a few tens of kV. Early accelerators were simply two electrodes enclosed in an evacuated tube with external connections to such high voltage source. A proton or electron source close to one electrode at a potential of *V* (or −*V* for electrons) provided the particles which were then accelerated towards the second electrode at earth potential. They emerged or were observed through a small hole in the earthed electrode. The energy acquired by each particle with charge, *e* Coulombs, was just *e*∗*V* Joules or, in the units commonly used for accelerated beams, *V* electron-Volts. An electron Volt is then just 1.6 × 10−19 Joules. If the particle is a fully stripped ion of an atom with atomic number *A* and charge *Z* then the energy is *ZV/A* electron Volts per nucleon.

The first high-voltage generator to approach 1 MeV was built by Cockcroft and Walton [1–3] in the 1930s to accelerate particles for their fission experiments. Their combination of diodes and capacitors, also known as rectifier circuit, is still used today to apply high voltage to the ion or proton source at the beginning of many linacs and synchrotrons although these are gradually being replaced by radio frequency quadrupoles.

The early 1930s also saw the invention by R.J. Van der Graaf [4] of an electrostatic generator which used a moving belt to carry charge into the high voltage terminal until it reaches a potential of several MV (Fig. 1.1). Van der Graaf accelerators have proved a useful source of low energy particles to this day but are inevitably limited by problems of voltage breakdown. Voltages up to 27 MV have been reached, putting the device in a discharge suppressing gas atmosphere (e.g. SF6). Although it is possible in theory to chain together several electrostatic accelerators, each with its cathode connected to the anode of the next, each stage increases the potential between the ends of the device and between the ends and ground and eventually electrical breakdown discharges the high voltage terminals.

# *1.2.2 The Ray Transformer*

The earliest idea of how to overcome the limitations of electrostatic acceleration involved using the time varying property of magnetic fields and came from the inventive mind of Rolf Wideröe.

Beginning his studies at Karlsruhe Technical University in 1923, he wondered if electrons in an evacuated ring would flow in the same way as the electrons in copper if they replaced the secondary winding of a transformer. His notebooks of that time contain sketches of a device he called a "ray transformer"; the first circular accelerator and the precursor of the "betatron" [5].

These sketches show a beam tube, in the form of an annulus, *R*, placed in the gap between the parallel poles or faces of a small electromagnet (on the left in Fig. 1.2). This magnet is in the form of a "C" and the field between the poles, *Bz*, guides particles in a circular orbit in the mid plane between the poles. A circular hole is cut in each pole through which the yoke of the transformer passes linking the beam tube. The primary winding of the transformer, labelled *W*1, is powered with alternative voltage from the mains. The beam tube is placed where one would normally expect the secondary winding. The beam within it carries the induced secondary current.

### 1 Accelerators, Colliders and Their Application 5

Unlike almost all accelerators that followed, the ray transformer relied entirely upon the inductive effect of a varying magnetic field. It is the rate of change of flux, *φ*, in the yoke which establishes an accelerating potential difference around the beam's path. The windings, that of the C-magnet and of the primary of the transformer, *W*1, give independent control of the guide field and accelerating flux.

Wideröe calculated that electrons circulating in a ring of only 10 or 20 cm diameter could reach several MeV within one quarter wave of the AC excitation of the transformer. He had to use Einstein's newly discovered theory of special relativity to correctly describe the motion of particles close to the speed of light. He also found an important principle which ensures that the beam radius does not change as it accelerates. To ensure constant radius during acceleration the total flux linking the beam including that generated by both sets of the coils, Ba, must be twice that generated by the left hand coil pair which produces the field keeping the beam in a circular orbit, Bg.

$$
\dot{B}\_a = 2\dot{B}\_g
$$

Unfortunately, Wideröe was dissuaded from building the ray transformer by difficulties with surface fields and by his professor, who wrongly assumed the beam would be lost because of gas scattering. However, his Ray Transformer and the 2 to 1 ratio, now known as the Wideröe principle, were important discoveries which were put into practice 15 years later when D.W. Kerst and R. Serber [6] built a series of betatrons.

Wideröe went on to develop a second basic acceleration method to overcome the electrostatic limitation: the drift tube linac.

# *1.2.3 Repetitive Acceleration*

There are two broad classes of accelerator characterized by the way they achieve repetitive acceleration and which overcome the insulation problems of the electrostatic machines. The simplest concept is that of the linear accelerator. Particles pass though cavities excited by radio frequency generators. They arrive on the threshold of each cavity with the energy they have already received and gain a further increment in energy from the electric field in the cavity which points in their direction of motion. Each cavity performs the function of the gap between the anode and cathode of an electrostatic accelerator but, unlike the electrostatic case, the increments of energy may be added together without developing a huge voltage to ground in any part of the apparatus. Of course, there is a limit to the voltage (energy increment) each cavity can apply and the length of the device becomes very long for energies above 1 GeV. Nevertheless, a linac has become the only way of accelerating highly relativistic electrons which radiate a large fraction of their energy when bent into a circular path.

As alternative concept, circular machines, like cyclotrons and synchrotrons use the same set of accelerating cavities over and over again as the particles make complete turns around the accelerator, being guided and focused by the magnet structure of the ring which is thus constraining their orbit. On each turn an increment of energy is added and, once accelerated, particles may be allowed to circulate indefinitely at their top energy. Two circulating beams of say protons and antiprotons or electrons and positrons can be sustained in the same ring and, colliding at experiments around the circumference, create new particles up to a mass (centre of mass energy as it is called) that is the sum of the two energies. Colliders are today the preferred configuration for a high-energy machine. Earlier, new particles were sought in the debris from a particles collision with a nucleon in a fixed target but such collisions are limited to a smaller centre of mass energy—which rises only as the square root of the accelerated beam energy.

# *1.2.4 Linear Accelerators*

Although disappointed by the rejection of his ray-transformer as a subject for his PhD, Wideröe was led to the idea of a linear accelerator by a paper by G. Ising [7] who tried to overcome the voltage breakdown problem of a single stage of acceleration by placing a series of hollow cylindrical electrodes one after another in a straight line to form what today we would call a 'drift tube linac' or linear accelerator. Wideröe realised that an oscillating potential applied to one drift tube flanked by two others which are earthed, accelerates at both gaps provided the oscillator's phase changes by 180◦ during the flight time between gaps.

In 1927 he built a three-tube model which accelerated sodium ions. At the wavelengths that radio transmitters generated at that time a particle travelling near the velocity of light would travel hundreds of meters in the time it would take for the r.f. to swing by half a sine wave. This would make the length of a drift tube impractically large. Sodium ions, being rather heavy compared with protons or electrons, travelled much slower than the velocity of light and this helped keep the apparatus down to table-top proportions. Although he realised that one might extend such a series of tubes indefinitely he did not take the idea any further as he was due to start his professional employment designing high voltage circuit breakers. Between 1931 and 1934, D. Sloan and E.O. Lawrence at Berkeley took up Wideröe's idea and constructed linacs with as many as 30 drift tubes to accelerate mercury ions but, these were never used for research.

Much later, in the mid-1940s, and when suitable high-power high-frequency oscillators had become available to meet the needs of war-time radar, L.W. Alvarez (1946) started to build the first serious proton linac at the Radiation Laboratory of the University of California. Figure 1.3 shows an Alvarez linac. A series of drift tubes are mounted within a copper-lined cylinder excited by a radio transmitter. As in Wideröe's linac, particles gain energy from the accelerating potential differences between the ends of the drift tube, but now the phase shift between drift tube

**Fig. 1.3** left: The concept of the drift tube linac (from [8]); right: CERN's Linac 1

gaps is 360◦. Each gap appears to the particle to be an identical field gradient which accelerates particles from left to right. The particles are protected from the decelerating phase while inside the metallic drift tubes. Although the particle gains energy steadily as it passes each gap, the total voltage between parts of the assembly and ground does not become larger along the length of the device as it would for an electrostatic machine.

The distances between gaps, or the lengths of the tubes, increase as the particle is accelerated since it travels an ever increasing distance during one swing of the radio frequency oscillation. At low energy, we would expect this distance to increase with the velocity or the root of the kinetic energy but when the energy is large we find the length of the drift tubes and their spacing no longer increases a practical demonstration of special relativity. The Alvarez structure is still widely used, especially for non-relativistic proton and ion beams.

It was well known at the time that waves might be propagated along a much simpler smooth waveguide and that some of the modes have an accelerating electric field in the direction of propagation. Closer examination however shows that the stumbling block is that the phase velocity of these modes in a wave guide is always greater than that of light and hence the particle sees a field which sometimes accelerates and then decelerates as the wave overtakes the particle. It was later found that the phase velocity could be reduced by a series of iris diaphragms in the pipe. Such a structure (Fig. 1.4) is very popular in electron linacs and also in storage rings in which the particle is close to the velocity of light and cavities need not be tuned to follow the acceleration cycle.

These diaphragm-loaded linac structures have been commonly used as injectors for circular accelerators to accelerate electrons and protons to energies in the range 10 to 1000 MeV. As compact high frequency structures they have also been widely used to accelerate electrons to, typically 10 MeV, as a source of X-rays for cancer therapy. An early and very successful adventure in the electron linac development was the "two-mile long" Stanford Linear Accelerator at SLAC in California which has been the work horse for a number of ground breaking fixed target experiments and circulating beam storage ring projects at 20 to 50 GeV. With the help of two semi-circular arcs it was used to bring beams of electrons and positrons into headon collision in the Stanford Linear Collider Project. This project, is forerunner for

**Fig. 1.4** Iris loaded structure (from [9]). The 'chimney' is the input waveguide

today's projected Linear Colliders in which linear accelerators accelerate positrons and electrons to energies approaching 1 TeV to collide them head on in a bid to overcome the very considerable energy lost by an electron to synchrotron radiation in circular lepton rings at high energy.

# *1.2.5 Cyclotrons*

Unlike a linac, whose length must be extended to reach a higher energy, the cyclotron, as it is called, is a relatively compact accelerator in which the energy is only limited by the diameter and field strength of the magnet. The cyclotron idea first occurred to E.O. Lawrence who, reading through Wideröe's thesis, ruminated on the possibility of using a magnetic field to recirculate the beam through two of drift tubes. The cyclotron idea was published in 1930 [10] and another colleague, M.S. Livingston, who was also later to contribute much to the field, was given the job of making a working model as his doctoral thesis.

In Fig. 1.5 we see the two 'Dee's' which comprise the positive and negative electrodes of the accelerating system between the poles of the magnet. These are like two halves of a closed cylinder divided along its diameter. A radio-frequency generator excites them with an alternating field of constant frequency. The potential difference between the 'Dee's' accelerates the ions as they pass the gap between the two halves of the structure. The fundamental trick is that the field oscillates at the particle's circulation frequency and hence the sign of the potential difference at each gap is always in the accelerating direction.

As long as cyclotrons accelerate ions to modest energies, classical rather than relativistic mechanics still applies. In Fig. 1.6 we see the balance between centripetal acceleration of motion in a circle and the force exerted by the vertical magnetic field,

$$evB = \frac{mv^2}{\rho}, \text{if } v \ll c,\tag{1.1}$$

**Fig. 1.6** Balance of forces in

a cyclotron

and, rearranging, we can define the magnetic rigidity—the reluctance of the beam to be bent in a curve:

$$
\mathcal{B}\varrho = \frac{mv}{e}\text{, if }v \ll c.\tag{1.2}
$$

In the relativistic regime if we replace the classical momentum, *mv,* by the relativistic momentum, *p* = *γ mv*, with *γ* being the Lorentz factor*,* we obtain the equation, valid in the relativistic regime:

$$B\varrho = \frac{p}{e}.\tag{1.3}$$

By good fortune the radius of the orbit in a cyclotron is proportional to the velocity and the frequency of revolution this being the inverse of the time of revolution—just the length of one turn divided by the particle's velocity

*f* = *v* 2*π* = *v* 2*π* · *eB mv .* (1.4)

has a numerator and denominator which are both proportional to *v*. This frequency remains constant as the particle is accelerated in the low energy, classical, regime. Thus, the circulating particles stay in synchronisation with the oscillating RF field and a continuous stream of ions injected in the centre will follow a spiral path to reach their highest energy at the rim of the poles.

Unfortunately, the synchronism between r.f. voltage and revolution frequency breaks down as the particles velocity begins to approach that of light and the relativistic mass in the above equation is no longer constant. This happens over 30 MeV for protons and at double this energy for deuterons. Electrons are much too light and relativistic to be accelerated in a cyclotron to any significant energy. For them other acceleration concepts are more adequate, like the disk loaded travelling wave linac or the betatron that both were described before.

The possible remedy of making the field stronger at the edge of the poles would have preserved synchronism and continuous beams but, as we shall see, was in conflict with the need to have a negative radial gradient to the field to provide vertical weak focusing. As a consequence a more powerful concept had to be developed to achieve highest particle beam energies: The synchrotron.

# *1.2.6 The Synchrotron*

Meanwhile, in the 1940s, still higher energies were needed to pursue the aims of physics and the stage was set for the discovery of the synchrotron principle which opened the way to the series of circular accelerators and storage rings which have served particles physics up to the present day. It was Australian physicist Mark Oliphant who synthesized three old ideas into a new concept—the synchrotron. The ideas were: accelerating between the gaps of resonators, varying the frequency, and pulsing the magnet. In 1943 he described his invention in a memo to the UK Atomic Energy Directorate (see [11]).

Particles should be constrained to move in a circle of constant radius thus enabling the use of an annular ring of magnetic field *...* which would be varied in such a way that the radius of curvature remains constant as the particles gain energy through successive accelerations by an alternating electric field applied between coaxial hollow electrodes.

Unlike the cyclotron, the synchrotron accelerates the beam as a series of discrete pulses or "bunches" as they are called. Each short pulse is injected at low field and then the field rises in proportion to the momentum of particles as they are accelerated. This ensures that the radius of the orbit remains constant. In contrast to cyclotrons and betatrons, the synchrotron needs no massive poles to support a

magnetic field within the beam's circular orbit. The guide field is instead provided by a slender ring of individual magnets. The fact that the machine is pulsed and the frequency must be controlled to track the increasing speed of particles is a complication, but it solves the difficulty that isochronous cyclotron builders had encountered in accelerating relativistic particles.

Instead of the Dees of a cyclotron acceleration is provided in a synchrotron by fields within a hollow cylindrical resonator or "pillbox" cavity, Fig. 1.7, excited by a radio transmitter. A particle passes from left to right as it completes each turn of the synchrotron receiving another increment in energy at each revolution.

The early synchrotrons, like the cyclotron before them, relied on a slight negative radial gradient in the vertical magnet field to produce field lines which belly outwards from the magnet gap. A small radial field component deflects any particles which head off towards the poles back to the median plane. Unfortunately, this field shape has the opposite (defocusing) effect horizontally but, up to a certain, rather weak, gradient strength focusing is assured by a slight imbalance between the central force and the centrifugal acceleration. The gradient cannot be too large hence the term "weak focusing".

Oliphant was the first to start building a proton synchrotron (at Birmingham University) but he was overtaken by Stan Livingston's 3 GeV Cosmotron at Brookhaven National Laboratory and later by the 6 GeV Bevatron at Berkeley.

Due to the weak focusing forces in these first synchrotrons, the particles' excursions, both horizontally and vertically are large and the magnet pole width and gap correspondingly so. Strong focusing changed this. It was invented at the Cosmotron, which was actually the first proton synchrotron to operate, whose weak focusing 'C' shaped magnet was open to the outside. The top energy of the Cosmotron was limited by the extra fall-off in field caused by the effect of saturation. Stan Livingston and E.D. Courant wanted to compensate this by re-installing some of the C magnets with their return yokes towards the outside. They were afraid of the variations in gradient around the ring but were surprised to calculate that the focusing seemed to improve as the strength of the alternating component of the gradient increased. Courant, Livingston, and H.S. Snyder [12, 13] were able to explain this retrospectively with an optical analogy of alternating focusing by equal

**Fig. 1.8** The CERN 25 GeV proton synchrotron

convex and concave lenses which will transport rays which pass through the centres of defocusing lenses.

Alternating gradient or strong focusing greatly reduces the beam's excursions and so the cross section of the magnet gap by more than an order of magnitude. Its discovery enabled Brookhaven and CERN to build the next generation of proton synchrotrons, AGS and PS, to reach 30 GeV—five times the energy of the Bevatron—yet use beam pipes of only a few centimetres height and width.

This was to lead to huge economies in the cost per unit length of the magnet system. Figure 1.8 shows how this was applied to the first of the two synchrotrons, AGS and PS that used this focusing system. From then on all synchrotrons and, later, storage ring colliders use this scheme. The history of synchrotrons has been always to seek methods of improving focusing and economizing on magnet aperture. The only other step function in their development to higher energies has been the use of superconducting magnets whose higher fields reduce the circumference of the machine by a factor between 3 and 5.

# *1.2.7 Phase Stability*

When the first synchrotrons were built it was by no means obvious that the circulating beam and the accelerating voltage would remain in step. There were those who thought that any slight mistiming of the sine wave of accelerating voltage in the cavity might build up over many turns until particles would begin to arrive within the negative, decelerating, phase of the sine wave and be left behind. Even if one succeeded in achieving synchronism for the ideal, *synchronous particle*, others of slightly different energy would not have the same velocity and take a different time to circulate around the machine. Would not these particles gradually get out of step until they were lost? After all, particles had to make many hundred thousand turns before reaching full energy and while transverse focusing was understood there was no apparent focusing available in the longitudinal direction. Fortunately the comforting principle of phase stability, which prevents this happening, was soon to be independently discovered by V. I. Veksler in Moscow in 1944 [14] and McMillan in Berkeley in 1945 [15], opening the way to the construction of the first synchrotrons. We shall return to this later.

When it came to the next generation of synchrotrons, interest focused on colliding two opposing beams of particles. It had been known for some time that the energy available in the centre of mass from a collision of particles, one in the beam with energy *E* and the other of mass *m*0 in a fixed target, only increased with the square root of the accelerators energy, √*m*0*E*. Two particles of the same mass and energy *E* colliding head on made available all their energy in the centre of mass, 2*E.* The difficulty was making the two bunches of particles of sufficient density to have a significant probability of collision or, in technical jargon a high enough luminosity. Once this problem was solved a series of colliders: ISR, SppS, LEP, Tevatron, HERA and finally LHC followed. Some of these (ISR, HERA and LHC) were two separate rings which intersected to collide particles at several points around the circumference. Others (SppS and LEP) collided protons with antiprotons and electrons with their anti-particles: positrons. These exploited the fact that beams of particles and antiparticles will circulate on identical trajectories, but in opposite directions, in a single ring of bending and focusing magnets.

At present several studies are ongoing, to pave the way to even higher energies, mainly increasing the size of the machine and using super conducting magnets with higher critical field, to gain more bending and focusing fields in the lattice. One example, the Future Circular Collider study, FCC, under the guidance of CERN, is studying a 100 km proton storage ring to achieve centre of mass energies of up to 100 TeV. The R & D effort of accelerators of this dimension and complexity, in any case, has to be done by a truly international, in other words worldwide effort.

# **References**

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# **Chapter 2 Beam Dynamics**

**E. Wilson and B. J. Holzer**

# **2.1 Linear Transverse Beam Dynamics**

Now let us look in detail at the motion of particles in the transverse coordinates of the coordinate system defined in Fig. 2.1.

# *2.1.1 Co-ordinate System*

The guide field of a synchrotron is usually vertically directed, causing the particle to follow a curved path in the horizontal plane (Fig. 2.1). The force guiding the particle in a circle is horizontal and is given by:

$$F = e \cdot \mathfrak{v} \times \mathfrak{B},\tag{2.1}$$

where:

*v* is the velocity of the charged particle in the direction tangential to its path, *B* is the magnetic guide field.

The guide field inside a dipole magnet is uniform and the ideal motion of the particle is simply a circle of (local) radius of curvature, *ρ*(*s*). The trajectory of an ideal particle (ideal in energy and without any amplitude) that is defined by the arrangement of the dipole magnets is called *design orbit*. The machine is usually designed with this orbit at the centre of its vacuum chamber. Now there is no such

E. Wilson · B. J. Holzer (-)

CERN (European Organization for Nuclear Research), Meyrin, Geneva, Switzerland e-mail<EMAIL_ADDRESS>
S. Myers, H. Schopper (eds.), *Particle Physics Reference Library*, https://doi.org/10.1007/978-3-030-34245-6\_2

**Fig. 2.1** Charged particle orbit in magnetic field

thing as an ideal particle. Still, we shall suppose that it is possible to find an orbit or curved path for the non-ideal particle which closes on itself around the synchrotron, which we call the *closed* or *equilibrium orbit* and it should be close enough to the ideal design orbit.

# *2.1.2 Displacement and Divergence*

A beam of particles enters the machine as a bundle of trajectories spread about the ideal orbit. At any instant a particle may be displaced horizontally by *x* and vertically by *z* from the ideal position and may also have divergence angles horizontally and vertically:

$$\mathbf{x}' = d\mathbf{x}/d\mathbf{s}, \quad \text{and } \mathbf{z}' = d\mathbf{z}/d\mathbf{s}. \tag{2.2}$$

The divergence would cause particles to leave the vacuum pipe except for the carefully shaped field which restores them back towards the beam centre so that they oscillate about the ideal orbit. The design of the restoring fields determines the transverse excursions of the beam and the size of the cross section of the magnets and is therefore of crucial importance to the cost of a project.

# *2.1.3 Bending Magnets and Magnetic Rigidity*

The design of a synchrotron; the diameter of the ring and its sheer size and cost for a given energy is driven by the fact that bending particle trajectories depends on a magnetic rigidity. The rigidity increases with momentum and is a function of the bending field which, for room temperature magnets, saturates at about 2 T.

**Fig. 2.2** Vector diagram showing differential changes in momentum for a particle trajectory

We will now briefly derive an expression for the magnetic rigidity of a relativistic. A particle has a relativistic momentum vector *p* and travels perpendicular to a field *B* which is into the plane of the diagram (Fig. 2.2).

We write the Lorentz force on the particle on its circular path as

$$F\_{Lorentz} = e \ast (\boldsymbol{\upsilon} \times \boldsymbol{B})$$

Assuming an idealized homogeneous dipole magnet along the particle orbit, having pure vertical field lines, the condition for a perfect circular orbit is defined as equality between this Lorentz force and the centrifugal force.

$$F\_{centrifugal} = \frac{\gamma mv^2}{\rho}$$

This yields the following condition for the idealized ring:

$$B\rho = \frac{p}{e}$$

where we are referring to protons and have accordingly set q = e. We conclude that the beam rigidity Bρ, given by the magnetic field and the size of the machine, defines the momentum of a particle that can be carried in the storage ring, or in other words, it ultimately defines, for a given particle energy, the magnetic field of the dipole magnets and the size of the storage ring.

We really should use the units Newton-second for *p* and express *e* in Coulombs to give (*Bρ*) in Tesla·metres. However, in charged particle dynamics we often talk in a careless way about the 'momentum' *pc.* This actually has the dimensions of an energy and is expressed in units of GeV.

**Fig. 2.3** Geometry of a particle trajectory in a bending magnet of length and deflecting angle *θ*

A useful rule of thumb formula based on these units is:

$$B\rho \text{ [T} \cdot \text{m]} = 3.3356 \text{ [pc [GeV]} \text{]}.\tag{2.3}$$

# *2.1.4 Particle Trajectory in a Dipole Bending Magnet*

The trajectory of a particle in a bending magnet or dipole of length is shown in Fig. 2.3. Usually the magnet is placed symmetrically about the arc of the particle's path. One may see from the geometry that:

$$\sin\left(\theta/2\right) = \frac{\ell}{2\varrho} = \frac{\ell B}{2\left(B\varrho\right)},\tag{2.4}$$

and, if *θ* << π/2

$$
\theta \approx \frac{\ell B}{(B\varrho)}.\tag{2.5}
$$

So the bending angle provided by a dipole magnet is given by the ration of its integrated field strength and the beam rigidity.

**Fig. 2.4** Two views of a sphere rolling down a gutter as it is focused by the walls

# *2.1.5 Weak Focusing*

We have mentioned that cyclotrons and early synchrotrons relied on weak focusing to constrain circulating particles within the vacuum chamber. In order to provide this, the vertical guide field has a slight negative gradient in the radial direction around the rim of the accelerator. The field lines belly out from the outer gap of the magnet. It can be shown, by applying ∇ × *B* = **0**, that there will be horizontal field components in this region. These produce vertically directed forces on errant particles causing them to oscillate about the median plane in a potential well (Fig. 2.4).

The motion is analogous to a small sphere rolling down a slightly inclined gutter with constant speed. Figure 2.4 shows two views of this motion and from the bottom view we recognise the motion as a sine wave. Note too that the sphere makes four complete oscillations along the gutter. In the language of accelerators, its motion has a wave number or "tune", *Q* = 4.

To complete the analogy of a weak focusing synchrotron we imagine that we bend the gutter into a circle rather like the brim of a hat. We provide the necessary instrumentation to measure the displacement of the sphere from the centre of the gutter each time it passes a given mark on the brim and we also have a means to measure its transverse velocity. With the aid of a computer, we might convert this information into the divergence angle, which is used as vertical axis in Fig. 2.5:

$$
\alpha' = \frac{d\mathbf{x}}{ds} = \frac{v\_\perp}{v\_\parallel}.\tag{2.6}
$$

We can make a ring shaped gutter out of a slightly different length of gutter than is shown so that *Q* is not an integer. We can plot a point for each arrival of

**Fig. 2.5** The elliptical locus of a particle's history in phase space as it circulates in a synchrotron

the sphere in a diagram of *x* against *x* which we call a 'phase space diagram' of transverse motion. The sphere has a large transverse velocity as it crosses the axis of the gutter and has almost zero transverse velocity as it reaches its maximum displacement.

If we plot these 'observations' they will be an ellipse (Fig. 2.5) and the phase of the oscillator will advance by *Q* evolutions each time the particle returns. Of course, only the fractional part of *Q* may be deduced from our observations since our measurements do not reveal what happens round the rest of the hat's brim.

Now let us use the analogy to define some of the transverse dynamical quantities of a particle beam. The area of the ellipse is a measure of how much the particle departs from the ideal trajectory, represented in the diagram by the origin.

$$\text{Area} = \pi \varepsilon \text{ [mm} \cdot \text{rad]} \,. \tag{2.7}$$

In accelerator notation we use *ε*, the product of the semi-axes of the ellipse as a measure of the area called the emittance. The emittance is usually quoted in units of π mm·mradians. Thus if the semi-axes are 1 mm and 1 mrad the emittance will be 1 mm·mradian but the area will be π mm·mradian. The maximum excursion in displacement, the major axis, of the ellipse is defined as:

$$
\hat{\boldsymbol{x}} = \sqrt{\boldsymbol{\varepsilon}\boldsymbol{\beta}},\tag{2.8}
$$

At locations where the beta function reaches an extremum, i.e. α = 0, we obtain hence

$$
\hat{\mathbf{x}}' = \sqrt{\varepsilon/\beta}.\tag{2.9}
$$

We shall see that *β* (later to be called the envelope or betatron function) is a property of the gutter, not the beam. In the synchrotron it varies around the ring and is the envelope function we have plotted in Fig. 2.10 and again in Fig. 2.11. By analogy, the "brim of the hat" which represents the alternating gradient focusing system shown in this figure will vary its width and curvature around the crown and *β* will follow this variation in some way.

# *2.1.6 Alternating Gradient Focusing*

In Chap. 1 we described a major break-through in the design of synchrotrons: the discovery of alternating gradient (AG) focusing (see [1] for an excellent summary of the dynamics of AG focussing). This allowed designers to use much stronger focusing systems with considerable savings in the space needed for the beam cross section.

The principle is shown in Fig. 2.6 which depicts an optical system in which each lens is concave in one plane while convex in the other and they alternate. It is possible, even with lenses of equal strength, to find a ray which is always on axis at the *D* lenses in the horizontal plane and therefore only sees the *F* lenses. To appear like Fig. 2.6 the spacing of the lenses would have to be 2*f*. If the ray is also central in the lenses which are vertically defocusing, the same condition will apply simultaneously in the vertical plane. At least one particular particle or trajectory corresponding to this ray will never be defocused and be contained indefinitely.

The alternating gradient idea will work even when the rays in the *D* lenses do not pass exactly at their centre and the lenses are not spaced by precisely 2*f.* In fact it is sufficient for the lens strengths and spacing to be chosen to ensure that the particle trajectories *tend* to be closer to the axis in *D* lenses than in *F* lenses as shown in Fig. 2.10.

**Fig. 2.6** Optical analogy with an alternating pattern of lenses

# *2.1.7 Quadrupole Magnets*

The first alternating gradient synchrotorns used alternating magnetic lenses formed by bending magnets having the same vertical guide field but a radial gradient of alternating sign. In a modern synchrotron the functions of guiding and focussing the beam are separated. The dipole magnets which do the guiding have no gradient. The principal focusing elements are quite a different kind of magnet with four poles which produce gradient but no bending. The poles of these quadrupole magnets are truncated rectangular hyperbolae and alternate in polarity around the aperture circle which just touches the poles.

Figure 2.7 shows a particle's view of the fields and forces in the aperture of a quadrupole as it passes through normal to the plane of the paper. The field shape is such that it is zero on the axis of the device but its strength rises linearly with distance from the axis. This can be seen from a superficial examination of Fig. 2.7 if we remember that the product of field and length of any field line joining the poles is a constant. Symmetry tells us that the field is vertical in the median plane (and purely horizontal in the vertical plane of asymmetry). The field must be downwards on the left of the axis if it is upwards on the right.

The horizontal focusing force, −*evBz*, has an inward direction on both sides and, like the restoring force on a weight suspended from a spring, rises linearly with displacement, *x*. The strength of the quadrupole is characterised by its gradient *dBz/dx* normalised with respect to magnetic rigidity:

$$k = \frac{1}{(B\varrho)} \frac{dB\_{\mathfrak{z}}}{dx}.\tag{2.10}$$

**Fig. 2.7** Components of field and force in a magnetic quadrupole. Positive ions approach the reader on paths parallel to the *s* axis (orthogonal to *x* and *z*) [2]

### 2 Beam Dynamics 23

The angular deflection given to a particle passing through a short quadrupole of length, and strength *k*, at a displacement *x* is therefore:

$$
\Delta \mathbf{x}' = \theta = \ell B / (B\rho) = \ell B^\prime \mathbf{x} / (B\rho) = \ell \mathbf{k} \mathbf{x}.\tag{2.11}
$$

The use of *x* to indicate the divergence angle of a trajectory is defined in Fig. 2.5. Compare this with a converging lens in optics:

$$
\Delta \mathbf{x}' = -\mathbf{x}/f \tag{2.12}
$$

and we see that the focal length of a horizontally focusing quadrupole is

$$f = -1/\left(k\ell\right)\tag{2.13}$$

The particular quadrupole shown in Fig. 2.7 would focus positive particles coming out of the paper or negative particles going into the paper in the horizontal plane. A closer examination reveals that such a quadrupole deflects particles with a vertical displacement away from the axis—vertical displacements are defocused. This can be seen if Fig. 2.7 is rotated through 90◦.

# *2.1.8 The Equation of Motion*

Earlier we derived an expression for the change in divergence of a particle passing through the quadrupole. A horizontally focusing quadrupole (which is at the same time vertically defocusing) has a negative *k*.

We first look at the vertical plane. The angular deflection given to a particle passing through a short quadrupole of length *ds* and strength *k* at a displacement *z* is therefore:

$$dz' = -kz ds.\tag{2.14}$$

From this we can deduce a differential equation for the motion

$$z'' + k(s)z = 0.\tag{2.15}$$

Here we would like to make a clear statement: While inside a lattice element, say a quadrupole lens, the normalised gradient *k* is constant and we get a equation that we know from Hook's law in classical mechanics, (see Eq. 2.12), the situation now is more general. We allow *k*(*s*) to change, while our particles are running through the accelerator. The corresponding equation (2.15) is called Hill's Equation, a second order linear equation with a periodic coefficient, *k*(*s*) which describes the distribution of focusing strength around the ring. The above form of Hill's equation applies to the motion in the vertical plane while in the horizontal plane the effect of the dipole magnets has to be included:

$$
\alpha'' + \left[\frac{1}{\rho^2(s)} - k(s)\right]x = 0.\tag{2.16}
$$

Here the sign in front of *k*(*s*) is reversed so that the quadrupole focuses. The extra focusing term *1/ρ2* due to the curvature of the orbit can be significant in small rings. In the old constant gradient synchrotrons, this weak focusing term was the only form of horizontal focusing.

We see in Fig. 2.10, the pattern of one cell of a simple synchrotron lattice—a pattern which is repeated many times around the circumference as may be seen in Fig. 2.11 which shows—in addition to the focusing and defocusing lenses also the bending magnets—bending magnets. Within this pattern of dipole and quadrupole focusing and defocusing (*F* and *D*), particles make betatron oscillations within the envelopes described by *β*x and *β*z, or more precisely, the square roots of these quantities (here we use the variable *y* to represent either the horizontal or the vertical coordinate, *x* or *z*)

$$\mathbf{y} = \sqrt{\varepsilon\beta(\mathbf{s})}\sin\left(\phi(\mathbf{s}) + \phi\_0\right). \tag{2.17}$$

If one tries to verify that this is the solution of Hills Equation an important and necessary condition emerges:

$$
\phi' = 1/\beta \tag{2.18}
$$

From which we see that 2π*β* is the local wavelength of the transverse oscillations.

# *2.1.9 Matrix Description*

Usually in alternating gradient (AG) machines, the ring is a repetitive pattern of focusing fields that we call the "lattice". Each lattice element may be expressed by a matrix and whole sections of the ring which transport the beam from place to place may be represented as the product matrix of the single element matrices involved, which makes the description of particle trajectories very simple and very elegant at the same time. Any linear differential equation, like Hill's Equation, has solutions which can be traced from one point, *s*1, to another, *s*2, by a 2 × 2 matrix, the transport matrix:

$$
\begin{pmatrix} \mathbf{y} \ (\mathbf{s}\_2) \\ \mathbf{y}' \ (\mathbf{s}\_2) \end{pmatrix} = \begin{pmatrix} a \ b \\ c \ d \end{pmatrix} \begin{pmatrix} \mathbf{y} \ (\mathbf{s}\_1) \\ \mathbf{y}' \ (\mathbf{s}\_1) \end{pmatrix} = M\_{21} \begin{pmatrix} \mathbf{y} \ (\mathbf{s}\_1) \\ \mathbf{y}' \ (\mathbf{s}\_1) \end{pmatrix}. \tag{2.19}
$$

### 2 Beam Dynamics 25

The transport matrix *M*21 has a rather simple form for each focusing quadrupole that the particle encounters and for the drift length between quadrupoles and it is easy to compute the four elements numerically once we define the length and focusing strength. We can trace particles by simply forming the product of these elementary matrices. But there is also a general relation between the elements *a*, *b*, *c*, *d* and the amplitude and phase of transverse motion between any two points. Each term in *M*21 can be expressed as a function of *β*(*s*) and *φ*(*s*). The functions of *β*(*s*) and *φ*(*s*) may be calculated by comparing the numerical result of multiplying the individual matrices for quadrupoles and drift lengths with what we know must be the general form of each element.

As a first step, we derive the general form of a periodic transport matrix.

To simplify the notation we drop the explicit dependence of *β* and *φ* on *s* from the expressions—we will just have to remember that they vary with *s*. We also introduce a new quantity:

$$
w = \sqrt{\beta}.\tag{2.20}$$

just to avoid too many terms in what follows.

In this new notation we can write the solution of the Hill Equation:

$$\mathbf{y} = \varepsilon^{1/2} w \,\,\cos\left(\varphi + \phi\_0\right). \tag{2.21}$$

Taking the derivative and substituting *ϕ* = 1/*β* = 1/*w*2 we have:

$$\mathbf{y}' = \varepsilon^{1/2} w' \cos \left(\varphi + \phi\_0 \right) - \frac{\varepsilon^{\frac{1}{2}}}{w} \sin \left(\varphi + \phi\_0 \right). \tag{2.22}$$

Next we substitute these explicit expressions for *y* and *y* in both sides of the matrix equation. We do this first with the initial condition *ϕ*0 = 0, this is the so-called 'cosine' solution, and then we do it again for the 'sine' solution with *ϕ*0 = *π*/2. This is exactly equivalent to tracing the paraxial and central rays through an optical lens. We write *φ*2 − *φ*1 = *φ* for each case. Each of the two solutions give us two equations for *y* and *y* and thus we obtain four simultaneous equations which can be solved for *a*, *b*, *c*, *d* in terms of *w*, *w* , and *ϕ*. The result is the most general form of the transport matrix between the positions s1 and s2 :

$$M\_{12} = \begin{pmatrix} \frac{w\_2}{w\_1}\cos\varphi - w\_2 w\_1' \sin\varphi & w\_1 w\_2 \sin\varphi\\ -\frac{1 + w\_1 w\_1' w\_2 w\_2'}{w\_1 w\_2} \sin\varphi - \left(\frac{w\_1'}{w\_2'} - \frac{w\_2}{w\_1}\right) \cos\varphi \frac{w\_1}{w\_2} \cos\varphi + w\_1 w\_2' \sin\varphi \end{pmatrix} . \tag{2.23}$$

This rather formidable looking expression simplifies a lot, if we refer to a full circle, in other words, if we restrict *M* to apply between two identical points in successive turns or cells of a periodic structure. Then *w*2 = *w*1, *w* 2 = *w* 1, and *ϕ* to become *μ*, the phase advance per cell. The matrix for one period is now:

$$M = \begin{pmatrix} \cos \mu - ww' \sin \mu & w^2 \sin \mu \\ -\frac{1 + w^2 w'^2}{w^2} \sin \mu & \cos \mu + ww' \sin \mu \end{pmatrix}. \tag{2.24}$$

Next we invent some new functions of *β*:

$$\begin{cases} \beta = w^2, \\ \alpha = -ww' = -\frac{\beta'}{2}, \\ \gamma = \frac{1 + \left(ww'\right)^2}{w^2} = \frac{1 + w^2}{\beta}. \end{cases} \tag{2.25}$$

These functions (which are not the same as the parameters used in special relativity!) are a complete and compact description of the dynamics. The matrix now becomes even simpler:

$$M = \begin{pmatrix} \cos \mu + \alpha \sin \mu & \beta \sin \mu \\ -\gamma \sin \mu & \cos \mu - \alpha \sin \mu \end{pmatrix} = \begin{pmatrix} a \ b \\ c \ d \end{pmatrix} . \tag{2.26}$$

This is the Twiss matrix. It is the basic matrix for periodic lattices and should be memorized.

We can imagine that if we can only find an independent way of computing the numerical values of the four elements we can solve and find:

$$\begin{aligned} \cos \mu &= (\text{Tr} \, \mathbf{M}) / 2 = (a+d) / 2, \\ \beta &= b / \sin \mu > 0, \\ \alpha &= (a-b) / (2 \sin \mu) \\ \nu &= -c / \sin \mu. \end{aligned} \tag{2.27}$$

These Twiss parameters, *μ*, *β*, *α*, and *γ* , are therefore rigorously determined by the overall effect of the focusing properties of the lattice elements. Still, they vary around the ring and apply to the point chosen in the period as a starting and finishing point. We shall see that each individual component, quadrupole, dipole, or drift space in the ring has its own matrix and this provides the independent method of calculation. We must first choose the starting point, the location, *s*, where we wish to know *β* and the other Twiss parameters. By starting there and multiplying the element matrices together for one turn we are able to find *a*, *b*, *c*, *d* numerically for that location. We can then apply the above four equations to find the Twiss matrix. If the machine has a natural symmetry in which there are a number of identical periods, it is sufficient to do the multiplication up to the corresponding point in the next period. The values of *α*, *β*, and *γ* would be the same if we went on for the whole ring. By choosing different starting points we can trace *β*(*s*) and *α*(*s*). We now give the matrices for the three basic lattice elements.

# *2.1.10 Transport Matrices for Lattice Components*

An empty space or drift length is the simplest of the lattice component matrices. Figure 2.8(a) shows the analogy between a particle trajectory and a diverging ray in optics. The angle of the ray and the divergence of the trajectory are related:

$$
\theta = \tan^{-1} \left( \mathbf{x}' \right). \tag{2.28}
$$

The effect of a drift length in phase space is a simple horizontal translation from (*x*, *x* ) to (*x*+*x* , *x* ) and can therefore be written as a matrix:

$$
\begin{pmatrix} x\_2 \\ x\_2' \end{pmatrix} = \begin{pmatrix} 1 \ \ell \\ 0 \ 1 \end{pmatrix} \begin{pmatrix} x\_1 \\ x\_1' \end{pmatrix} . \tag{2.29}
$$

The next case is that of a thin quadrupole magnet of infinitely small length but finite integrated gradient:

$$
\ell k = \frac{1}{(B\rho)} \frac{\partial B\_{\mathbb{Z}}}{\partial x}. \tag{2.30}
$$

The optical analogy of a thin quadrupole with a converging lens is illustrated in Fig. 2.8(b). A ray, diverging from the focal point arrives at the lens at a displacement, *x*, and is turned parallel by a deflection:

$$
\theta \approx \frac{1}{f} \text{x.}\tag{2.31}
$$

This deflection will be the same for any ray at displacement *x* irrespective of its divergence. This behaviour can be expressed by a simple matrix, the thin lens matrix:

$$
\begin{pmatrix} x\_2 \\ x\_2' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix} \begin{pmatrix} x\_1 \\ x\_1' \end{pmatrix} . \tag{2.32}
$$

**Fig. 2.8** The effect of a drift—(**a**), left side—and a focusing quadrupole lens—(**b**) right side—on a particle trajectory. The mathematical expressions are given in Eqs. (2.29) and (2.32)

A particle arriving at a quadrupole lens at a displacement *x* obeys Hill's equation

$$
\lambda'' + kx = 0.\tag{2.33}
$$

Hence the small deflection *θ* is just:

$$
\Delta \mathbf{x}' = -k \mathbf{x} \boldsymbol{\ell}. \tag{2.34}
$$

Comparing quadrupoles with optical lenses we remember that *k* = 1/*f* and is the power of the lens and that the matrix, for a thin lens, can be written:

$$
\begin{pmatrix} 1 & 0 \\ -k\ell \ 1 \end{pmatrix}.\tag{2.35}
$$

Under the influence of these focusing and defocusing fields, a particle trajectory will finally look like a more or less zig-zag shaped curve; which for the example of eight regular cells it is shown in Fig. 2.9.

Quadrupoles are sometimes not short compared to their focal length. One must therefore use the matrices for a long quadrupole when one comes to compute the final machine:

$$\begin{array}{l} M\_F = \begin{pmatrix} \cos\ell\sqrt{k} & \frac{1}{\sqrt{k}}\sin\ell\sqrt{k} \\ -\sqrt{k}\sin\ell\sqrt{k} & \cos\ell\sqrt{k} \end{pmatrix}, \text{and} \\\ M\_D = \begin{pmatrix} \cosh\ell\sqrt{k} & \frac{1}{\sqrt{k}}\sinh\ell\sqrt{k} \\ -\sqrt{k}\sinh\ell\sqrt{k} & \cosh\ell\sqrt{k} \end{pmatrix}. \end{array} \tag{2.36}$$

**Fig. 2.9** A single particle trajectory in a ring: At each part of the lattice the amplitude and angle, (x, x ) of the particle are described by a matrix transformation, according to Eq. (2.32). The blue line corresponds to an ideal particle, with x = x = 0 and so refers to the ideal orbit

**Fig. 2.10** The focusing effect of trajectory length in a pure sector dipole magnet

We can compare this with the solutions of Hill's equations within *F* and *D* quadrupoles:

$$\begin{cases} z = \cos\sqrt{k}\ell \ z\_0 + \frac{1}{\sqrt{k}} \sin\sqrt{k}\ell \ z'\_0, \\ x = \cosh\sqrt{k}\ell \ x\_0 + \frac{1}{\sqrt{k}} \sinh\sqrt{k}\ell \ x'\_0. \end{cases} \tag{2.37}$$

We have so far ignored the bending that takes place in dipole magnets and these may be thought of as drift lengths in a first approximation. An exact calculation should include the focusing effect of their ends. A pure sector magnet, whose ends are normal to the beam will give more deflection to a ray which passes at a displacement *x* away from the centre of curvature (Fig. 2.10). This particle will have a longer trajectory in the magnet. The effect is exactly like a lens which focuses horizontally but not vertically. The matrices for a sector magnet are:

$$\begin{aligned} M\_{\rm H} &= \begin{pmatrix} \cos \theta & \rho \sin \theta \\ -(1/\rho)\sin \theta & \cos \theta \end{pmatrix}, \\\ M\_{\rm V} &= \begin{pmatrix} 1 & \rho \theta \\ 0 & 1 \end{pmatrix}. \end{aligned} \tag{2.38}$$

Some bending magnets are not sector magnets as in Fig. 2.9, but have end faces which are parallel. It is easier to stack laminations this way than on a curve. The entry and exit angles are therefore, *θ*/2, and the horizontal focusing effect is reduced but there is an additional focusing effect for a particle whose trajectory is displaced vertically. In the computer model one may convert a pure sector magnet into a parallel faced magnet by simply adding two thin lenses at each face. They are horizontally defocusing and vertically focusing and their strength is:

$$k\ell = -\frac{\tan\left(\theta/2\right)}{\rho}.\tag{2.39}$$

Unlike early lattice designers we have computers to help when we come to multiply these elements together to form the matrix for a ring or a period of the lattice [3–5]. A lattice program such as MAD [6] does all the matrix multiplication to obtain (*a*, *b*, *c*, *d*) from each specified point, *s*, and back again. It prints out *β* and *ϕ* and other lattice variables in each plane, and we can plot the result to find the beam envelope around the machine. This is the way machines are designed. Lengths, gradients, and numbers of FODO normal periods are varied to match the desired beam sizes and *Q* values.

In Fig. 2.8 we saw the trajectory of a particle, oscillating in a pattern of alternating focussing and defocusing quadrupoles (FODO). The trajectories in general all lie within an envelope which has the general features of the optical model in Fig. 2.6. If we were to repeat the observation of the displacement and divergence of a particle on successive turns we would find the elliptical locus of its motion (Fig. 2.5). The aspect ratio of this ellipse would depend upon where in the ring we choose to make the observation. The ellipse would be squat near *D* lenses and elongated near *F*'s. The figure would appear just the same if we were to plot it between what are *F* quadrupoles in the vertical plane. Of course, the whole pattern of quadruples and the envelope is shifted by the distance between adjacent quadrupoles because *F*quadrupoles in one plane are *D* in the other (et vice versa).

# *2.1.11 The Betatron Envelopes*

To recapitulate, a modern synchrotron consists of pure bending magnets and quadrupole magnets or lenses which provide focusing. These are interspersed among the bending magnets of the ring in a pattern called the lattice. By suitable choice of strength and spacing of the lenses the envelope function *β*(*s*) can be made periodic in such a way that it is large at all *F* quadrupoles and small at all *D*'s. Symmetry will ensure this is true also in the vertical plane. Particles oscillating within this envelope will always tend to be further off axis in *F* quadrupoles than in *D* quadrupoles and there will therefore be a net focusing action. We have already seen that *β* is the aspect ratio of the phase space ellipse (see also [7, 8]).

In Fig. 2.11 we see an example of such a magnet pattern which is one cell, or about 1% of the circumference, of the 400 GeV SPS at CERN. Although the SPS is now considered a rather old fashioned machine its simplicity leads us to use it as an example. The focusing structure is FODO and in this pattern half of the quadrupoles (*F*) focus, while the other half, defocus (*D*) the beam. Bending magnets, which in a first approximation do no focussing are represented together with other nonfocussing elements by the letter "O". The envelope of these oscillations follows a function *β*(*s*) which has waists near each defocusing magnet and has a maximum at the centres of *F* quadrupoles. Since *F* quadrupoles in the horizontal plane are *D* quadrupoles vertically, and vice versa, the two functions *β*h(*s*) and *β*v(*s*) are one half-cell out of register in the two transverse planes. The function *β* has the dimensions of length but the units bear no relation at this stage to physical beam

size. The reader should be persuaded that particles do not follow the *β*(*s*) curves but oscillate within them in a form of modified sinusoidal motion whose phase advance is described by *φ*(*s*). The phase change per cell in the example shown is close to π/2 but the rate of phase advance is modulated throughout the cell.

# **2.2 Coupling**

Until now we have considered the motion in the vertical and horizontal direction to be orthogonal and independent. This is the ideal case. Now we look at what happens when there is a skew quadrupole or solenoidal field in the machine which couples horizontal motion into vertical and vice versa. This is rather a special case affecting mainly electron synchrotrons and the reader may choose to skip to Sect. 2.3 and leave coupling to a second reading.

In a fully coupled machine the betatron oscillations in the two transverse directions are like two harmonic oscillators which transfer energy from one to the other with a frequency which is just the difference between their *Q*'s. They act like coupled pendula. In this way all the horizontal "emittance" can add to the vertical emittance and the beam exceeds the available vertical aperture.

The phenomenon is particularly important in electron rings. The electron beam would damp to zero emittance were it not for quantum emission in the horizontal plane exciting betatron oscillations. There is no comparable excitation in the vertical plane and only coupling of the horizontal oscillations into the vertical plane gives the beam any vertical dimensions. Vertical emittance, and the magnet gap needed to accept it, is directly proportional to coupling.

# *2.2.1 Coupling Fields*

There are two principal configurations of field which excite coupling. The first we shall consider is a skew quadrupole, i.e. a quadrupole whose poles lie symmetrically in the horizontal and vertical planes (Fig. 2.12).

A particle with horizontal position *x*, experiences not a *Bz* as would be the case in a normal quadrupole and which would change its *x* , but a *Bx* which together with the paraxial velocity deflects vertically in the direction of *v* × *B*. Of course once the particle has acquired a vertical displacement *z* after a number of turns it experiences a vertical field, for in a skew quadrupole the field is:

$$\begin{aligned} B\_{\chi} \, / \, (B\rho) &= kx, \\ B\_{\mathbb{Z}} \, / \, (B\rho) &= -kz. \end{aligned} \tag{2.40}$$

Thus a horizontal displacement couples into the vertical plane leading to a vertical divergence and displacement. The vertical displacement goes on to couple back into the horizontal plane modifying the horizontal displacement and divergence and so it proceeds transferring transverse momentum back and forth from one plane to the other.

A solenoid is the other field configuration that can couple the two planes but this kind of coupling is less important in synchrotrons and we leave it to the reader to consult a more exhaustive treatment of coupling in [9].

**Fig. 2.12** The magnetic field and force in a skew quadrupole

# *2.2.2 Qualitative Treatment of Coupling*

In our treatment the theory is deliberately simplified to reveal the physical mechanisms at work. We assume that the coupling is driven by a single skew quadrupole at the centre of one of the existing lattice machine quadrupoles where *β* is maximum and its derivative zero. We ignore the changes in betatron phase of one plane with respect to the other within a single turn.

The skew quadrupole gradient is normalized:

$$\begin{array}{l} k = \frac{1}{\left(B\rho\right)} \left(\frac{\partial B\_{\chi}}{\partial \chi}\right)\_{z=0}, \\ l = \text{length of the quadrupole.} \end{array} \tag{2.41}$$

Figure 2.13 on the left shows the betatron motion in the horizontal plane. We have normalized the elliptical phase space trajectory into a circle at the location of the skew quadrupole by multiplying the divergence by *βx*. On the right we have done the same for the vertical plane. The angular kick *p*, on passing the skew quadrupole is calculated from a similar diagram for the vertical plane and

$$
\Delta p\_{\overline{\lambda}} = \beta\_{\overline{\lambda}} k \ell w \cos \mathcal{Q} \mathbf{v} \theta,\tag{2.42}
$$

where *w* = √*ε*V*βz* is the radius of the circle for vertical motion, and *u* = √*ε*H*βx* is the radius horizontally.

**Fig. 2.13** Phase space diagram

The kick, projected as an amplitude increment becomes:

$$
\delta\mu = w\beta\_{\text{x}}k\ell\sin\mathcal{Q}\_{\text{H}}\theta\cos\mathcal{Q}\_{\text{V}}\theta.\tag{2.43}
$$

When we use:

$$
\sin A \cos B = \frac{1}{2}\sin(A - B) + \frac{1}{2}\sin(A + B)
$$

and ignore the second, high frequency term; we obtain the coupled equations for a single passage:

$$\begin{split} \frac{\delta w}{w} &= -\sqrt{\frac{\varepsilon\_{\rm H}}{\varepsilon \mathrm{v}}} \sqrt{\frac{\beta\_{\rm x} \beta\_{\rm z}}{2}} k \ell \sin \left( Q\_{\rm H} - Q\_{\rm V} \right) \theta, \\\frac{\delta u}{u} &= \sqrt{\frac{\varepsilon \mathrm{v}}{\varepsilon\_{\rm H}}} \sqrt{\frac{\beta\_{\rm x} \beta\_{\rm z}}{2}} k \ell \sin \left( Q\_{\rm H} - Q\_{\rm V} \right) \theta. \end{split} \tag{2.44}$$

These are incremental equations which we must sum over the *n* turns as the coupling enhances *u* at the expense of *w*.

Figure 2.14 shows diagrammatically the coupled motion. The vertical betatron amplitude decrease from *w* + *w* to *w* in one quarter period of the slow oscillation which takes 1/4|*Q*H − *Q*v| turns. The mean value of the cosine is taken as 2/π. We then arrive at the expressions for the maximum excursions in amplitude:

$$\begin{array}{lcl}\frac{\Delta w}{w} = \sqrt{\frac{\varepsilon\_{\rm H}}{\varepsilon\_{\rm V}}} \frac{\sqrt{\beta\_{\rm x} \beta\_{\rm z}}}{4\pi |Q\_{\rm H} - Q\_{\rm v}|} k\ell, \\\frac{\Delta u}{u} = \sqrt{\frac{\varepsilon\_{\rm V}}{\varepsilon\_{\rm H}}} \frac{\sqrt{\beta\_{\rm x} \beta\_{\rm z}}}{4\pi |Q\_{\rm H} - Q\_{\rm v}|} k\ell. \end{array} \tag{2.45}$$

We now move from the phase plane into real space. Some machines were designed to have a rectangular "vacuum chamber" which would accept particles which simultaneously have large horizontal and vertical "emittances". In this sense emittance is defined for a single particle

$$
\varepsilon\_{\rm H} = \pi \mu^2 / \beta\_{\rm x}, \varepsilon\_{\rm V} = \pi \, w^2 / \beta\_{\rm y}. \tag{2.46}
$$

In the presence of coupling, the particle motion is a series of Lissajous figures filling the rectangular cross-section but always touching it somewhere on each turn (Fig. 2.15). It is inevitable therefore that if coupling increases either amplitude by *u/u* or *w/w*, some fraction of particles will be lost.

A rigorous treatment of coupling is too lengthy to include here but the reader may consult [9–11] for a complete description. This lengthier treatment leads to a model in which the modes of betatron oscillations are no longer about the vertical and horizontal planes but about two orthogonal principal planes inclined with respect to the vertical and horizontal frame.

**Fig. 2.14** Coupled betatron oscillations for 1/|*Q*H − *Q*v| turns

# **2.3 Liouville's Theorem**

Now let us return to the 'mainstream' of transverse dynamics. Liouville's theorem is a conservation law that applies to the area occupied by a number of particles plotted in phase space.

We should think of a beam of particles as a cloud of points within a closed contour in a transverse phase space diagram (Fig. 2.16). Liouville's theorem tells us that this area within the contour is conserved. The contour is usually, but not always, an ellipse. In Fig. 2.5 we came across such an elliptical contour—the locus of a particle's motion plotted in phase space (x,x ) and we call its area, the emittance. We could also think of it as a limiting contour enclosing all the particles in the beam which we would again call the emittance—not of the particle but of the beam as a particle ensemble.

We express beam emittance in units of π mm·milliradians. According to Liouville the emittance area will be conserved as the beam passes down a transport line or circulates in a synchrotron whatever magnetic focusing or bending operation we do on the beam—provided that only conservative forces are taken into account.

Even though the ellipse may appear to have many shapes around the accelerator its phase space area will not change (Fig. 2.17). The aspect ratio of the ellipse will change however. At a narrow waist, near a *D* quadrupole (a) in Fig. 2.17, its

**Fig. 2.17** How the conserved phase space appears at different points in a FODO cell

**Fig. 2.18** The parameters of a phase-space ellipse containing an emittance *ε* at a certain point in the lattice. The shape and orientation of the ellipse are determined by the Twiss parameters at the given location

divergence will be large, while in an *F* quadrupole (d) where the betatron function is maximum, its divergence will be small. The beam is also seen at a broad waist or maximum in the beta function and a place where the beam is diverging.

In Fig. 2.18 we see how the various features of the ellipse are related to the Twiss parameters. The equation of the ellipse, often called the Courant and Snyder invariant, has the form

$$
\lambda \mathbf{y}(\mathbf{s}) \mathbf{y}^2 + 2\alpha(\mathbf{s}) \mathbf{y} \mathbf{y}' + \beta(\mathbf{s}) \mathbf{y}'^2 = \varepsilon. \tag{2.47}
$$

Here *y* is used to mean either of the transverse displacements, *x* or *z*. It is straight forward determine the relation between the shape and orientation of the (x,x ) ellipse and the Twiss parameters α, β, γ as indicated in Fig. 2.18.

The invariance of the (x,x ) space area, as we move to different points in the ring is an alternative statement of Liouville's theorem.

A word of caution—another, stricter, version of Liouville's theorem states that:

In the vicinity of a particle, the particle density in phase space is constant if the particles move in an external magnetic field or in a general field in which the forces do not depend upon velocity.

This rules out the application of Liouville's theorem to situations in which space charge forces within the beam play a role or when there is a velocity dependent effect such as when particles emit synchrotron light. However we may apply Liouville to proton beams which do not normally emit synchrotron light and to electrons travelling for a few turns in a synchrotron. This is usually too short a time for electrons to emit enough synchrotron light energy to affect their transverse motion.

Liouville's theorem does not apply as a proton beam is accelerated. Observations tell us this is not the case. The beam appears to shrink. This is because the coordinates we have used so far, *y* and *y* , are not 'canonical' in the sense defined by Hamiltonian in his mechanics, which is part and parcel of Liouville's mathematical theory of dynamics. We should therefore express emittance in Hamilton's canonical phase space and relate this carefully to the co-ordinates, displacement, *y*, and divergence, *y* , which we have been using so far. We can then define an emittance which is conserved even as we accelerate.

We shall have to be particularly careful not to confuse Twiss parameters, and the parameters of special relativity: In special relativity we use *β* as the ratio of the particles velocity and the speed of light and the Lorentz factor *γ* describes the total energy divided by the rest energy. The reader will have to examine the context to be sure. For those who have not met Hamiltonian mechanics, it is sufficient to know that the canonical co-ordinates of relativistic mechanics are:

$$p = \frac{m\_0 \dot{\mathbf{y}}}{\sqrt{1 - v^2/c^2}}, q = \mathbf{y}. \tag{2.48}$$

Here *q* or *y* is a general transverse co-ordinate, *p* its conjugate momentum and we define *β* and *γ* when used in the context of special relativity to be:

$$\begin{array}{l} \beta = v/c, \\ \gamma = 1/\sqrt{1 - \beta^2}, \\ m\_0 = \text{rest mass}, \\ c = \text{velocity of light.} \end{array} \tag{2.49}$$

### 2 Beam Dynamics 39

We may find the relationship between canonical momentum and divergence from the substitution:

$$p\_\mathcal{V} = m\_0 \frac{d\mathbf{y}}{dt} \mathcal{V} = m\_0 \frac{ds}{dt} \frac{d\mathbf{y}}{ds} \mathcal{V} = mc \left(\beta \mathcal{V}\right) \mathbf{y}'.\tag{2.50}$$

Writing Liouville's Theorem expressed in canonical coordinates we can use the above expression to define a conserved quantity and relate it to the area in (*y*, *y* ) space

$$
\int p\_\mathbf{y} d\mathbf{y} = m\_0 c \left(\beta \mathbf{y}\right) \int \mathbf{y}' d\mathbf{y} = p\_0 \int \mathbf{y}' d\mathbf{y} \tag{2.51}
$$

where *p*0 is the momentum in the direction of motion of the particle.

This invariant is the emittance, *ε*, of our transverse phase space multiplied by the relativistic *βγ* which is proportional to momentum. Accelerator physicists often call this the invariant or 'normalised' emittance:

$$
\varepsilon^\* = \beta \gamma \varepsilon \text{ [}\pi \text{ mm} \cdot \text{mrad]} \tag{2.52}
$$

This normalised emittance, *ε*∗, is conserved as acceleration proceeds in a synchrotron and the physical emittance within the right-hand side of the equation must fall inversely with momentum if the whole term is to be conserved. Close to the velocity of light this implies that it is inversely proportional to energy.

$$Emitance = \pi \varepsilon = \int \mathbf{y}' d\mathbf{y} = \pi \varepsilon^\* / (\beta \gamma) \propto 1/p\_0. \tag{2.53}$$

We therefore expect the beam dimensions to shrink as (Fig. 2.19) a phenomenon called 'adiabatic damping'.

# *2.3.1 Chains of Accelerators*

As a consequence of the adiabatic shrinking, the beam emittance is largest at low energy, and so is the beam dimension. Proton accelerators need their full aperture at injection and it is then that their design is most critical. For this reason it is economic to split a single large ring into a chain of accelerators—the smaller radius rings having a large aperture while the higher energy rings with large radius can have smaller apertures. In these chains of proton accelerators, such as the CERN accelerator complex, Linac – Booster – PS – SPS, the invariant emittance, determined by the parameters of the beam as it leaves the ion source at the beginning of the linac, may be conserved to several hundred GeV. Of course one must guard against mismatches between machines or non-linear fields which dilate the emittance.

# *2.3.2 Exceptions to Liouville's Theorem*

The invariance of normalised emittance of a proton beam and the shrinking of its physical emittance with energy is quite the opposite of what happens in an electron machine. Liouville's theorem only applies to conservative systems, where particles are guided by external fields and not to electron machines where particles emit some of their own energy. Electrons, being lighter than protons and hence more relativistic, emit quanta of radiation as they are accelerated. This quantised emission causes particles to jump around in momentum, leading to changes in the trajectories amplitude and angle. These changes couple into both planes of transverse phase space. At the same time, there is a steady tendency for particles near the edge of the emittance to lose transverse energy and fall back towards the centre. In an electron machine the emittance is determined not by the Liouville but by the equilibrium between these two effects. In fact, it grows with *E*2.

Consider a number of protons which have the maximum amplitude present in the beam. They follow trajectories at the perimeter of the ellipse but at any instant have a random distribution of initial phases *φ*0. If we were able to measure *y* and y for each and plot them in phase space, they would lie around the ellipse of area *πε* and their co-ordinates would lie in the range of

$$\begin{array}{ll} -\sqrt{\beta\varepsilon} \le \mathbf{y} \le \sqrt{\beta\varepsilon}, \\ -\sqrt{\varepsilon\boldsymbol{\nu}} \le \mathbf{y}' \le \sqrt{\varepsilon\boldsymbol{\nu}}. \end{array} \tag{2.54}$$

Particles in a beam are usually distributed in a population which appears Gaussian when projected on a vertical or horizontal plane. In a proton machine the emittance boundary used to be conventionally chosen to be that of a proton with amplitude 2σ. This would include about 90% (strictly 87%) of a Gaussian beam where σ is the standard distribution. In an electron machine a 2σ boundary would be too close to the beam and an aperture stop placed at this distance would rather rapidly absorb most of the beam as particles redistribute themselves, moving temporarily into the tails due to quantum emission and damping. The safe physical boundary for electrons depends on the lifetime required but is in the region of 6σ to 10σ. The emittance which is normally quoted for an electron beam corresponds to an electron with the amplitude of σ in the Gaussian projection. We are then free to choose how many σ's we must allow. There is consequently a factor 4 between emittance defined by electron and proton experts.

# **2.4 Momentum Dependent Transverse Motion**

In the previous chapters, we have studied the motion of a particle as it swings from side to side about the ideal orbit around the synchrotron: the transverse motion. Nothing is perfect, however, and so we cannot assume that each and every proton in a large ensemble of up to 1011 particles will have exactly the ideal momentum. Instead we expect a certain momentum spread in the beam and therefore we have to study how transverse motion depends on small departures, *p*/*p*0, from the synchronous momentum *p*0.

# *2.4.1 Dispersion*

The central closed orbit of a synchrotron is matched to an ideal (synchronous) momentum *p*0. A particle of this momentum and of zero betatron amplitude will pass down the centre of each quadrupole, be bent by exactly 2π by the bending magnets in one turn of the ring and remain synchronous with the r.f. frequency. Its path is called the central (or synchronous) momentum closed orbit. In Fig. 2.8 this ideal orbit is the horizontal axis and we see particles executing betatron oscillations about it but these oscillations do not replicate every turn. The synchronous orbit, however, closes on itself so that *x* and *x* remain zero.

We now consider a closed orbit which is distorted in the horizontal plane by non-ideal bends in the dipole. Figure 2.20 shows a particle with a lower momentum *p*/*p* < 0 and which is bent horizontally more in each dipole of a FODO lattice. We could argue that the total deflection, being more than 2π would cause it to spiral inwards and hit the vacuum chamber wall. On the other hand there is a closed orbit for this lower momentum in which the extra bending forces are compensated by extra focusing forces as the orbit is displaced inwards in the *F* quadrupoles and less so in the defocusing in the *D*'s (Fig. 2.20). We may describe the shape of this new closed orbit for a particle of unit *p*/*p* by a *dispersion* function *D*(*s*). The

displacement of the closed orbit is:

$$
\alpha(s) = D(s) \frac{\Delta p}{p\_0}.\tag{2.55}
$$

In Fig. 2.21 we see how the effect of dispersion for off momentum orbits adds to the betatron motion to widen the beam cross section. The betatron motion of each of the three particles: *p*/*p* < 0, *p*/*p* = 0, and *p*/*p* > 0, is within an ellipse in physical (*x*, *z*) space. The ellipses for each momentum are separated by a distance *D*(*s*)*p*/*p*. The semi-aperture required will be:

$$a\_{\rm V} = \sqrt{\beta\_{\rm V} \varepsilon\_{\rm V}}, a\_{\rm H} = \sqrt{\beta\_{\rm H} \varepsilon\_{\rm H}} + D(\rm s) \frac{\Delta p}{p}. \tag{2.56}$$

# *2.4.2 Chromaticity*

This effect is equivalent to the chromatic aberration in a lens. It is defined as a quantity *Q* :

$$
\Delta \mathcal{Q} = \mathcal{Q}' \frac{\Delta p}{p}. \tag{2.57}
$$

### 2 Beam Dynamics 43

The chromaticity [12] arises because the focusing strength of a quadrupole has (*Bρ*) in the denominator and is therefore inversely proportional to momentum:

$$k = \frac{1}{(B\rho)} \frac{dB\_{\mathbb{Z}}}{dx}.\tag{2.58}$$

A small spread in momentum in the beam, ± *p*/*p*, causes a spread in focusing strength:

$$\frac{\Delta k}{k} = \mp \frac{\Delta p}{p}.\tag{2.59}$$

Integrated over all focusing (and defocusing) elements in the ring, we obtain a change in the tune of the machine

$$
\Delta Q = \frac{1}{4\pi} \int \beta(s)\delta k(s)ds. \tag{2.60}
$$

This enables us to calculate *Q* :

$$
\Delta Q = \frac{1}{4\pi} \int \beta(\mathbf{s}) \delta k(\mathbf{s}) d\mathbf{s} = \left[ \frac{-1}{4\pi} \int \beta(\mathbf{s}) k(\mathbf{s}) d\mathbf{s} \right] \frac{\Delta p}{p}. \tag{2.61}
$$

The quantity in square brackets is the chromaticity *Q* . To be clear, this is called the natural chromaticity. For most alternating gradient machines, its value is about −1.3*Q.* Of course there are two *Q* values relating to horizontal and vertical oscillations and therefore two chromaticities. Chromaticity may be corrected with sextupole magnets (see Chap. 3, and Sects. 6.1 and 8.1).

# **2.5 Longitudinal Motion**

# *2.5.1 Stability of the Lagging Particle*

Suppose two particles are well below the velocity of light. A particle *A,* that arrives at the right moment to in the RF resonator and thus will obtain exactly the right acceleration voltage. We call this particle "*synchronous*" (see Fig. 2.22). A second particle, *B*, arrives late, and so receives an extra energy increment which will cause it to speed up and overtake the synchronous particle, *A*. In so doing, its energy defect, *E*, grows and, provided the amplitude is not too large, its trajectory will follow an ellipse in phase space. This describes this motion up and down the r.f. wave (Fig. 2.22) and may remind some readers of the representation of a simple harmonic oscillator, or pendulum. When plotted in a phase space diagram of velocity versus longitudinal displacement we indeed obtain a shape that is elliptical. The trajectory

**Fig. 2.22** The limiting trajectory for a particle in a 'moving' or accelerating bucket when the stable phase is not zero

of this longitudinal movement in phase space is closed and over many turns the average deviation from the synchronous energy is zero. This *phase stability* depends upon the fact that δ*E* is positive when *φ* − *φ*s is small and positive [13, 14].

When a particle reaches the non-linear part of the r.f. wave and over the top of the wave, it will still be restored and oscillate about the stable phase provided it does not reach and pass the point where it receives less incremental energy than the synchronous particle. On this non-linear part of the curve the motion is no longer an ellipse but is distorted into a fish-shape but its trajectory is still closed and stable. However, if a particle, *C*, oscillates with such large amplitude that it falls below the synchronous voltage, an increase in *φ* will cause a negative *E* which in turn causes *φ* to move further away from the synchrotron particle. This particle is clearly unstable and will be continuously decelerated. There is a particle which, starting at *φ* = π − *φ*s, would trace out a limiting fish-shaped trajectory which is the boundary or *separatrix* between stable and unstable motion. The region within this separatrix is called the r.f. bucket and is shown in the lower half of Fig. 2.22. Formulae for the calculation of the parameters of moving buckets are to be found in [15].

Let us look more carefully at the argument that a particle, arriving late because of its lower energy, would see a higher RF voltage from the rising waveform and, accelerated to a higher velocity, would catch up with the synchronous particle. Dispersion may make the situation more complicated. Giving the errant particle more energy will speed it up but may also send it on an orbit of larger radius.

The path length that the particle, *B*, must travel around the machine, or more correctly, the change in path length with momentum, must depend upon the dispersion function. The closed orbit will have a mean radius:

$$R = R\_0 + \overline{D}\frac{\Delta p}{p}.\tag{2.62}$$

Close to the velocity of light where acceleration can increase momentum but not velocity, the longer path length will more than cancel the small effect of velocity and the particle, instead of catching up with its synchronous partner, will arrive even later than it did on the previous turn. This seems to defeat the whole idea of phase stability. Depending on how the synchrotron is designed and which particles it accelerates, there can be a certain energy where our initial ideas of phase stability break down. This is called the transition energy. Fortunately there is also a way of ensuring stability above transition.

# *2.5.2 Transition Energy*

The rigorous argument to resolve the question of velocity versus path length is to examine how the revolution time (or its reciprocal, the revolution frequency) varies as the particle is given extra acceleration. The revolution frequency is:

$$f = \frac{\beta c}{2\pi R}, (\beta = v/c) \,. \tag{2.63}$$

This revolution frequency, *f*, depends on two momentum dependent variables, the relativistic *β*=*v/c* and *R*, the mean radius. The penultimate equation gives the change in the radius. The momentum dependence of *β* is determined by:

$$p = \frac{E\_0 \beta}{\sqrt{1 - \beta^2}}.\tag{2.64}$$

The rate of "catching up" depends upon a "slip factor", *η*, which is defined as logarithmic differential of frequency as a function of momentum. The procedure of partial derivatives tells us there must be two terms. Hence:

$$\eta\_{\rm rf} = \frac{\Delta f/f}{\Delta p/p} = \frac{p}{\beta} \frac{d\beta}{dp} - \frac{p}{R} \frac{dR}{dp} = \frac{1}{\chi^2} - \frac{D}{R\_0}.\tag{2.65}$$

The first term on the right-hand side describes the increase in speed with *p* and the other (negative), how the path to be completed increases with *p*.

The second term is energy independent while the first term shrinks as acceleration proceeds. At low energy this is largest and *η* is >0. But, since *γ* = *E*/*E*0, the first term becomes smaller than the second at high energy so that *η* changes sign from positive to negative. During the acceleration process there is a certain energy, the transition energy, at which *η* is momentarily zero. At transition, the value of *γ* satisfies:

$$\frac{1}{\left|\chi\_{\rm tr}^{2}\right.} = \frac{D}{R}.\tag{2.66}$$

In proton synchrotron design this condition tends to be encountered mid–way through the acceleration cycle and can only be avoided with some ingenuity in the design of the lattice. This was a worry to the designers of the PS and AGS, the

first proton synchrotrons of high enough energy to encounter this problem during acceleration but it was then realised that one could, almost instantaneously, change the phase of the voltage wave in the RF cavities to be falling rather than rising at the moment of the synchronous particles arrival (see Fig. 2.23). With such a reversed slope, particles arriving late are given less than their ration of energy and take a inner circular path—a short cut—to arrive earlier next time.

Electron machines are fortunate in that due to the small rest mass their Lorentz factor *γ* , being 2000 times higher, ensures that the first term may be neglected and such machines operate always above transition.

# *2.5.3 Synchrotron Motion*

If we consider the motion of a particle on the linear part of the voltage wave of an r.f. cavity it is not difficult to imagine that it approximates rather closely to a harmonic oscillator. Unlike to the situation in the transverse plane, however, the motion becomes more complicated when the oscillation amplitude is larger, and the particle feels the non-linear part of the sinusoidal RF wave, or even more, for part of its motion it finds itself over the crest of the wave. But first let us focus on a small amplitude solution.

It is not hard to deduce from special relativity that the momentum may be written

$$p = m\_0 c \left(\beta \varproj} \right). \tag{2.67}$$

The quantity (*βγ* ) serves as the momentum co-ordinate in longitudinal phase space. The other co-ordinate is the particle's arrival phase, *φ,* with respect to the zero crossing of the r.f. voltage at the cavity. Let us consider the simplest case of a small oscillation in a stationary bucket, *φ*s = 0 (when the particle is not being accelerated).

### 2 Beam Dynamics 47

A particle with a small phase error will describe an ellipse in phase space which one may write parametrically as -

$$\begin{array}{l} \Delta(\beta \underline{\gamma}) = \Delta(\bar{\beta}\underline{\gamma}) \sin 2\pi f\_{\rm s} t, \\ \phi = \hat{\phi} \cos 2\pi f\_{\rm s} t, \end{array} \tag{2.68}$$

where *f*s is the frequency of execution of these oscillations in phase which we call the synchrotron frequency.

To reveal the differential equation behind this motion we must first remember that the angular frequency 2π*f* of an oscillator is nothing other than the rate of change of phase, *φ*˙ or to be exact −*φ*˙. (The negative sign stems from the fact that *φ* is a phase lag.) We may therefore relate the rate of change in arrival phase to the difference in revolution frequency of the particle, compared to that of the synchronous particle.

$$\dot{\phi} = -2\pi h \left[ f \left( \Delta \beta \gamma \right) - f(0) \right] = -2\pi h \Delta f. \tag{2.69}$$

We have multiplied by, *h*, the harmonic number of the r.f. since *φ* is the phase angle of the r.f. swing while *f*(*βγ* ) is the revolution frequency. Here we can use the definition of the slip factor *η* and then simply use some standard relativistic relations to end up with *f* as a function of *E*, the energy defect with respect to the synchronous particle:

$$
\Delta f = \eta f \frac{\Delta p}{p} = \eta f \frac{\Delta \left(\beta \chi\right)}{\left(\beta \chi\right)} = \frac{\eta f}{\beta^2} \frac{\Delta \chi}{\chi} = \frac{\eta f}{E\_0 \beta^2 \chi} \Delta E. \tag{2.70}
$$

where *E*0 the total energy (including its rest mass) of the synchronous particle.

We differentiate once more to obtain a second order differential equation which we hope to resemble a simple oscillator.

$$
\ddot{\phi} = -\frac{2\pi h\eta f}{E\_0 \rho^2 \nu} \left(\Delta \dot{E}\right). \tag{2.71}
$$

The extra energy given per turn to a particle whose arrival phase is *φ* will be

$$
\Delta E = eV\_0 \left(\sin\phi - \sin\phi\_\text{s}\right),
\tag{2.72}
$$

and the rate of change of energy will be this times, *f*, the revolution frequency. So we can write

$$\ddot{\phi} = -\frac{2\pi e V\_0 h \eta f^2}{E\_0 \theta^2 \mathcal{Y}} \left(\sin\phi - \sin\phi\_\text{s}\right). \tag{2.73}$$

This is a fundamental and exact description of the motion provided the parameters should change slowly (the adiabatic assumption). We can simply integrate to find its solution numerically but to see an analytic solution for small amplitudes we set *φ*s = 0 and *φ* ≈ sin*φ* :

$$
\ddot{\phi} + \frac{2\pi e V\_0 h \eta f^2}{E\_0 \beta^2 \gamma} \phi = 0. \tag{2.74}
$$

The frequency of these synchrotron oscillations in longitudinal phase space is

$$f\_s = \sqrt{\frac{|\eta| \, he \, V\_0}{2\pi \, E\_0 \beta^2 \, \mathcal{Y}}} f,\tag{2.75}$$

or writing *f*rf = *hf* we could equally express

$$f\_s = \sqrt{\frac{|\eta| \, eV0}{2\pi \, E\_0 \beta^2 \gamma h}} f\_{\text{rf}}.\tag{2.76}$$

In analogy to the transverse plane, we define a synchrotron tune, *Q*s, as the number of such oscillations per revolution of the machine. This is analogous to *Q* in transverse phase space.

$$Q\_s = \frac{f\_\text{s}}{f} = \sqrt{\frac{|\eta| \, ehV\_0 \cos \phi\_\text{s}}{2\pi E\_0 \beta^2 \nu}}. \tag{2.77}$$

Usually *Q*s is less than 10% of the revolution frequency. It drops down to zero at *γ* transition where *η* is zero and then rises again. In large proton machines it can be in the region 0 to 100 Hz and, but for the vacuum, one might hear it!

Close to *γ* tr we cannot strictly assume that *β*, *γ* , *η*, and *f* vary slowly in comparison with the synchrotron oscillation which this equation describes. Hence we should use a more exact form of the equation of motion and approximate only when it seems that this is justified:

$$\frac{d}{dt}\left[\frac{E\_0 \beta^2 \gamma \dot{\phi}}{2\pi \eta h f^2}\right] + e \, V\_0 \left(\sin\phi - \sin\phi\_8\right) = 0. \tag{2.78}$$

In a stationary bucket, when *φ*s = 0, this exact differential equation for large amplitude motion is identical to that for a rigid pendulum:

$$
\ell \frac{d^2 \theta}{dt^2} + \mathbf{g} \sin \theta = 0. \tag{2.79}
$$

There is an extra term, sin*φ*s, on the right hand side of the synchrotron equation which is not there in the pendulum case but it could be introduced too for the pendulum by using a magnetic 'bob' and biasing its equilibrium position to one side by attaching a weight on a cantilever at right angles to the rod of the pendulum.

In fact the unbiased pendulum corresponds to synchrotron motion when there is no acceleration—we say the bucket is stationary. In Fig. 2.22 we saw how particles close to the edge of the stable area of the bucket follow a fish-shaped trajectory when *φ*s = 0; before acceleration starts or when the beam is held at the same energy in collider mode (see Fig. 2.24).

In order to accelerate *φ*s must be made finite, in which case the figure changes somewhat. The stable area becomes smaller and shaped like a fish—or rather a series of fish chasing each other's tails. Small amplitude motion will still be sinusoidal but the ellipse will be centred on the stable phase *φ*s and not on *φ* = 0.

# *2.5.4 Stationary Buckets*

The size of the bucket depends on how close the stable phase, *φ*s is to the crest of the sine-wave. It shrinks to zero if *φ*s = 90◦. There is a special case if *φ*s is zero. This is often the case as a beam injected into a synchrotron before acceleration has started or in a collider where the r.f. simply holds the bunches together. The bucket is then said to be 'stationary' stretching over all phases from −π to π. Its height is the range of energies 2*E* which the r.f. wave can constrain and this turns out to be dependent on √*V* for a given *φ*s. If *V* is reduced, the more energetic particles spill out of the bucket.

Very often the particles are injected as a continuous ribbon without any longitudinal structure crosshatched in Fig. 2.24. Usually acceleration has not yet started, the magnetic field *B* is constant, and *φ*s is zero. If *V* is increased slowly, the height of the stationary bucket grows, and more and more of the energy spread in the beam, *E,* is trapped (Fig. 2.24). This is called "adiabatic trapping".

# **References**

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# **Chapter 3 Non-linear Dynamics in Accelerators**

**Werner Herr and Etienne Forest**

# **3.1 Introduction**

Non-linear effects in accelerator physics are important both during the design stage and for successful operation of accelerators. Since both of these aspects are closely related, they will be treated together in this overview. Some of the most important aspects are well described by methods established in other areas of physics and mathematics. Given the scope of this handbook, the treatment will be focused on the problems in accelerators used for particle physics experiments. Although the main emphasis will be on accelerator physics issues, some of the aspects of more general interest will be discussed. In particular to demonstrate that in recent years a framework has been built to handle the complex problems in a consistent form, technically superior and conceptually simpler than the traditional techniques. The need to understand the stability of particle beams has substantially contributed to the development of new techniques and is an important source of examples which can be verified experimentally. Unfortunately the documentation of these developments is often poor or even unpublished, in many cases only available as lectures or conference proceedings.

This article is neither rigorous nor a complete treatment of the topic, but rather an introduction to a limited set of contemporary tools and methods we consider useful in accelerator theory.

W. Herr (-)

E. Forest KEK, High Energy Research Organisation, Tsukuba, Japan

CERN (European Organization for Nuclear Research), Meyrin, Genève, Switzerland e-mail<EMAIL_ADDRESS>
# *3.1.1 Motivation*

The most reliable tools to study (i.e. description of the machine) are simulations (e.g. tracking codes).

# *3.1.2 Single Particle Dynamics*

The concepts developed here are used to describe single particle transverse dynamics in rings, i.e. circular accelerators or storage rings. This is not a restriction for the application of the presented tools and methods. In the case of linear betatron motion the theory is rather complete and the standard treatment [2] suffices to describe the dynamics. In parallel with this theory the well known concepts such as closed orbit and Twiss parameters are introduced and emerged automatically from the Courant-Snyder formalism [2]. The formalism and applications are found in many textbooks (e.g. [3–5]).

In many new accelerators or storage rings (e.g. LHC) the description of the machine with a linear formalism becomes insufficient and the linear theory must be extended to treat non-linear effects. The stability and confinement of the particles is not given a priori and should rather emerge from the analysis. Non-linear effects are a main source of performance limitations in such machines. A reliable treatment is required and the progress in recent years allows to evaluate the consequences. Very useful overview and details can be found in [6–8].

# *3.1.3 Layout of the Treatment*

Following a summary of the sources of non-linearities in circular machine, the basic methods to evaluate the consequences of non-linear behaviour are discussed. Since the traditional approach has caused misconception and the simplifications led to wrong conclusions, more recent and contemporary tools are introduced to treat these problems. An attempt is made to provide the physical picture behind these tools rather than a rigorous mathematical description and we shall show how the new concepts are a natural extension of the Courant-Snyder formalism to non-linear dynamics. An extensive treatment of these tools and many examples can be found in [7]. In the last part we summarize the most important physical phenomena caused by the non-linearities in an accelerator.

# **3.2 Variables**

For what follows one should always use canonical variables!

In Cartesian coordinates:

$$R = (X, P\_X, Y, P\_Y, Z, P\_Z, t)\tag{3.1}$$

If the energy is constant (i.e. *PZ* = const*.*), we use:

$$(X, P\_X, Y, P\_Y, Z, t)\tag{3.2}$$

This system is rather inconvenient, what we want is the description of the particle in the neighbourhood of the reference orbit/trajectory:

$$R\_d = (X, P\_X, Y, P\_Y, Z, t) \tag{3.3}$$

which are considered now the deviations from the reference and which are zero for a particle on the reference trajectory

It is very important that it is the reference not the design trajectory!

(so far it is a straight line along the *Z*-direction)

# *3.2.1 Trace Space and Phase Space*

A confusion often arises about the terms Phase Space (*x,px ,...*) or Trace Space (*x,x ,...*)

It is not laziness nor stupidity to use one or the other:

It comes back to a remark made at the beginning, i.e. that we shall use rings for our arguments. In single pass machine, e.g. linac, beam lines, spectrometers, the beam is not circulating over many turns and several hours, therefore there is no interest in stability issues. Instead for most of these applications what counts is the coordinates and angles at a given position (*x,x ,y,y* ), e.g. at the end of a beam line or a small spot one an electron microscope. When "accelerator physicists" talk about concepts such as tune, resonances, *β*-functions, equilibrium emittances etc., all these are irrelevant for single pass machine. There is no need to study iterating systems. In these cases the use of the trace space is fully adequate, in fact preferred because the quantities can be measured. In the end, the mathematical tools are very different from the ones discussed in this article.

# *3.2.2 Curved Coordinate System*

For a "curved" trajectory, in general not circular, with a local radius of curvature *ρ(s)* in the horizontal (*X*–*Z* plane), we have to transform to a new coordinate system *(x, y, s)* (co-moving frame) with:

$$\begin{cases} X = (\mathbf{x} + \rho) \cos \left( \frac{s}{\rho} \right) - \rho \\ Y = \mathbf{y} \\ Z = (\mathbf{x} + \rho) \sin \left( \frac{s}{\rho} \right) \end{cases} \tag{3.4}$$

The new canonical momenta become:

$$\begin{aligned} p\_{\chi} &= \,^P P\_{\chi} \cos \left( \frac{s}{\rho} \right) \,^+ + P\_Z \sin \left( \frac{s}{\rho} \right) \\ p\_{\chi} &= \,^P P\_Y \\ p\_s &= \,^P P\_Z \left( 1 + \frac{\chi}{\rho} \right) \cos \left( \frac{s}{\rho} \right) \,^- - P\_X \left( 1 + \frac{\chi}{\rho} \right) \sin \left( \frac{s}{\rho} \right) \end{aligned} \tag{3.5}$$

# **3.3 Sources of Non-linearities**

Any object creating non-linear electromagnetic fields on the trajectory of the beam can strongly influence the beam dynamics. They can be generated by the environment or by the beam itself.

# *3.3.1 Non-linear Machine Elements*

Non-linear elements can be introduced into the machine on purpose or can be the result of field imperfections. Both types can have adverse effects on the beam stability and must be taken into account.

### **<IP_ADDRESS> Unwanted Non-linear Machine Elements**

The largest fraction of machine elements are either dipole or quadrupole magnets. In the ideal case, these types of magnets have pure dipolar or quadrupolar fields and behave approximately as linear machine elements. Any systematic or random deviation from this linear field introduces non-linear fields into the machine lattice. These effects can dominate the aperture required and limit the stable region of the beam. The definition of tolerances on these imperfections is an important part of any accelerator design.

Normally magnets are long enough that a 2-dimensional field representation is sufficient. The components of the magnetic field can be derived from the potential and in cylindrical coordinates (*r,* , *s* = 0) can be written as:

$$B\_{\rm r}(r,\Theta) = \sum\_{n=1}^{\infty} (B\_{\rm n} \sin(n\Theta) + A\_{\rm n} \cos(n\Theta)) \left(\frac{r}{R\_{ref}}\right)^{n-1},\tag{3.6}$$

$$B\_{\Theta}(r,\Theta) = \sum\_{n=1}^{\infty} (B\_n \cos(n\Theta) - A\_n \sin(n\Theta)) \left(\frac{r}{R\_{ref}}\right)^{n-1},\tag{3.7}$$

where *Rref* is a reference radius and *Bn* and *An* are constants. Written in Cartesian coordinates we have:

$$B(z) = \sum\_{n=1}^{\infty} (B\_n + iA\_n) \left(\frac{r}{R\_{ref}}\right)^{n-1} \tag{3.8}$$

where *z* = *x* + *iy* = *rei*. The terms *n* correspond to 2*n*-pole magnets and the *Bn* and *An* are the normal and skew multipole coefficients. The beam dynamics set limits on the allowed multipole components of the installed magnets.

### **<IP_ADDRESS> Wanted Non-linear Machine Elements**

In most accelerators the momentum dependent focusing of the lattice (chromaticity) needs to be corrected with sextupoles [3, 4]. Sextupoles introduce non-linear fields into the lattice that are larger than the intrinsic non-linearities of the so-called linear elements (dipoles and quadrupoles). In a strictly periodic machine the correction can be done close to the origin and the required sextupole strengths can be kept small. For colliding beam accelerators usually special insertions are foreseen to host the experiments where the dispersion is kept small and the *β*-function is reduced to a minimum. The required sextupole correction is strong and can lead to a reduction of the dynamic aperture, i.e. the region of stability of the beam. In most accelerators the sextupoles are the dominant source of non-linearity. To minimize this effect is an important issue in any design of an accelerator.

Another source of non-linearities can be octupoles used to generate amplitude dependent detuning to provide Landau damping in case of instabilities.

# *3.3.2 Beam–Beam Effects and Space Charge*

A strong source of non-linearities are the fields generated by the beam itself. They can cause significant perturbations on the same beam (space charge effects) or on the opposing beam (beam-beam effects) in the case of a colliding beam facility.

As an example, for the simplest case of round beams with the line density *n* and the beam size *σ* the field components can be written as:

$$E\_r = -\frac{ne}{4\pi\epsilon\_0} \cdot \frac{\partial}{\partial r} \int\_0^\infty \frac{\exp(-\frac{r^2}{(2\sigma^2 + q)})}{(2\sigma^2 + q)} \mathrm{d}q,\tag{3.9}$$

and

$$B\_{\Phi} = -\frac{ne\beta c\mu\_0}{4\pi} \cdot \frac{\partial}{\partial r} \int\_0^\infty \frac{\exp(-\frac{r^2}{(2\sigma^2 + q)})}{(2\sigma^2 + q)} \mathrm{d}q. \tag{3.10}$$

In colliding beams with high density and small beams these fields are the dominating source of non-linearities. The full treatment of beam-beam effects is complicated due to mutual interactions between the two beams and a self-consistent treatment is required. in the presence of all other magnets in the ring.

# **3.4 Map Based Techniques**

In the standard approach to single particle dynamics in rings, the equations of motion are introduced together with an ansatz to solve these equations. In the case of linear motion, this ansatz is due to Courant-Snyder [2]. However, this treatment must assume that the motion of a particle in the ring is stable and confined. For a non-linear system this is a priori not known and the attempt to find a complete description of the particle motion must fail.

The starting point for the treatment of the linear dynamics in synchrotrons is based on solving a linear differential equation of the Hill type.

$$\frac{d^2\mathbf{x}(\mathbf{s})}{ds^2} + \underbrace{\left(a\_0 + 2\sum\_{n=1}^{\infty} a\_n \cdot \cos(2ns)\right)\mathbf{x}(\mathbf{s})}\_{K(\mathbf{s})} \mathbf{x}(\mathbf{s}) = \mathbf{0} \text{ .} $$

Each element at position *s* acts as a source of forces, i.e. we must write for the forces *K* → *K(s)* which is assumed to be a periodic function, i.e. *K(s* + *C)* = *K(s)r ing*

The solution of this Boundary Value Problem must be periodic too!

It is therefore not applicable in the general case (e.g. Linacs, Beamlines, FFAG, Recirculators, . . . ), much better to treat it as an Initial Value Problem.

In a more useful approach we do not attempt to solve such an overall equation but rather consider the fundamental objects of an accelerators, i.e. the machine elements themselves.  All elements have a well defined action on a particle which can be described independent of other elements or concepts such as closed orbit or *β*-functions. Mathematically, they provide a "map" from one face of a building block to the other, i.e. a description of how the particles move inside and between elements. In this context, a map can be anything from linear matrices to high order integration routines.

A map based technique is also the basis for the treatment of particle dynamics as an Initial value Problem (IVP).

It follow immediately that for a linear, 1st order equation of the type

$$\frac{d\mathbf{x}(\mathbf{s})}{ds} = \begin{array}{c} K(\mathbf{s}) \ x(\mathbf{s}) \\ \end{array} \quad \text{(and initial values at} \quad \mathbf{s}\_0).$$

the solution can always be written as:

$$\begin{array}{ll} \mathbf{x}(\mathbf{s}) &= a \cdot \mathbf{x}(\mathbf{s\_0}) + b \cdot \mathbf{x'}(\mathbf{s\_0})\\ \mathbf{x'}(\mathbf{s}) &= c \cdot \mathbf{x}(\mathbf{s\_0}) + d \cdot \mathbf{x'}(\mathbf{s\_0}) \end{array} \Longrightarrow \begin{pmatrix} \mathbf{x} \\ \mathbf{x'} \end{pmatrix}\_s = \overbrace{\begin{pmatrix} a & b \\ c & d \end{pmatrix}} \begin{pmatrix} \mathbf{x} \\ \mathbf{x'} \end{pmatrix}\_{s\_0} $$

where the function *K(s)* does not have to be periodic. Furthermore, the determinant of the matrix *A* is always 1. Therefore it is an advantage to use maps (matrices) for a linear systems from the start, without trying to solve a differential equation.

The collection of all machine elements make up the ring pr beam line and it is the combination of the associated maps which is necessary for the description and analysis of the physical phenomena in the accelerator ring or beam line.

For a circular machine the most interesting map is the one which describes the motion once around the machine, the so-called One-Turn-Map. It contains all necessary information on stability, existence of closed orbit, and optical parameters. The reader is assumed to be familiar with this concept in the case of linear beam dynamics (Chap. 2) where all maps are matrices and the Courant-Snyder analysis of the corresponding one-turn-map produces the desired information such as e.g. closed orbit or Twiss parameters.

It should therefore be the goal to generalize this concept to non-linear dynamics. The computation of a reliable one-turn-map and the analysis of its properties will provide all relevant information.

Given that the non-linear maps can be rather complex objects, the analysis of the one-turn-map should be separated from the calculation of the map itself.

# **3.5 Linear Normal Forms**

# *3.5.1 Sequence of Maps*

Starting from a position *s*0 and combining all matrices to get the matrix to position *s*0 + *L* (shown for 1D only):

$$
\begin{pmatrix} \boldsymbol{\chi} \\ \boldsymbol{\chi'} \end{pmatrix}\_{s\_0+L} = \underbrace{\boldsymbol{M}\_N \circ \ \boldsymbol{M}\_{N-1} \circ \dots \circ \ \boldsymbol{M}\_1}\_{M(s\_0,L)} \circ \begin{pmatrix} \boldsymbol{\chi} \\ \boldsymbol{\chi'} \end{pmatrix}\_{s\_0} \tag{3.11}
$$

For a ring with circumference *C* one obtains the One-Turn-Matrix (OTM) at *s*0

$$
\begin{pmatrix} \mathbf{x} \\ \mathbf{x'} \end{pmatrix}\_{s\_0 + \mathcal{C}} = \underbrace{\begin{pmatrix} m\_{11} \ m\_{12} \\ m\_{21} \ m\_{22} \end{pmatrix}}\_{\mathbf{MortM}} \circ \begin{pmatrix} \mathbf{x} \\ \mathbf{x'} \end{pmatrix}\_{s\_0} \tag{3.12}
$$

Without proof, the scalar product:

$$\left(\begin{array}{c} \chi\\\chi'\end{array}\right)\_{s0} \cdot M\_{OTM} \left(\begin{array}{c} \chi\\\chi'\end{array}\right)\_{s0} = \begin{array}{c} \mathsf{const.} \ = \ \mathsf{J} \end{array} \tag{3.13}$$

is a constant of the motion: invariant of the One Turn Map.

With this approach we have a strong argument that the construction of the One Turn Map is based on the properties of each element in the machine. It is entirely independent of the purpose of the machine and their global properties. It is not restricted to rings or in general to circular machine.

Once the One Turn Map is constructed, it can be analysed, but this analysis does not depend on how it was constructed.

As a paradigm: the construction of a map (being for a circular machine or not) and its analysis are conceptual and computational separated undertakings.

# *3.5.2 Analysis of the One Turn Map*

The key for the analysis is that matrices can be transformed into **Normal Forms.** Starting with the One-Turn-Matrix, and try to find a (invertible) transformation *A* such that:

*AMA*−1 = *R (*or : *A*−1*RA* = *M)*

	- A "Normal Form", (or at least a very simplified form of the matrix)
	- For example (most important case): *R* becomes a pure rotation
	- All coordinates are transformed by *A*
	- This transformation *A* "analyses" the complexity of the motion, it contains the structure of the phase space

$$M = A \diamond R \diamond A^{-1} \quad \text{or} \; : \quad R = A^{-1} \diamond M \diamond A^{-1}$$

The motion on an ellipse becomes a motion on a circle (i.e. a rotation): *R* is the simple part of the map and its shape is dumped into the matrix *A*. *R* can be obtained by the evaluation of the Eigenvectors and Eigenvalues.

One finds for the two components of the original map:

$$A = \begin{pmatrix} \sqrt{\beta(s)} & 0\\ -\frac{\alpha(s)}{\sqrt{\beta(s)}} \frac{1}{\sqrt{\beta(s)}} \end{pmatrix} \quad \text{and} \quad R = \begin{pmatrix} \cos(\Delta\mu) & \sin(\Delta\mu) \\\\ -\sin(\Delta\mu)\cos(\Delta\mu) \end{pmatrix} \tag{3.14}$$

Please note that the normal form analysis gives the eigenvectors (3.14) without any physical picture related to their interpretation. The formulation using *α* and *β* is due to Courant and Snyder. Amongst other advantages it can be used to "normalise" the position *x*: the normalised position *xn* is the "non-normalized" divided by √*β*. The variation of the normalised position *xn* is then smaller than in the non-normalized case. This is also better suited for analytical calculation, e.g. involving perturbation theory.

The Normal Form transformation together with this choice gives the required information:

# *3.5.3 Action-Angle Variables*

More appropriate for studies of beam dynamics is the use of Action-Angle variables.

Once the particles "travel" on a circle, the motion is better described by the canonical variables action *Jx* and angle *x*:

with the definitions and the choice is (3.14):

$$\begin{aligned} \chi\_{\chi} &= \sqrt{2J\_{\chi}\beta\_{\chi}} \quad \cos(\Psi\_{\chi})\\ p\_{\chi} &= -\sqrt{\frac{2J\_{\chi}}{\beta\_{\chi}}} \left( \sin(\Psi\_{\chi}) + a\_{\chi}\cos(\Psi\_{\chi}) \right) \\ J\_{\chi} &= \frac{1}{2} (\wp\_{\chi}\mathbf{x}^{2} + 2a\_{\chi}\wp\_{\chi} + \beta\_{\chi}p\_{\chi}^{2}) \end{aligned} \tag{3.15}$$

# *3.5.4 Beam Emittance*

A sad and dismal story in accelerator physics is the definition of the emittance. Most foolish in this context is to relate emittance to single particles. This is true in particular when we have a beam line which is not periodic. In that case the Courant-Snyder parameters can be determined from the beam. These parameters are related to the moments of the beam, e.g. the beam size is directly related to the second order moment *< x*2 *>*. Using the expression above for the action and angle, we can write for this expression:

$$<\chi^2>\quad =\quad<2J\_\chi\beta\_\chi\cdot\cos^2(\Psi\_\chi)>\quad =\quad2\beta\_\chi\. \tag{3.16}$$

The average of cos2 can immediately be evaluated as 0.5 and defining the emittance as:

$$
\epsilon\_{\chi} \; = \; \; \; < \; J\_{\chi} \; > , \tag{3.17}
$$

we write

$$<\mathbf{x}^2 > \quad = \; \beta\_{\mathbf{x}} \cdot \epsilon\_{\mathbf{x}}.\tag{3.18}$$

Using a similar procedure (details and derivation in e.g. [3], and to a much lesser extent in [1]) one can determine the moments

$$\quad =\quad \chi\_{\chi} \cdot \epsilon\_{\chi},\tag{3.19}$$

and

$$ \quad = \quad -a\_{\chi} \cdot \epsilon\_{\chi}.\tag{3.20}$$

Using these expressions, the emittance becomes readily

$$\epsilon\_{\chi} \quad = \sqrt{<\chi^2>\quad - \quad ^2} \tag{3.21}$$

Therefore, once the emittance is measured, the Courant-Snyder parameters are determined by Eqs. (3.18), (3.19), and (3.20).

Since other definitions often refer to the treatment by Courant and Snyder, here a quote from Courant himself in [9]:

### *Interlude 1*

The invariant *J* is simply related to the area enclosed by the ellipse:

$$\text{Area } \texttt{enclosed} \; = \; \; 2\pi J. \tag{3.22}$$

In accelerator and storage ring terminology there is a quantity called the *emittance* which is closely related to this invariant. The emittance, however, is a property of a distribution of particles, not a single particle. Consider a Gaussian distribution in amplitudes. Then the (rms) emittance, , is given by:

$$\left(\mathbf{x}\_{rms}\right)^{2} = \ \beta\_{x}(\mathbf{s}) \cdot \boldsymbol{\epsilon}\_{x}.\tag{3.23}$$

In terms of the action variable, *J* , this can be rewritten

$$
\epsilon\_{\chi} := \quad \prec J \, > \,. \tag{3.24}
$$

where the bracket indicates an average over the distribution in *J* .

Other definitions based on handwaving arguments or those approximately valid only in special cases, should be discarded, in particular those relying on presumed distributions, e.g. Gaussian.

# **3.6 Techniques and Tools to Evaluate and Correct Non-linear Effects**

The key to a more modern approach shown in this section is to avoid the prejudices about the stability and other properties of the ring. Instead, we must describe the machine in terms of the objects it consists of with all their properties, including the non-linear elements. The analysis will reveal the properties of the particles such as e.g. stability. In the simplest case, the ring is made of individual machine elements such as magnets which have an existence on their own, i.e. the interaction of a particle with a given element is independent of the motion in the rest of the machine. Also for the study of non-linear effects, the description of elements should be independent of concepts such as tune, chromaticity and closed orbit. To successfully study single particle dynamics, one must be able to describe the action of the machine element on the particle as well as the machine element.

# *3.6.1 Particle Tracking*

The ring being a collection of maps, a particle tracking code, i.e. an integrator of the equation of motion, is the most reliable map for the analysis of the machine. Of course, this requires an appropriate description of the non-linear maps in the code. It is not the purpose of this article to describe the details of tracking codes and the underlying philosophy, such details can be found in the literature (see e.g. [6]). Here we review and demonstrate the basic principles and analysis techniques.

### **<IP_ADDRESS> Symplecticity**

If we define a map through *z*2 = *M*12*(z*1*)* as a propagator from a location "1" to a location "2" in the ring, we have to consider that not all possible maps are allowed. The required property of the map is called "symplecticity" and in the simplest case where *M*12 is a matrix, the symplecticity condition can be written as:

$$M \Rightarrow M^T \cdot S \cdot M = S \quad \text{where} \quad S = \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \tag{3.25}$$