Particle Accelerator Physics pp 3-41 | Cite as

# Introduction to Accelerator Physics

## Abstract

The development of charged particle accelerators and it’s underlying principles has its basis on the theoretical and experimental progress in fundamental physical phenomena. While active particle accelerator experimentation started seriously only in the twentieth century, it depended on the basic physical understanding of electromagnetic phenomena as investigated both theoretically and experimentally mainly during the nineteenth and beginning twentieth century. In this introduction we will recall briefly the history leading to particle accelerator development, applications and introduce basic definitions and formulas governing particle beam dynamics.

## Keywords

Synchrotron Radiation Lorentz Force Linear Accelerator Storage Ring Lorentz TransformationThe development of charged particle accelerators and it’s underlying principles has its basis on the theoretical and experimental progress in fundamental physical phenomena. While active particle accelerator experimentation started seriously only in the twentieth century, it depended on the basic physical understanding of electromagnetic phenomena as investigated both theoretically and experimentally mainly during the nineteenth and beginning twentieth century. In this introduction we will recall briefly the history leading to particle accelerator development, applications and introduce basic definitions and formulas governing particle beam dynamics.

## 1.1 Short Historical Overview

The history and development of particle accelerators is intimately connected to the discoveries and understanding of electrical phenomena and the realization that the electrical charge comes in lumps carried as a specific property by individual particles. It is reported that the Greek philosopher and mathematician Thales of Milet, who was born in 625 BC first observed electrostatic forces on amber. The Greek word for amber is electron or \(\eta \lambda \epsilon \kappa \tau \rho o\nu\)and has become the origin for all designations of electrical phenomena and related sciences. For more than 2000 years this observation was hardly more than a curiosity. In the eighteenth century, however, electrostatic phenomena became quite popular in scientific circles and since have been developed into a technology which by now completely embraces and dominates modern civilization as we know it.

It took another 100 years before the carriers of electric charges could be isolated. Many systematic experiments were conducted and theories developed to formulate the observed electrical phenomena mathematically. It was Coulomb, who in 1785 first succeeded to quantify the forces between electrical charges which we now call Coulomb forces. As more powerful sources for electrical charges became available, glow discharge phenomena were observed and initiated an intensive effort on experimental observations during most of the second half of the nineteenth century. It was the observations of these electrical glow discharge phenomena that led the scientific community to the discovery of elementary particles and electromagnetic radiation which are both basic ingredients for particle acceleration.

- 1815
The physician and chemist W. Proust postulates, initially anonymous, that all atoms are composed of hydrogen atoms and that therefore all atomic weights come in multiples of the weight of a hydrogen atom.

- 1839
M. Faraday [1] publishes his experimental investigations of electricity and described various phenomena of glow discharge.

- 1858
J. Plücker [2] reports on the observation of cathode rays and their deflection by magnetic fields. He found the light to become deflected in the same spiraling direction as Ampere’s current flows in the electromagnet and therefore postulated that the electric light, as he calls it, under the circumstances of the experiment must be magnetic.

- 1867
L. Lorenz working in parallel with J.C. Maxwell on the theory of electromagnetic fields formulates the concept of retarded potentials although not yet for moving point charges.

- 1869
J.W. Hittorf [3], a student of Plücker, started his thesis paper with the statement (translated from german):

*“The undisputed darkest part of recent theory of electricity is the process by which in gaseous volumes the propagation of electrical current is effected”*. Obviously observations with glow discharge tubes displaying an abundance of beautiful colors and complicated reactions to magnetic fields kept a number of researchers fascinated. Hittorf conducted systematic experiments on the deflection of the light in glow discharges by magnetic fields and corrected some erroneous interpretations by Plücker.- 1871
C.F. Varley postulates that cathode rays are particle rays.

- 1874
H. von Helmholtz postulates atomistic structure of electricity.

- 1883
J.C. Maxwell publishes his

*Treatise on Electricity and Magnetism*.- 1883
T.A. Edison discovers thermionic emission.

- 1886
E. Goldstein [4] observed positively charged rays which he was able to isolate from a glow discharge tube through channels in the cathode. He therefore calls these rays Kanalstrahlen.

- 1887
H. Hertz discoveries transmission of electromagnetic waves and photoelectric effect.

- 1891
G.J. Stoney introduces the name electron.

- 1895
H.A. Lorentz formulates electron theory, the Lorentz force equation and Lorentz contraction.

- 1894
P. Lenard builts a discharge tube that allows cathode rays to exit to atmospheric air.

- 1895
W. Röntgen discovers x-rays.

- 1895
E. Wiedemann [5] reports on a new kind of radiation studying electrical sparks.

- 1897
J.J. Thomson measures the e/m-ratio for kanal and cathode rays with electromagnetic spectrometer and found the e/m ratio for cathode rays to be larger by a factor of 1,700 compared to the e/m ratio for kanal rays. He concluded that cathode rays consist of free electricity giving evidence to free electrons.

- 1897
J. Larmor formulates concept of Larmor precession.

- 1898
A. Liènard calculates the electric and magnetic field in the vicinity of a moving point charge and evaluated the energy loss due to electromagnetic radiation from a charged particle travelling on a circular orbit.

- 1900
E. Wiechert derives expression for retarded potentials of moving point charges.

- 1901
W. Kaufmann, first alone, and in 1907 together with A.H. Bucherer measure increase of electron mass with energy. First experiment in support of theory of special relativity.

- 1905
A. Einstein publishes theory of special relativity.

- 1906
J.J. Thomson [6] explains the emission of this radiation as being caused by acceleration occurring during the collision of charged particles with other atoms and calculated the energy emitted per unit time to be \((2e^{2}f\,^{2})/(3V )\), where

*e*is the charge of the emitting particle,*f*the acceleration and*V*the velocity of light.- 1907
G.A. Schott [7, 8] formulated the first theory of synchrotron radiation in an attempt to explain atomic spectra.

- 1909
R.A. Millikan starts measuring electric charge of electron.

- 1913
First experiment by J. Franck and G. Hertz to excite atoms by accelerated electrons.

- 1914
E. Marsden produces first proton beam irradiating paraffin with alpha particles.

- 1920
H. Greinacher [9] builts first cascade generator.

- 1922
R. Wideroe as a graduate student sketches ray transformer (betatron).

- 1924
G. Ising [10] invents as a student the electron linac with drift tubes and spark gap excitation.

- 1928
R. Wideroe [11] reports first operation of linear accelerator with potassium and sodium ions. Discusses operation of betatron and failure to get beam for lack of focusing.

- 1928
P.A.M. Dirac predicts existence of positrons.

- 1931
R.J. Van de Graaff [12] builts first high voltage generator.

- 1932
Lawrence and Livingston [13] accelerate first proton beam from 1.2 MeV cyclotron\(<IndexTerm> <Primary>Cyclotron</Primary> </IndexTerm>\) employing weak focusing.

- 1932
J.D. Cockcroft and E.T.S. Walton [14] use technically improved cascade generator to accelerate protons and initiate first artificial atomic reaction: \(\mathrm{Li} + p \rightarrow \, 2\,\mathrm{He}.<IndexTerm> <Primary>Cascadegenerator</Primary> </IndexTerm>\)

- 1932
in the same year, C.D. Andersen discovers positrons, neutrons were discovered by J. Chadwick, and H.C. Urey discoveries deuterons.

- 1939
W.W. Hansen, R. Varian and his brother S. Varian invent klystron microwave tube at Stanford.

- 1941
D.W. Kerst and R. Serber [15] complete first functioning betatron.

- 1941
B. Touschek and R. Wideroe formulate storage ring principle.

- 1944
D. Ivanenko and I.Ya. Pomeranchuk [16] and J. Schwinger [17] point out independently an energy limit in circular electron accelerators due to synchrotron radiation losses.

- 1945
V.I. Veksler [18] and E.M. McMillan [19] independently discover the principle of phase focusing.

- 1945
J.P. Blewett [20] experimentally discovers synchrotron radiation by measuring the energy loss of electrons.

- 1947
L.W. Alvarez [21] designs first proton linear accelerator at Berkeley.

- 1948
E.L. Ginzton et al. [22] accelerate electrons to 6 MeV with Mark I at Stanford.

- 1949
McMillan et al. commissioned 320 MeV electron synchrotron.

- 1950
N. Christofilos [23] formulates concept of strong focusing.

- 1952
M.S. Livingston et al. [24] describe design for 2.2 GeV

*Cosmotron*in Brookhaven.- 1951
H. Motz [25] builds first wiggler magnet to produce quasi monochromatic synchrotron radiation.

- 1952
E. Courant et al. [26] publish first paper on strong focusing.

- 1954
R.R. Wilson et al. operate first AG electron synchrotron in Cornell at 1.1 GeV.

- 1954
Lofgren et al. accelerate protons to 5.7 GeV in

*Bevatron.*- 1955
M. Chodorow et al. [27] complete 600 MeV

*MARK III*electron linac.- 1955
M. Sands [28] define limits of phase focusing due to quantum excitation.

- 1959
E. Courant and Snyder [29] publish their paper on the

*Theory of the Alternating-Gradient Synchrotron*.

Research and development in accelerator physics blossomed significantly during the 1950s supported by the development of high power radio frequency sources and the increased availability of government funding for accelerator projects. Parallel with the progress in accelerator technology, we also observe advances in theoretical understanding, documented in an increasing number of publications. It is beyond the scope of this text to only try to give proper credit to all major advances in the past 60 years and refer the interested reader to more detailed references.

## 1.2 Particle Accelerator Systems

Particle accelerators come in many forms applying a variety of technical principles. All are based on the interaction of the electric charge with static and dynamic electromagnetic fields and it is the technical realization of these interactions that leads to the different types of particle accelerators. Electromagnetic fields are used over most of the available frequency range from static electric fieldsto ac magnetic fields in betatrons oscillating at 50 or 60 Hz, to radio frequency fields in the MHz to GHz range and ideas are being explored to use laser beams to generate high field particle acceleration.

In this text, we will not discuss the different technical realization of particle acceleration but rather concentrate on basic principles which are designed to help the reader to develop technical solutions for specific applications meeting basic beam stability requirements. For particular technical solutions we refer to the literature. Further down we will discuss briefly basic accelerator types and their theoretical back ground. Furthermore, to discuss basic principles of particle acceleration and beam dynamics it is desirable to stay in contact with technical reality and reference practical and working solutions. We will therefore repeatedly refer to certain types of accelerators and apply theoretical beam dynamics solutions to exhibit the salient features and importance of the theoretical ideas under discussion. In these references we use mostly such types of accelerators which are commonly used and are extensively publicized.

### 1.2.1 Main Components of Accelerator Facilities

In the following paragraphs we describe components of particle accelerators in a rather cursory way to introduce the terminology and overall features. Particle accelerators consist of two basic units, the particle source or injector and the main accelerator. The particle source comprises all components to generate a beam of desired particles.

Generally glow discharge columns are used to produce proton or ion beams*,* which then are first accelerated in electrostatic accelerators like a *Van de Graaff* or *Cockcroft-Walton* accelerator and then in an *Alvarez-type* linear accelerator. To increase the energy of heavy ion beams the initially singly charged ions are, after some acceleration, guided through a thin metal foil to strip more electrons off the ions. More than one stripping stage may be used at different energies to reach the maximum ionization for most efficient acceleration.

Much more elaborate measures must be used to produce antiprotons. Generally a high energy proton beam is aimed at a heavy metal target*,* where, through hadronic interactions with the target material, among other particles antiprotons are generated. Emerging from the target, these antiprotons are collected by strong focusing devices and further accelerated.

Electrons are commonly generated from a heated cathode, also called a thermionic gun, which is covered on the surface by specific alkali oxides or any other substance with a low work function to emit electrons at technically practical temperatures. Another method to create a large number of electrons within a short pulse uses a strong laser pulse directed at the surface of a photo cathode. Systems where the cathode is inserted directly into an accelerating rf field are called rf guns. Positrons are created the same way as antiprotons by aiming high energy electrons on a heavy metal target where, through an electromagnetic shower and pair production, positrons are generated. These positrons are again collected by strong magnetic fields and further accelerated.

Whatever the method of generating particles may be, in general they do not have the time structure desired for further acceleration or special application. Efficient acceleration by rf fields occurs only during a very short period per oscillation cycle and most particles would be lost without proper preparation. For high beam densities it is desirable to compress the continuous stream of particles from a thermionic gun or a glow discharge column into a shorter pulse with the help of a chopper device and/or a prebuncher. The chopper may be a mechanical device or a deflecting magnetic or rf field moving the continuous beam across the opening of a slit. At the exit of the chopper we observe a series of beam pulses, called bunches, to be further processed by the prebuncher. Here early particles within a bunch are decelerated and late particles accelerated. After a well defined drift space, the bunch length becomes reduced due to the energy dependence of the particle velocity. Obviously this compression works only as long as the particles are not relativistic while the particle velocity can be modulated by acceleration or deceleration.

No such compression is required for antiparticles, since they are produced by high energetic particles having the appropriate time structure. Antiparticle beams emerging from a target have, however, a large beam size and beam divergence. To make them suitable for further acceleration they are generally stored for some time in a cooling or damping ring. Such cooling rings are circular “accelerators” where particles are not accelerated but spend just some time circulating. Positrons circulating in such storage rings quickly lose their transverse momenta and large beam divergence through the emission of synchrotron radiation. In the case of antiprotons, external fields are applied to damp the transverse beam size or they circulate against a strong counterrotating electron beam loosing transverse momentum through scattering.

Antiparticles are not always generated in large quantities. On the other hand, the accelerator ahead of the conversion target can often be pulsed at a much higher rate than the main accelerator can accept injection. In such cases, the antiparticles are collected from the rapid cycling injector in an accumulator ring and then transferred to the main accelerator when required.

Particle beams prepared in such a manner may now be further accelerated in linear or circular accelerators. A linear accelerator consists of a linear sequence of many accelerating units where accelerating fields are generated and timed such that particles absorb and accumulate energy from each acceleration unit. Most commonly used linear accelerators consist of a series of cavities excited by radio frequency sources to high accelerating fields. In the induction accelerator , each accelerating unit consists of a transformer which generates from an external electrical pulse a field on the transformer secondary which is formed such as to allow the particle beam to be accelerated. Such induction accelerators can be optimized to accelerate very high beam currents to medium beam energies.

For very high beam energies linear accelerators become very long and costly. Such practical problems can be avoided in circular accelerators where the beam is held on a circular path by magnetic fields in bending magnets and passing repeatedly every turn through accelerating sections, similar to those in a linear accelerator. This way, the particles gain energy from the accelerating cavities at each turn and reach higher energies while the fields in the bending magnets are raised.

The basic principles to accelerate particles of different kind are similar and we do not need to distinguish between protons, ions, and electrons. Technically, individual accelerator components differ more or less to adjust to the particular beam parameters which have mostly to do with the particle velocities. For highly relativistic particles the differences in beam dynamics vanish. Protons and ions are more likely to be nonrelativistic and therefore vary the velocity as the kinetic energy is increased, thus generating problems of synchronism with the oscillating accelerating fields which must be solved by technical means.

After acceleration in a linear or circular accelerator the beam can be directed onto a target, mostly a target of liquid hydrogen, to study high energy interactions with the target protons. Such fixed target experimentation dominated nuclear and high energy particle physics from the first applications of artificially accelerated particle beams far into the 1970s and is still a valuable means of basic research. Obviously, it is also the method in conjunction with a heavy metal target to produce secondary particles like antiparticles for use in colliding beam facilities and mesons for basic research.

To increase the center-of-mass energy for basic research, particle beams are aimed not at fixed targets but to collide head on with another beam. This is one main goal for the construction of colliding beam facilities or storage rings. In such a ring, particle and antiparticle beams are injected in opposing directions and made to collide in specifically designed interaction regions. Because the interactions between counter orbiting particles is very rare, storage rings are designed to allow the beams to circulate for many turns with beam life times of several hours to give the particles ample opportunity to collide with other counter rotating particles. Of course, beams can counter rotate in the same magnetic fields only if one beam is made of the antiparticles of the other beam while two intersecting storage rings must be employed to allow the collision of unequal particles.

The circulating beam in an electron storage ring emits synchrotron radiation due to the transverse acceleration during deflection in the bending magnets. This radiation is highly collimated in the forward direction, of high brightness and therefore of great interest for basic and applied research, technology, and medicine.

Basically the design of a storage ring is the same as that for a synchrotron allowing some adjustment in the technical realization to optimize the desired features of acceleration and long beam lifetime, respectively. Beam intensities are generally very different in a synchrotron from that in a storage ring. In a synchrotron, the particle intensity is determined by the injector and this intensity is much smaller than desired in a storage ring. The injection system into a storage ring is therefore designed such that many beam pulses from a linear accelerator, an accumulator ring or a synchrotron can be accumulated. A synchrotron serving to accelerate beam from a low energy preinjector to the injection energy of the main facility, which may be a larger synchrotron or a storage ring, is also called a booster synchrotron or short a booster.

Although a storage ring is not used for particle acceleration it often occurs that a storage ring is constructed long after and for a higher beam energy than the injector system. In this case, the beam is accumulated at the maximum available injection energy. After accumulation the beam energy is slowly raised in the storage ring to the design energy by merely increasing the strength of the bending and focusing magnets.

Electron positron storage rings have played a great role in basic high-energy research. For still higher collision energies, however, the energy loss due to synchrotron radiation has become a practical and economic limitation. To avoid this limit, beams from two opposing linear accelerators are brought into head on collision at energies much higher than is possible to produce in circular accelerators. To match the research capabilities in colliding beam storage rings, such linear colliders must employ sophisticated beam dynamics controls, focusing arrangements and technologies similar to X-ray laser systems now operating.

### 1.2.2 Applications of Particle Accelerators

Particle accelerators are mainly known for their application as research tools in nuclear and high energy particle physics requiring the biggest and most energetic facilities. Smaller accelerators, however, have found broad applications in a wide variety of basic research and technology, as well as medicine. In this text, we will not discuss the details of all these applications but try to concentrate only on the basic principles of particle accelerators and the theoretical treatment of particle beam dynamics and instabilities. An arbitrary and incomplete listing of applications for charged particle beams and their accelerators is given for reference to the interested reader:

| | |

Electron/proton accelerators | Inertial fusion | |

Ion accelerators/colliders | Reactor fuel breeding | |

Continuous beam facility | | |

| Radiography by x-rays | |

Fixed target accelerator | Ion implantation | |

Colliding beam storage rings | Isotope production/separation | |

Linear colliders | Materials testing/modification | |

Food sterilization | | |

X-ray lithography | Free electron lasers, X-FEL | |

| Microprobe | |

Basic atomic and molecular physics | Holography | |

Condensed matter physics | | |

Earth sciences | Radiotherapy | |

Material sciences | Health physics | |

Chemistry | Microsurgery with tunable FEL | |

Molecular and cell biology | Sterilization | |

Surface/interface physics |

This list is by no means exhaustive and additions must be made at an impressive pace as the quality and characteristics of particle beams become more and more sophisticated, predictable and controllable. Improvements in any parameter of particle beams create opportunities for new experiments and applications which were not possible before. More detailed information on specific uses of particle accelerators as well as an extensive catalogue of references has been compiled by Scharf [30].

## 1.3 Definitions and Formulas

Particle beam dynamics can be formulated in a variety units and it is therefore prudent to define the units used in this text to avoid confusion. In addition, we recall fundamental relations of electromagnetic fields and forces as well as some laws of special relativity to the extend that will be required in the course of discussions.

### 1.3.1 Units and Dimensions

A set of special physical units, selected primarily for convenience, are most commonly used to quantify physical constants in accelerator physics. The use of many such units is often determined more by historical developments than based on the choice of a consistent set of quantities useful for accelerator physics.

*e*would gain while being accelerated between two conducting plates at a potential difference of 1 V. Therefore, 1 eV is equivalent to \(1.60217733 \times 10^{-19}\) J. Specifically, we will often use derivatives of the basic units to express actual particle energies in a convenient form:

In an effort to simplify the technical jargon used in accelerator physics the term energy is used for all three quantities although mathematically the momentum is then multiplied by the velocity of light for dimensional consistency. There are still numerical differences which must be considered for all but very highly relativistic particles. Where we need to mention the pure particle momentum and quote a numerical value, we generally use the total energy divided by the velocity of light with the unit eV/c. With this definition a particle of energy *cp* = 1 eV would have a momentum of *p* = 1 eV/c.

An additional complication arises in the case of composite particles like heavy ions, consisting of protons and neutrons. In this case, the particle energy is not quoted for the whole ion but in terms of the energy per nucleon.

*Q*passing a point during the time

*t*. Depending on the time duration one gets an instantaneous current or some average current. Therefore a quotation of the particle current requires also the definition of the time structure of the beam. In circular accelerators, for example, the average beam current

*I*relates directly to the beam intensity or the number of circulating particles

*N*. If

*β c*is the velocity of the particle and

*Z*the charge multiplicity, we get for the relation of beam current and beam intensity

*C*is the circumference of the circular accelerator. This is the average circulating current to be distinguished from the bunch current or peak bunch current, which is the charge per bunch

*q*divided by the duration of the bunch.

For a linear accelerator or beam transport line where particles come by only once, the definition of the beam current is more subtle. We still have a simple case if the particles come by in a continuous stream in which case the beam current is proportional to the particle flux \(\dot{N}\) or \(I = eZ\,\dot{N}\). This case, however, occurs very rarely since particle beams are generally accelerated by rf fields. As a consequence there is no continuous flux of particles reflecting the time varying acceleration of the rf field. The particle flux therefore is better described by a series of equidistant particle bunches separated by an integral number of wavelengths of the accelerating rf field. Furthermore, the acceleration often occurs only in bursts or pulses producing either a single bunch of particles or a string of many bunches. In these cases we distinguish between different current definitions. The peak current is the peak instantaneous beam current for a single bunch, while the average current is defined as the particle flux averaged over the duration of the beam pulse or any other given time period, e.g. 1 s.

Numerical conversion factors

Quantity | Replace cgs parameter by practical units | ||
---|---|---|---|

Potential | 1 esu | 300 V | |

Electrical field | 1 esu | 3 10 | |

Current | 1 esu | 0. 1 ⋅ | |

Charge | 1 esu | 0. 3333 10 | |

Force | 1 dyn | 10 | |

Energy | 1 eV | 1. 602 10 | |

1 eV | \(1.602\,10^{-12}\) erg |

Conversion factors for equations

Replace cgs-parameter | |||
---|---|---|---|

Quantity | by mks-parameter | ||

Potential | \(V _{\text{cgs}}\) | \(\sqrt{4\pi \varepsilon _{0}}V _{\text{mks}}\) | |

Electric field |
| \(\sqrt{4\pi \varepsilon _{0}}E_{\text{mks}}\) | |

Current |
| \(\frac{1} {\sqrt{4\pi \varepsilon _{0}}} I_{\text{mks}}\) | |

Current density |
| \(\frac{1} {\sqrt{4\pi \varepsilon _{0}}} j_{\text{mks}}\) | |

Charge |
| \(\frac{1} {\sqrt{4\pi \varepsilon _{0}}} q_{\text{mks}}\) | |

Charge density |
| \(\frac{1} {\sqrt{4\pi \varepsilon _{0}}} \rho _{\text{mks}}\) | |

Conductivity |
| \(\frac{1} {\sqrt{4\pi \varepsilon _{0}}} \sigma _{\text{mks}}\) | |

Inductance |
| \(4\pi \varepsilon _{0}\,L_{\text{mks}}\) | |

Capacitance |
| \(\frac{1} {4\pi \varepsilon _{0}} C_{\text{mks}}\) | |

Magnetic field | \(H_{\text{cgs}}\) | \(\sqrt{4\pi \mu _{0}}H_{\text{mks}}\) | |

Magnetic induction |
| \(\frac{4\pi } {\mu _{0}} B_{\text{mks}}\) |

### 1.3.2 Maxwell’s Equations

consistent with the SI-system of units by inclusion of the unit scale factors *ε*_{0} and *μ*_{0}. The quantities *ε* and *μ*are the relative dielectric constant and magnetic permeability of the surrounding materials, respectively. Integration of one or the other of Maxwell’s equations results, for example, in the fields from singly charged particles or those of an assembly of particles travelling along a common path and forming a beam. Applying Maxwell’s equations, we will make generous use of algebraic relations which have been collected in Appendix A.

## 1.4 Primer in Special Relativity

In accelerator physics the dynamics of particle motion is formulated for a large variety of energies from nonrelativistic to highly relativistic values and the equations of motion obviously must reflect this. Relativistic mechanics is therefore a fundamental ingredient of accelerator physics and we will recall a few basic relations of relativistic particle mechanics from a variety of more detailed derivations in generally available textbooks. Beam dynamics is expressed in a laboratory by a fixed system of coordinates but some specific problems are better discussed in the moving coordinate system of particles. Transformation between the two systems is effected through a Lorentz transformation.

### 1.4.1 Lorentz Transformation

Physical phenomena can appear different for observers in different systems of reference. Yet, the laws of nature must be independent of the reference system. In classical mechanics, we transform physical laws from one to another system of reference by way of the Galileo transformation \(z^{{\ast}} = z - vt\) assuming that one system moves with velocity *v* along the *z*-axis of the other system.

As the velocities of bodies under study became faster, it became necessary to reconsider this simple transformation leading to Einstein’s special theory of relativity. Maxwell’s equations result in electromagnetic waves expanding at a finite velocity and do not contain any reference to a specific system of reference. Any attempt to find a variation of the “velocity of light” with respect to moving reference systems failed, most notably in Michelson’s experiments. The expansion velocity of electromagnetic waves is therefore independent of the reference system and is finite.

*z*-axis with respect to the stationary system \(\mathcal{L}\).

*β*changes sign \(\left (v \rightarrow -v\right )\).

#### Lorentz Transformation of Fields

Again, for the inverse transformation only the sign of the relative velocity must be changed, \(\beta _{z} \rightarrow -\)*β*_{z}. According to this transformation of fields, a pure static magnetic field in the laboratory system \(\mathcal{L}\), for example, becomes an electromagnetic field in the moving system \(\mathcal{L}^{{\ast}}\). An undulator field, therefore, looks to an electron like a virtual photon with an electromagnetic field like a laser field and both interactions can be described by Compton scattering.

#### Lorentz Contraction

*z*-coordinate with a length \(\ell= z_{2} - z_{1}\). In the system \(\mathcal{L}^{{\ast}}\), which is moving with the velocity \(v_{z}\) in the positive

*z*-direction with respect to \(\mathcal{L}\), the rod appears to have the length \(\ell^{{\ast}} = z_{2}^{{\ast}}- z_{1}^{{\ast}}.\) By a Lorentz transformation we can relate that to the length in the \(\mathcal{L}\)-system. Observing both ends of the rod at the same time the lengths of the rod as observed from both systems relate like \(\ell= z_{2} - z_{1} =\gamma (z_{2}^{{\ast}} + v_{z}t_{2}^{{\ast}}) -\gamma (z_{1}^{{\ast}} + v_{z}t_{1}^{{\ast}}) =\gamma \ell ^{{\ast}}\) or

*γ*and is always longest in it’s own rest system. For example, the periodicity of an undulator

*λ*

_{p}becomes Lorentz contracted to

*λ*

_{p}∕

*γ*as seen by relativistic electrons. Because of the Lorentz contraction, the volume of a body at rest in the system \(\mathcal{L}\) appears also reduced in the moving system \(\mathcal{L}^{{\ast}}\) and we have for the volume of a body in three dimensional space

*γ*. As a consequence, the charge density

*ρ*of a particle bunch with the volume

*V*is lower in the laboratory system \(\mathcal{L}\) compared to the density in the system moving with this bunch and becomes

#### Time Dilatation

For a particle at rest in the moving system \(\mathcal{L}^{{\ast}}\) the time \(t^{{\ast}}\) varies slower than the time in the laboratory system. This is the mathematical expression for the famous twin paradox where one of the brothers moving in a space capsule at relativistic speed would age slower than his twin brother staying back. This phenomenon gains practical importance for unstable particles. For example, high-energy pions, observed in the laboratory system, have a longer lifetime by the factor *γ* compared to low-energy pions with *γ* = 1. As a consequence, high energy unstable particles, like pions and muons, live longer and can travel farther as measured in the laboratory system, because the particle decay time is a particle property and is therefore measured in its own moving system. This is important. For example, in medical applications when a beam of pions has to be transported from the highly radioactive target area to a radiation free environment for the patient for cancer treatment.

### 1.4.2 Lorentz Invariance

#### Invariance to Lorentz Transformations

The length of 4-vectors is the same in all reference systems and is therefore open to measurements and comparisons independent of the location of the experimenter. In fact, it can be shown (exercise) that even the product of two arbitrary 4-vectors is Lorentz invariant. Take two 4-vectors in an arbitrary frame of reference \(\boldsymbol{\tilde{a}}^{{\ast}}\mathbf{=}\left (a_{1}^{{\ast}},a_{2}^{{\ast}},a_{3}^{{\ast}},\text{i}a_{4}^{{\ast}}\right )\) and \(\boldsymbol{\tilde{b}}^{{\ast}}\mathbf{=}\left (b_{1}^{{\ast}},b_{2}^{{\ast}},b_{3}^{{\ast}},\text{i}b_{4}^{{\ast}}\right )\) and form the product \(\boldsymbol{\tilde{a}}^{{\ast}}\boldsymbol{\tilde{b}}^{{\ast}}\) in component form. A Lorentz transformation on both 4-vectors gives \(\boldsymbol{\tilde{a}}^{{\ast}}\boldsymbol{\tilde{b}}^{{\ast}} =\boldsymbol{\tilde{ a}\tilde{b}},\) which is the same in any reference system and is therefore Lorentz invariant. Specifically, the length of any 4-vector is Lorentz invariant.

#### Space-Time

*t*, the edge of this expanding light flash has expanded with the velocity of light to

*c*as has been experimentally verified by Michelson and Morley in 1887. The velocity of light is the same in all reference systems and its value is

*c*to give all components the same dimension. From the Lorentz invariant world time

*τ*, defined as

*t*.

Other 4-vectors can be formulated and often become relevant in accelerator physics as, for example, those listed below. More 4-vectors are listed in Appendix B.

#### Four-Velocity

Evaluating the square of the velocity 4-vector we find \(\boldsymbol{\tilde{v}}^{2} =\gamma \boldsymbol{ v}^{2} -\gamma c^{2} = -c^{2}\) in the rest frame and in any other reference frame. The velocity of light is the same in any reference system as experimentally verified by Michelson and Morley.

#### Four-Acceleration

#### Momentum-Energy 4-Vector

*E*

*E*

_{0}=

*Amc*

^{2}for a particle with atomic mass

*A*. From this the total energy is

*γ*in terms of energies. Sometimes, authors attach this relativistic factor to the mass and assume thereby an increasing moving mass. Einstein’s point of view is expressed in the following quote: “It is not good to introduce the concept of the mass of a moving body

*M*=

*γ m*

_{0}for which no clear definition can be given. It is better to introduce no mass concept other than the ‘rest mass’

*m*

_{0}. Instead of introducing

*M*it is better to mention the expression for the momentum and energy of a body in motion.” In this book, we take the rest mass

*m*

_{0}as an invariant.

*E*

_{0}=

*Amc*

^{2}is the rest energy of the particle and

*A*the atomic mass. For electrons we assume that

*A*= 1 and

*m*=

*m*

_{e}. Since in this text we concentrate mainly on electrons and protons, we assume

*A*= 1. The kinetic energy is defined as the total energy minus the rest energy

*L*

_{acc}is the path length through the accelerating field. In discussions of energy gain through acceleration, we consider only energy differences and need therefore not to distinguish between total and kinetic energy. The particle momentum finally is defined by

*cp*. In electron accelerators the numerical distinction between energy and momentum is insignificant since we consider in most cases highly relativistic particles. For proton accelerators and even more so for heavy ion accelerators the difference in both quantities becomes, however, significant.

*γ*from (1.33) we get

#### Photon 4-Vector

An analogous 4-vector can be formulated for photons using deBroglie’s relations \(\boldsymbol{p} = \hslash \boldsymbol{k}\) and \(E = \hslash \omega\) for \(c\boldsymbol{\tilde{k}} = (ck_{x},ck_{y},ck_{z},\mathrm{i}\omega )\,.\) Since the energy-momentum 4-vector is derived from the canonical momentum, we will have to modify this 4-vector when electromagnetic fields are present.

#### Force 4-Vector

The force 4-vector is the time derivative of the energy-momentum 4-vector \(\left (c\boldsymbol{\dot{p}},\text{i}\dot{E}\right ),\) which is consistent with the observation (so far) that the rest mass does not change with time.

#### Electro-magnetic 4-Vector

The electromagnetic-potential 4-vector is \(\left (c\boldsymbol{A},\text{i}\phi \right ).\)

### 1.4.3 Spatial and Spectral Distribution of Radiation

*ω*. Since the space-time coordinates are independent from each other, we may equate their coefficients on either side of the equation separately.

#### Spectral Distribution

*ct*-coefficients define the transformation of the oscillation frequency

*Θ*with respect to the direction of motion of the source. In these cases \(n_{z}^{{\ast}} =\cos \varTheta ^{{\ast}}\) and the frequency shift can be very large for highly relativistic particles with

*γ*≫ 1.

#### Spatial Distribution

This angle is very small for highly relativistic electrons like those in a storage ring, where *γ* is of the order of 10^{3}–10^{4}.

### 1.4.4 Particle Collisions at High Energies

*m*

_{1}and

*m*

_{2}and velocities

**v**

_{1}and

**v**

_{2}, respectively,

*γ*colliding with a proton at rest in a target. For a target proton at rest with \(\gamma _{2} = 1, m_{2} = m_{\mathrm{p}}, \beta _{2} = 0\) and \(\beta \gamma = \sqrt{\gamma ^{2 } - 1}\), the center of mass energy is

## 1.5 Principles of Particle-Beam Dynamics

Accelerator physics relates primarily to the interaction of charged particles with electromagnetic fields. Detailed knowledge of the functionality of this interaction allows the design of accelerators meeting specific goals and the prediction of charged particle beam behavior in those accelerators. The interplay between particles and fields is called beam dynamics. In this section, we recall briefly some features of electromagnetic fields and fundamental processes of classical and relativistic mechanics as they relate to particle beam dynamics.

### 1.5.1 Electromagnetic Fields of Charged Particles

Predictable control of charged particles is effected only by electric and magnetic fields and beam dynamics is the result of such interaction. We try to design and formulate electromagnetic fields in a way that can be used to accurately predict the behavior of charged particles. To describe the general interaction of fields based on electric currents in specific devices and free charged particles which we want to preserve, guide and focus, we use as a starting point Maxwell’s equations (1.4).

#### Electric Field of a Point Charge

*q*at rest. The natural coordinate system is the polar system because the fields of a point charge depend only on the radial distance from the charge. We integrate Coulomb’s law (1.4) over a spherical volume containing the charge

*q*at its center. With d

*V*= 4

*π r*

^{2}d

*r*the integral becomes \(\int \nabla \boldsymbol{E}\) d\(V \mathbf{=}\int _{0}^{R} \frac{1} {r^{2}} \frac{\partial } {\partial r}\left (r^{2}E_{ r}\right )\) d\(V = 4\pi R^{2}E_{ r}\left (R\right )\), where

*R*is the radial distance from the charge. On the r.h.s. of Coulomb’s law (1.4), an integration over all the charge

*q*gives \(\int \frac{\rho }{\epsilon _{ 0}\epsilon }\) d\(V = \frac{q} {\epsilon _{0}\epsilon }\) and the electric field of a point charge at distance

*R*is

*R*.

#### Fields of a Charged Particle Beam

*ρ*

_{0}is the charge density in the particle beam. We assume a uniform continuous beam and expect therefore no azimuthal or longitudinal dependence, leaving only the radial dependence. Radial integration over a cylindrical volume of unit length collinear with the beam gives with the volume element d

*V*= 2

*π r*d

*r*, on the l.h.s. \(\left \vert rE_{r}\right \vert _{0}^{r}2\pi\). The r.h.s. is \(\frac{\rho _{0}} {\epsilon _{0}\epsilon } \pi r^{2}\) and the electric field for a uniformly charged particle beam with radius

*R*is

The fields increase linearly within the beam and decay again like 1∕*r* outside the beam. Real particle beams do not have a uniform distribution and therefore a form function must be included in the integration. In most cases, the radial particle distribution is bell shaped or Gaussian. Both distributions differ little in the core of the beam and therefore a convenient assumption is that of a Gaussian distribution for which the fields will be derived in Problem 1.3.

### 1.5.2 Vector and Scalar Potential

*V*which does not alter the validity of Maxwell’s equations for all fields so defined. To summarize, both, electric and magnetic fields can be derived from a scalar

*V*and vector \(\boldsymbol{A}\) potential

### 1.5.3 Wave Equation

*V*such that it meets the condition

*P*results in the definition of the vector and scalar potential at the point

*P*. Both electric and magnetic fields may be derived as discussed in the last section.

#### Lienard-Wiechert Potentials

*e*at rest, we can integrate (1.57) readily to get

*A*(

*R*,

*t*) = 0 and \(V (R,t) = \frac{e} {4\pi \epsilon _{0}\epsilon R}\). On the other hand, in case of a moving point charge the potentials cannot be obtained by simply integrating over the “volume” of the point charge. The motion of the charge must be taken into account and the result of a proper integration (see Chap. 25) are the Liénard-Wiechert potentials for moving charges [31, 32]

### 1.5.4 Induction

*S*, which is equivalent to a voltage. On the r.h.s. the magnetic flux passing through the surface

*S*is integrated and

By virtue of Faraday’s law, the time varying magnetic flux *Φ* through the area *S* generates an electromotive force along the boundaries of *S*. In accelerator physics this principle is applied in the design of a betatron. Similarly, from the second term on the right hand side in Ampère’s law (1.4), we get a magnetic induction from a time varying electric field. Both phenomena play together to form the principle of induction or, in a particular example, that of a transformer.

### 1.5.5 Lorentz Force

*v*≈

*c*and to get the same force from an electric field as from, say a 1 Tesla magnetic field, one would have to have an unrealistic high field strength of \(\boldsymbol{E} \approx 300\) MV/m. For this reason, magnetic fields are used to deflect or focus relativistic charged particles. For sub-relativistic particles like ion beams with velocities \(v \ll c,\) on the other hand, electric fields may be more efficient.

### 1.5.6 Equation of Motion

Accelerator physics is to a large extend the description of charged particle dynamics in the presence of external electromagnetic fields or of fields generated by other charged particles. We use the Lorentz force to formulate particle dynamics under the influence of electromagnetic fields. Whatever the interaction of charged particles with electromagnetic fields and whatever the reference system may be, we depend in accelerator physics on the invariance of the Lorentz force equation under coordinate transformations. All acceleration and beam guidance in accelerator physics will be derived from the Lorentz force. For simplicity, we use throughout this text particles with one unit of electrical charge *e* like electrons and protons. In case of multiply charged ions the single charge *e* must be replaced by *eZ* where *Z* is the charge multiplicity of the ion. Both components of the Lorentz force are used in accelerator physics where the force due to the electrical field is mostly used to actually increase the particle energy while magnetic fields are used to guide particle beams along desired beam transport lines. This separation of functions, however, is not exclusive as the example of the betatron accelerator shows where particles are accelerated by time dependent magnetic fields. Similarly electrical fields are used in specific cases to guide or separate particle beams.

It becomes obvious that the kinetic energy of a particle changes whenever it travels in an accelerating electric field \(\boldsymbol{E}\) and the acceleration occurs in the direction of the electric field. This acceleration is independent of the particle velocity and acts even on a particle at rest \(\boldsymbol{v} = 0\). The second component of the Lorentz force in contrast depends on the particle velocity and is directed normal to the direction of propagation and normal to the magnetic field direction. We find therefore from (1.66) the result that the kinetic energy is not changed by the presence of magnetic fields since the scalar product \(\left (\boldsymbol{v \times B}\right )\boldsymbol{v} = 0\) vanishes. The magnetic field causes only a deflection of the particle trajectory.

*Z*is the charge multiplicity of the charged particle and

*A*the atomic mass. For simplicity we drop from here on the factors

*A*and

*Z*since they are different from unity only for ion beams. For ion accelerators we note therefore that the particle charge

*e*must be replaced by

*eZ*and the mass by

*Am*.

Both relations in (1.65) can be used to describe the effect of the Lorentz force on particles. However, ease of mathematics makes us use one or the other. We use the first equation for dynamics in magnetic fields and the second for that in accelerating fields. Since the energy or the particle velocity does not change in a magnetic field it is straightforward to calculate \(\varDelta \boldsymbol{p.}\) On the other hand, accelerating fields do change the particle’s velocity which must be included in the time integration to get \(\varDelta \boldsymbol{p.}\) Calculating *Δ E*_{kin}, we need to know only the spatial extend and magnitude of the accelerating fields to perform the integration.

It is obvious from (1.71) and (1.72) how differently the dynamics of particle motion is affected by the direction of the Lorentz force. Specifically the dynamics of highly relativistic particles under the influence of electromagnetic fields depends greatly on the direction of the force with respect to the direction of particle propagation. The difference between parallel and perpendicular acceleration will have a great impact on the design of electron accelerators. As we will see later, the acceleration of electrons is limited due to the emission of synchrotron radiation. This limitation, however, is much more severe for electrons in circular accelerators where the magnetic forces act perpendicularly to the propagation compared to the acceleration in linear accelerators where the accelerating fields are parallel to the particle propagation. This argument is also true for protons or for that matter, any charged particle, but because of the much larger particle mass the amount of synchrotron radiation is generally negligibly small.

### 1.5.7 Charged Particles in an Electromagnetic Field

*ρ*the local curvature and

*m*the mass of the particle with charge

*e*. For a magnetic field orthogonal to the velocity vector of the particle the vector product is always parallel and opposite to \(\boldsymbol{n}\) and (1.73) reduces to

Here, some caution is appropriate, because during the deflection the unit vector \(\boldsymbol{n}\) is changing direction while the electric field may not change direction as in the case of a field between parallel straight plates. However, if the electrodes are bend along the expected particle path, the direction of the electric field is changing with \(\boldsymbol{n}\) or the deflection of the beam.

### 1.5.8 Linear Equation of Motion

*B*

_{0y}is the bending field and

*g*the field gradient \(g = \partial B_{y}/\partial x.\) The particle beam is not perfectly monochromatic and we account for this by expanding the particle energy to first order \(\frac{1} {cp} \approx \frac{1} {cp_{0}} \left (1-\delta \right )\), where \(\delta =\varDelta p/p_{0}.\) With this we get the equation of motion

*x*and \(\delta\)

*z*. However, the solutions will be of oscillatory nature describing the particle motion in the restoring fields of the focusing devices. Actual analytical solutions will be discussed in great detail later in this text.

### 1.5.9 Energy Conservation

This equation expresses the conservation of energy relating the change in field energy and particle acceleration with a new quantity describing energy loss or gain through radiation.

#### Poynting Vector

Knowing the electric fields we may determine the Poynting vector describing electro-magnetic waves or synchrotron radiation.

### 1.5.10 Stability of a Charged-Particle Beam

Individual particles in an intense beam are under the influence of strong repelling electrostatic forces creating the possibility of severe stability problems. Particle beam transport over long distances could be greatly restricted unless these space-charge forces can be kept under control. First, it is interesting to calculate the magnitude of the problem.

If all particles would be at rest within a small volume, we would clearly expect the particles to quickly diverge from the center of charge under the influence of the repelling space charge forces from the other particles. This situation may be significantly different in a particle beam where all particles propagate in the same direction. We will therefore calculate the fields generated by charged particles in a beam and derive the corresponding Lorentz force due to these fields. Since the Lorentz force equation is invariant with respect to coordinate transformations, we may derive this force either in the laboratory system or in the moving system of the particle bunch.

*γ*

^{−2}for higher energies. Obviously this repelling space charge force is much stronger for proton and especially for ion beams because of the smaller value for

*γ*and, in the case of ions, because of the larger charge multiplicity which increases the space-charge force by a factor of

*Z*.

We obtained the encouraging result that at least relativistic particle beams become stable under the influence of their own fields. For lower particle energies, however, significant diverging forces must be expected and adequate focusing measures must be applied. The physics of such space charge dominated beams is beyond the scope of this book and is treated elsewhere, for example in considerable detail in [33].

## Problems

**1.1 (S).** Use the definition for *β*, the momentum, the total and kinetic energy and derive expressions \(p\left (\beta,E_{\text{kin}}\right )\), \(p\left (E_{\text{kin}}\right )\) and \(E_{\text{kin}}\left (\gamma \right )\). Simplify the expressions for very large energies, *γ* ≫ 1. Derive from these relativistic expressions the classical nonrelativistic formulas.

**1.2 (S).** Prove the validity of the field equations \(E_{r} = \frac{1} {2\epsilon _{0}} \rho _{0}r\) and \(B_{\varphi } = \frac{1} {2}\mu _{0}\beta \rho _{0}r\) for a uniform cylindrical particle beam with constant charge density *ρ*_{0} within a radius *r* < *R*. Derive the field expressions for *r* > *R*.

**1.3 (S).** Derive the electric and magnetic fields of a beam with a radial charge distribution \(\rho \left (r,\varphi,z\right ) =\rho \left (r\right )\). Derive the field equations for a Gaussian charge distribution with standard deviation *σ* given by \(\rho \left (r\right ) =\rho _{0}\exp \left [-r^{2}/\left (2\sigma ^{2}\right )\right ]\). What are the fields for *r* = 0 and *r* = *σ*?

**1.4 (S).** A circular accelerator with a circumference of 300 m contains a uniform distribution of singly charged particles orbiting with the speed of light. If the circulating current is 1 amp, how many particles are orbiting? We instantly turn on an ejection magnet so that all particles leave the accelerator during the time of one revolution. What is the peak current at the ejection point? How long is the current pulse duration? If the accelerator is a synchrotron accelerating particles at a rate of 10 acceleration cycles per second, what is the average ejected particle current?

**1.5 (S).** A proton with a kinetic energy of 1 eV is emitted parallel to the surface of the earth. What is the bending radius due to gravitational forces? What are the required transverse electrical and magnetic fields to obtain the same bending radius? What is the ratio of electrical to magnetic field? Is this ratio different for a proton energy of say 10 TeV? Why? (gravitational constant \(6.67259 \times 10^{-11}\,\mathrm{m}^{3}\mathrm{kg}^{-1}\mathrm{s}^{-2}\)).

**1.6 (S).** Consider a highly relativistic electron bunch of *n* = 10^{10} uniformly distributed electrons. The bunch has the form of a cylindrical slug, *ℓ* = 1 mm long and a radius of *R* = 0. 1 μm. What is the electrical and magnetic field strength at the surface of the beam. Calculate the peak electrical current of the bunch. If two such beams in a linear collider with an energy of 500 GeV pass by each other at a distance of 10 μm (center to center), what is the deflection angle of each beam due to the field of the other beam?

**1.7 (S).** Show that for plane waves\(\boldsymbol{n} \times \boldsymbol{ E} = c\boldsymbol{B}\) .

**1.8 (S).** Show that the product of two 4-vectors is Lorentz invariant.

**1.9 (S).** Prove that the 4-acceleration is indeed given by (1.23).

**1.10 (S).** Using 4-vectors, derive the frequency of an outgoing photon from a head-on Compton scattering process of an electron with a photon of frequency \(\omega _{\text{L}}.\)

**1.11 (S).** Using 4-vectors, derive the frequency of an outgoing photon from a head-on Compton scattering process of an electron with the field of an undulator with period *λ*_{u}.

**1.12.** Protons are accelerated to a kinetic energy of 200 MeV at the end of the Fermilab Alvarez linear accelerator. Calculate their total energy, their momentum and their velocity in units of the velocity of light \(\left (m_{\text{p}}c^{2} = 938.27\,\text{ MeV}\right ).\)

**1.13.** Consider electrons to be accelerated in the *L* = 3 km long SLAC linear accelerator with a uniform gradient of 20 MeV/m. The electrons have a velocity \(v = \frac{1} {2}c\) at the beginning of the linac. What is the length of the linac in the rest frame of the electron? Assume the particles at the end of the 3 km long linac would enter another 3 km long tube and coast through it. How long would this tube appear to be to the electron?

**1.14 (S).** A positron beam of energy *E* accelerated in a linac hits a fixed hydrogen target. What is the available energy from a collision with a target electron assumed to be at rest? Compare this available energy with that obtained in a linear collider where electrons and positrons from two similar linacs collide head on at the same energy.

**1.15 (S).** The SPEAR colliding beam storage ring has been constructed originally for electron and positron beams to collide head on with an energy of up to 3.5 GeV. At 1.55 GeV per beam a new particle, the *ψ*∕*J*-particle, was created. In a concurrent fixed target experiment at BNL, such *ψ*∕*J*-particle have been produced by protons hitting a hydrogen target. What proton energy was required to produce the new particle? Determine the positron energy needed to create *ψ*∕*J*-particles by collisions with electrons in a fixed target.

**1.16.** A charged pion meson has a rest energy of 139. 568 MeV and a mean life time of \(\tau _{0\pi } = 26.029\) ns in its rest frame. What are the life times *τ*_{π}, if accelerated to a kinetic energy of 20 MeV? and 100 MeV? A pion beam decays exponentially like e\(^{-t/\tau _{\pi }}\). At what distance from the source will the pion beam intensity have fallen to 50 *%*, if the kinetic energy is 20 MeV? or 100 MeV?

**1.17 (S).** Assume you want to produce antiprotons by accelerating protons and letting them collide with other protons in a stationary hydrogen target. What is the minimum kinetic energy the accelerated protons must have to produce antiprotons? Use the reaction \(p + p \rightarrow p + p + p +\bar{ p}\).

**1.18.** Use the results of Problem 1.3 and consider a parallel beam at the beginning of a long magnet free drift space. Follow a particle under the influence of the beam self fields starting at a distance *r*_{0} = *σ* from the axis. Derive the radial particle distance from the axis as a function of *z*.

**1.19.** Show that (1.57) is indeed a solution of (1.55).

**1.20.** Express the equation of motion (1.67) for *Z* = 1 in terms of particle acceleration, velocity and fields only. Verify from this result the validity of (1.71) and (1.72).

**1.21.** Plot on log-log scale the velocity *β*, total energy as a function of the kinetic energy for electrons, protons, and gold ions Au^{+14}. Vary the total energy from 0. 01*mc*^{2} to \(10^{4}mc^{2}.\) Why does the total energy barely change at low kinetic energies.

**1.22.** The design for a Relativistic Heavy Ion Collider calls for the acceleration of completely ionized gold atoms in a circular accelerator with a bending radius of *ρ* = 242. 78 m and superconducting magnets reaching a maximum field of 34.5 kg. What is the maximum achievable kinetic energy per nucleon for gold ions Au^{+77} compared to protons? Calculate the total energy, momentum, and velocity of the gold atoms (A_{Au}=197).

**1.23.** Gold ions Au^{+14} are injected into the Brookhaven Alternating Gradient Synchrotron AGS at a kinetic energy per nucleon of 72 MeV/u. What is the velocity of the gold ions? The AGS was designed to accelerate protons to a kinetic energy of 28.1 GeV. What is the corresponding maximum kinetic energy per nucleon for these gold ions that can be achieved in the AGS? The circulating beam is expected to contain 6 ⋅ 10^{9} gold ions. Calculate the beam current at injection and at maximum energy assuming there are no losses during acceleration. The circumference of the AGS is \(C_{\text{AGS}} = 807.1\) m. Why does the beam current increase although the circulating charge stays constant during acceleration?

**1.24.** Particles undergo elastic collisions with gas atoms. The rms multiple scattering angle is given by \(\sigma _{\theta } \approx Z\frac{20\,\left (\text{MeV/c}\right )} {\beta p} \sqrt{\frac{s} {\ell_{\text{r}}}}\), where *Z* is the charge multiplicity of the beam particles, *s* the distance travelled and \(\ell_{\text{r}}\) the radiation length of the scattering material (for air the radiation length at atmospheric pressure is \(\ell_{\text{r}} = 500\) m or 60. 2 g/cm^{2}). Derive an approximate expression for the beam radius as a function of *s* due to scattering. What is the approximate tolerable gas pressure in a proton storage ring if a particle beam is supposed to orbit for 20 h and the elastic gas scattering shall not increase the beam size by more than a factor of two during that time?

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