Introduction to Particle Accelerators and their Limitations
PProceedings of the 2019 CERN–Accelerator–School course on
High Gradient Wakefield Accelerators, Sesimbra, (Portugal)
Introduction to Particle Accelerators and their Limitations
Massimo Ferrario , Bernhard J. Holzer Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati, Rome, Italy; CERN, Geneva,Switzerland
Abstract
The paper gives a short overview of the principles of particle accelerators, theirhistorical development and the typical performance limitations. After an in-troduction to the basic concepts, the main emphasis is to sketch the layout ofmodern storage rings and their limitations in energy and machine performance.Examples of existing machines - among them clearly the LHC at CERN -demonstrate the basic principles and the technical and physical limits that weface today in the design and operation of these particle colliders. Pushing forever higher beam energies motivates the design of the future collider studiesand beyond that, the development of more efficient acceleration techniques.
Keywords
Accelerator physics; synchrotrons and storage rings; particle colliders;acceleration gradients; performance limits. Introduction
The study of matter, starting from the structure of the atom, the discovery of the nucleus and beyondthat the discovery of the variety of elementary particles and their interactions, summarised in a scientificpicture that we like to call "Standard Model", came along and drove the development of powerful toolsto create high energetic particle beams, the so-called particle accelerators.These machines are used nowadays world wide in engineering, life science and physics. More than35000 accelerators exist today for studies in a manifold of applications in physics, chemistry, medicineand structure analysis. The up to now largest accelerator, the Large Hadron Collider, LHC, at CERN inGeneva is operating on a level of 7 TeV energy per beam, corresponding to an available centre of massenergy of E cm = 14 TeV. It is standing in a long tradition of technical as well as physics progress inthe creation of high energy particle beams, their acceleration and the successful collision of the micrometer small beam sizes. This article gives a very basic introduction into the physics of these acceleratorsand (some of) their limitations. They have been selected arbitrarily by the authors, de-facto there aremany more that can be found and studied by the reader in a number of decent publications ( . . . andsomewhen hopefully overcome). We define a few key parameters to describe the performance of an ac-celerator. They clearly depend on the application, however, on a quite general basis we can summarizethem as:– energy of the accelerated particles– particle intensity, or current of the stored or accelerated particle beam– beam quality, the experts call it emittance – in the case of lepton beams, used for the production of synchrotron radiation, brightness of the emit-ted light– luminosity in case of particle colliders.As a matter of fact, it depends on the application of the particle beam, which of these parametersare the (most) relevant ones and how we have to optimise the design of our accelerator to reach highestperformance. Out of the 35000 particle accelerators that exist and operate worldwide, the largest fractionAvailable online at https://cas.web.cern.ch/previous-schools a r X i v : . [ phy s i c s . acc - ph ] J u l s used for medical applications and ion implantation. Figure 1 shows the distribution of the existingmachines over their field of operation. We must admit, that for those, working in basic research, like e.g., Fig. 1:
Particle accelerators and their applications. the authors of this article, it is sometimes brings us down to earth to realise that the number of machinesworking at the energy frontier is "close to negligible". Still, and be it for the bare impression of sucha large piece of scientific infrastructure, we show in Fig. 2 a photo of the LHC in the Geneva valley.
Fig. 2:
The Large Hadron Collider, LHC, located in the Geneva valley.
In this paper we would like to follow a little bit the path that has been paved by the ingenious dis-coveries and developments that date back to the time of the discovery of the nucleus by Ernest Rutherford.Using alpha particles of some MeV from a natural source, he discovered in scattering experimentsthe nucleus of the atom. Now, a radioactive source is not an ideal tool for precise, trigger-able andhealthy experiments. So it was indeed Rutherford who discussed the possibility of artificially acceleratedparticles with two colleagues, Cockcroft and Walton. Following this idea, they developed within onlyfour years the first particle accelerator ever built: In 1932 they could demonstrated the first splitting ofa nucleus (Li) using a 400 keV proton beam.Their acceleration mechanism was based on a rectifier- or Greinacher-circuit, consisting of a num-ber of diodes and capacitors that transformed a relatively small AC-voltage to a DC potential that corre-sponds to a multiple of the applied basic potential, according to the number of diode/capacitor units thatare used. The particle source was a standard hydrogen discharge source, connected to the high-voltage2art of the system and the particle beam was accelerated to ground potential, hitting the Lithium tar-get [1]. Even if of modest energy at that time, their beam could split the Li nucleus and . . . needles tosay: Cockcroft and Walton got the Nobel-prize for it.
Fig. 3:
A Cockcroft-Walton Generator used at CERN as pre-accelerator for the proton beams. The device ismeanwhile replaced by the more compact and efficient Radio Frequency Quadrupole (RFQ) technique.
In a parallel approach, but based on a completely different technique, another type of DC accel-erator had been invented: Van-de-Graaf designed a DC-accelerator [2], that used a mechanical transportsystem to carry charges, sprayed on a belt or chain, to a high voltage terminal. In general these machinesreach higher voltages than the Cockcroft-Walton devices but they are more limited in particle intensity.Common to all DC accelerators is the limitation of the achievable beam energy due to high voltage break-down effects (discharges). Without insulating gas ( SF in most cases) electrical fields will be limited toabout 1 MV/m and even using most sophisticated devices, like the one in Fig. 4, acceleration voltages ofsome MV cannot be overcome. Fig. 4:
A typical example of a Tandem-van-de-Graff Accelerator. These are very reliable machines for precisionmeasurements in atom and nuclear physics. (Photo: Max Planck Institute for Nuclear Physics, Heidelberg)
In fact, the example in the last figure shows an option that is applied in a number of places:Injecting a negative ion beam (even H − is used) and stripping the ions in the middle of the high voltageterminal allows to profit from the potential difference twice and thus to gain another step in beam energy.3 .2 Limit II: The size of the accelerating structure Given the obvious limitations of the above described DC machines, the next step forward is a naturalchoice:In 1928 Wideroe developed the concept of an AC accelerator. Instead of rectifying the AC voltage,he connected a series of acceleration electrodes in alternating order to the output of an AC supply.The schematic layout is shown in Fig. 5 where - for a moment the direction of the electric field isindicated by the arrows.
Fig. 5:
Schematic view of the Wiederoe principle as fundamental concept for AC (or "RF") acceleration.
In principle this device can produce step by step a multiple of the acceleration voltage, as longas for the negative half wave of the AC voltage the particles are shielded from the decelerating field.The energy gain after the n th step therefore is E n = n · q · U · sin ( ψ s ) , (1)where n denotes the n th acceleration step, q the charge of the accelerated particle, U the applied voltageper gap and ψ s the phase between the particle and the changing AC-voltage, as indicated in Fig. 6The key-point in such a Wideroe-structure is the length of the drift tubes that will protect the par-ticles from the negative half wave of the sinusoidal AC voltage. For a given frequency of the applied RFvoltage the length of the drift tube is defined by the speed of the particle and the duration of the negativehalf wave of the sinusoidal voltage: Fig. 6:
The frequency, and so the period of the RF system, and the particle speed determine the length of the drifttubes in the structure.
The time span of the negative half wave is defined by the applied frequency, ∆ t = τ rf / , where τ rf is the RF period. And so we get for the length of the n th drift tube l n = v n · τ rf . (2)It depends only on the period of the RF system and the velocity of the particle v n when traversingthe n th acceleration gap. Given the kinetic energy of the particle of mass m and velocity v , E kin = 12 · mv (3)4e obtain directly l n = 1 ν rf · (cid:114) nqU sinψ s m , (4)which defines the design concept of the machine.Figure 7 shows a photograph of such a device, the Unilac at the GSI institute in Darmstadt. Fig. 7:
Unilac at GSI, Darmstadt; the structure of the drift tubes and their increasing length as a function ofthe particle energy is clearly visible.
Two remarks should be made in this context:– This short derivation here is based on the classical approach. And in fact: These accelerators areusually optimum for "low energy" proton or heavy ion beams. Typical beam energies - referringto protons - are in the range of some 10 MeV. E.g. the Linac 2 at CERN delivers the protons forLHC operation with an energy of 50 MeV, corresponding to a relativistic beta of β = 0 . .– For higher energies, even in the case of protons or ions the speed some-when will approachthe speed of light and the length of the drift tubes and so the dimension of the whole accelera-tor reaches considerable sizes that might not be feasible anymore. A more advanced concept isneeded in order to keep the machine within reasonable dimensions. And the next logic step inthe historical development was to introduce magnetic fields to bend the trajectory of the particlebeam onto a circle. Beam Dynamics in Synchrotrons and Storage Rings
Before we continue in our rush for higher and higher energies, it seems adequate to motivate this neverending effort by two trivial (... just another word for "well known") statements: A high energetic particlebeam allows us to study new manifestations of matter. Having a certain amount of energy E availableallows to create a given amount of mass m : E = mc , (5)where c is the speed of light. As easy as that. And the quest for understanding the underlying structureof matter, means the creation and study of its fundamental constituents or building blocks: the mesons,baryons, and finally the quarks and leptons.There is a second motivation to achieve highest possible beam energies that is somehow de-coupled from the bare need to create new particles: The resolution that we can achieve in particlescattering experiments. Quantum mechanics tells us that a high energetic particle can and has to be5onsidered as a particle-wave with a wavelength defined by the momentum of the particle. The so-calledde-Broglie wave length is given by λ = hp , (6)where h is the Planck constant, h = 6 . ∗ − Js and p the particle momentum. For example, andto put the things in the right perspective: the famous green light of the visible spectrum correspondsto a wavelength of about λ = 500 nm. On the other side, the SLAC linear accelerator in Stanford,pushed the electrons to an energy of 50 GeV and on this level their de-Broglie wavelength correspondsto λ = 1 . ∗ − m. Now, while atoms have a typical radius of a few = 10 − m, nuclei are inthe order of − m. It is no surprise that Kendal, Friedman and Taylor could discover the quarksusing the SLAC electron beam, but they never would have had a chance to get their Nobel prize usinga conventional microscope.Having made this point, we can consider the next step forward in achievable beam energy, whichrequires circular structures. In order to apply over and over again the accelerating fields we will try tobend the particles on a circle and so bring them back to the RF structure where they receive the next stepin energy.As a consequence we will have to introduce magnetic ( (cid:126)B ) or electric ( (cid:126)E ) fields that deflect the par-ticles and keep them during the complete acceleration process on a well defined orbit. The Lorentz-forcethat acts on the particles thus will have to compensate exactly the centrifugal force that the particles willfeel on their bent orbit.In general we can write for a particle of charge q : (cid:126)F = q · ( (cid:126)E + (cid:126)v × (cid:126)B ) . (7)Now, talking about high energy particle beams the velocity v is close to the speed of light and so repre-sents a nice amplification factor whenever we apply a magnetic field. As a consequence magnetic fieldsfor bending and focusing of high energy charged particles are much more convenient than electric fields.Neglecting electrical fields, therefore, at the moment we write the Lorentz force and the centrifugal forceof the particle on its circular path as F Lorentz = e · v · B (8) F centrifugal = γm v ρ , (9)where m stands for the particle’s rest mass, ρ describes the bending radius of the trajectory and γ is the relativistic Lorentz factor. Assuming an idealised homogeneous dipole filed along the particleorbit we define the condition for a perfect circular orbit as equality between these two forces to obtainthe condition for the idealised ring: pe = B · ρ, (10)where we refer to protons and accordingly have set q = e . This condition relates the so-called beamrigidity Bρ to the particle momentum that can be carried in the storage ring, and it will in the end define– for a given magnetic field of the dipoles – the size of the storage ring.Now in reality instead of a continuous dipole field the storage ring will be built out of severaldipoles, powered in series to define the geometry of the ring. For a single dipole magnet the particletrajectory is shown schematically in Fig. 8.In the free space outside the dipole magnet the particle trajectory is following a straight line. Assoon as the particle enters the magnet it is bent on a circular path until it leaves the magnet at the otherside. 6 ig. 8: Dipole field of a storage ring and the schematic path of the particles.
The overall effect of the main bending (or "dipole") magnets in the ring will define a more of lesscircular path that we call design orbit. By definition this design orbit has to be closed upon itself andthus the overall effect of the main dipole magnets in the ring has to define a bending angle of exactly π .If α defines the bending angle of one single magnet α = dsρ = BdsB · ρ (11)we require therefore (cid:82) Bdlp/e = (cid:82) BdlB · ρ = 2 π. (12)In the true sense of the word a storage ring, therefore, is not a ring but more a polygon, where "poly"stands for the number of dipole magnets installed in the "ring".In the case of the LHC the dipole field has been pushed to the highest achievable values. 1232super conducting dipole magnets, with a length of 15 m each, define the geometry of the ring and viaEq. (12) the maximum momentum for the stored proton beam. Following the equation given above weobtain for a maximum momentum of p = 7 T eV a required B-field of B = 2 π · eV · m · . · m/s (13) B = 8 . T (14)that is needed to bend the beams. For convenience we have expressed the particle momentum in units ofGeV/c. Figure 9 shows a photo of the LHC dipole magnets, built out of super conducting NbTi filaments,that are operated at a temperature of T=1.9 K.It is clear from Eq. (12) that in case we are limited to a certain maximum achievable field ofthe bending magnets, the only way to accelerate the beam to higher energies is . . . the size of the machine.Figure 2 at the very beginning of this paper gives a nice impression about that fact. In addition to the main bending magnets that guide the beam on a closed orbit, focusing fields are neededto keep the particles close together. In modern storage rings and light sources the particles are kept formany hours in the machine and a carefully designed focusing structure is needed to provide the necessarybeam size at different locations in the ring and guarantee stability of the motion.Following the example of classical mechanics, linear restoring forces are needed, just as in the caseof a harmonic pendulum. Quadrupole magnets provide the corresponding field: They create a magnetic7 ig. 9:
Superconducting dipole magnets of the LHC. field that depends linearly on the particle amplitude, i.e. the distance of the particle from the design orbit. B x = g · y B y = g · x (15)The constant g is called gradient of the magnetic field and characterises the focusing strength of the mag-netic lens in both transverse planes. For convenience it is - as the dipole field - normalised to the particlemomentum. The normalised gradient is called k and defined by k = gp/e = gBρ . (16)The technical layout of such a quadrupole is shown in Fig. 10, that - again in the case of the LHC dipoles- shows a super conducting magnet. Fig. 10:
Super conducting quadrupole of the LHC storage ring.
Now, that we defined the basic building blocks of a storage ring, the task of the designer willbe to arrange these in a so-called magnet lattice and to optimise the field strengths in a way to obtainthe required beam parameters.A general design principle of modern synchrotrons or storage rings should be pointed out here: Ingeneral these machines are built following a so-called separate function scheme. Schematically this isshown in Fig. 11.An example of how such a magnet lattice looks like in reality is given in Fig. 12. It showsthe dipole (orange) and quadrupole (red) magnets in the TSR storage ring in Heidelberg. Eight dipolesare used to bend the beam on a "circle" and the quadrupole lenses in between them provide the focusingto keep the particles within the aperture limits of the vacuum chamber. Every magnet is designed andoptimised for a certain task: bending, focusing, chromatic correction, etc. We separate the magnetsin the design according to the job they are supposed to do and only in rare cases a combined functionscheme is chosen, where different magnet properties are combined in one piece of hardware. Expressed8 ig. 11:
Schematic layout of a synchrotron. mathematically, we refer to the general Taylor expansion of the magnetic field: B ( x ) p/e = 1 ρ + k · x + 12! mx + 13! nx + ... (17)Following the arguments above, we take at the moment only constant (dipole) or linear terms (quadrupole)in Eq. (17) into account. The higher order field contributions will be treated afterwards as - hopefully -small perturbations. Fig. 12:
TSR storage ring, Heidelberg, as a typical example of a separate function strong focusing storage ring.
The particle will now follow the "circular path”, defined by the dipole fields, and in addition willperform harmonic oscillations in both transverse planes. Schematically the situation is shown in Fig. 13.An ideal particle will follow the design orbit that is idealised in the plot by a circle. Any other particlewill perform transverse oscillations under the influence of the external focusing fields and the amplitudeof these oscillations will finally define the beam size.Unlike a classical harmonic oscillator, however, the equation of motion in the horizontal and ver-9 ig. 13:
Coordinate system used in particle beam dynamics. The longitudinal coordinate s is moving aroundthe ring with the particle considered. tical plane differ a little bit: Assuming a horizontal focusing magnet, the equation of motion is x (cid:48)(cid:48) + x · ( 1 ρ + k ) = 0 , (18)where the derivative refers to the s -coordinate, x (cid:48) = dx/ds , which is nothing else than the angle ofthe trajectory, and x (cid:48)(cid:48) = d x/ds . k describes the normalised gradient, as introduced above and the /ρ term reflects the so-called weak focusing, which is a property of the bending magnets. In the orthogonal,vertical plane however, due to the orientation of the field lines – which clear enough results in the endfrom Maxwell’s equations – the forces turn into a defocusing effect. And – also clear enough – the weakfocusing term disappeared, as usually the machine will be built in the horizontal plane and no verticalbends will be considered. y (cid:48)(cid:48) − y · k = 0 (19)The principle problem, arising from the different directions of the Lorentz force in the two transverseplanes of a quadrupole field is sketched in Fig. 14. It is the task of the machine designer to find an ade-quate solution to this problem and define a magnet pattern that will provide an overall focusing effect inboth transverse planes. Fig. 14:
Field configuration in a quadrupole magnet and the direction of the focusing and defocusing forces inboth planes.
Following quite closely the example of the classical harmonic oscillator, we can write down the so-lutions of the above mentioned equations of motion. We refer for simplicity to the situation in the hori-zontal plane; a "focusing" magnet is thus focusing in this horizontal plane and at the same time defocus-ing in the vertical one. Starting with initial conditions for the particle amplitude x and angle x (cid:48) in frontof the magnet element we obtain the following relations for the trajectory inside the magnet: x ( s ) = x · cos ( (cid:112) | K | s ) + x (cid:48) · (cid:112) | K | sin ( (cid:112) | K | s ) (20) x (cid:48) ( s ) = − x · (cid:112) | K | sin ( (cid:112) | K | s ) + x (cid:48) · cos ( (cid:112) | K | s ) . (21)10ere - for convenience and following the usual treatment in literature - the strong and weak focus-ing effects, k and /ρ , are merged in one parameter, K = k + 1 /ρ . Usually these two equations arecombined in a more elegant and convenient matrix form. (cid:18) xx (cid:48) (cid:19) s = M foc · (cid:18) xx (cid:48) (cid:19) , (22)where the matrix M foc contains all relevant information about the magnet element: M foc = (cid:32) cos ( (cid:112) | K | ) s √ | K | sin ( (cid:112) | K | ) s − (cid:112) | K | sin ( (cid:112) | K | ) s cos ( (cid:112) | K | ) s (cid:33) . Schematically the situation is visualised in Fig. 15.
Fig. 15:
Schematic principle of the effect of a focusing quadruole magnet.
In the case of a defocusing magnet we obtain in full analogy, (cid:18) xx (cid:48) (cid:19) s = M defoc · (cid:18) xx (cid:48) (cid:19) (23)with M defoc = (cid:32) cosh ( (cid:112) | K | s ) √ | K | sinh ( (cid:112) | K | s (cid:112) | K | sinh ( (cid:112) | K | ) s cosh ( (cid:112) | K | ) s (cid:33) , see Fig. 16. Fig. 16:
Schematic principle of the effect of a de-focusing quadrupole magnet.
For completeness we include finally the case of a field free drift of length s . Putting K = 0 weobtain M drift = (cid:18) s (cid:19) . This matrix formalism allows in a quite elegant way to combine the elements of a storage ringand calculate straight forward the particle trajectories. As an example we refer here to the simple caseof an alternating focusing and defocusing lattice, a so-called FODO lattice. Knowing the properties of11very single element in the accelerator, we can establish the corresponding matrices and calculate stepby step the amplitude and angle of the single particle trajectory around the ring. Even more convenientand mathematically simple, we can multiply out the different matrices and – given the initial conditions x , x (cid:48) – get directly the trajectory at any location in the ring, e.g., M total = M foc · M drift · M dipole · M drift · M defoc . . . (24)Schematically the trajectory obtained is shown in Fig. 17 . Fig. 17:
Calculated particle trajectory in a simple storage ring.
Several facts have to be emphasised in this context:– At each moment, or in each lattice element, the trajectory follows a harmonic oscillation.– However due to the different restoring or defocusing forces, the solution will look different at eachplace.– In the linear approximation that we refer to in this context, each and every particle will feelthe same external fields and their trajectories differ only due to their different initial conditions.– There seems to be an overall oscillation in both transverse planes while the particle is travellingaround the ring. Its amplitude stays well within the boundaries set by the vacuum chamber and itsfrequency, which in the example of Fig. 17 is roughly 1.4 transverse oscillations per revolution,corresponds to the eigen-frequency of the particle under the influence of the external fields.Coming closer to a real existing machine, we show in Fig. 18 the orbit that has been measuredduring one of the first injections into the LHC storage ring. In the upper part of the figure the horizontal,in the lower part the vertical orbit oscillations are plotted on a scale of ± mm. Every histogram barindicates the value of a beam position monitor at a certain location in the ring: The orbit oscillations areclearly visible. Counting (or better fitting) the number of oscillations in both transverse planes we obtainthe following values of Q x = 64 . (25) Q y = 59 . . (26)These values, which describe the eigen-frequencies of the particles, are called horizontal and verticaltune. Knowing the revolution frequency we easily can calculate the transverse oscillation frequencies,which for this type of machine usually lies in the range of kHz.As the tune characterises the particle oscillations under the influence of all external fields it is oneof the most important parameters of the storage ring. Therefore, usually it is displayed and controlledat any time in the control system of such a machine. Figure 19 shows as an example the tune signal ofthe HERA proton ring [3]. It is obtained via a Fourier analysis of the spectrum measured from the signalof the complete particle ensemble. The peaks obtained indicate the two tunes in the horizontal andvertical plane of the machine and in a sufficiently linear machine a quite narrow spectrum is obtained.12 ig. 18: Horizontal (top) and vertical (bottom) closed orbit oscillations, measured in LHC during the commission-ing of the machine.
Fig. 19:
Tune signal of a proton storage ring (HERA-p).
Referring for a milli-second again to Fig. 17, the question arises, how the trajectory of the particlewill look like for the second turn, or for the third .... or for an arbitrary number of them. Now, as we aredealing with a circular machine the amplitude and angle, x, x’, at the end of the first turn are the initialconditions of the second turn and so on. And after many turns the overlapping trajectories begin to forma pattern, like in Fig. 20, that looks indeed like a beam having here and there a larger and smaller beamsize but still being well defined in its amplitude by the external focusing forces.And to make a long story short [4], a mathematical function, called β or amplitude function, canbe defined that describes the envelope of the single particle trajectories. Fig. 20:
Many single particle trajectories form in the end a pattern that corresponds to the beam size in the ring.
Referring to this new variable, β , we can re-write the equation for a particle trajectory in its transverseoscillations as x ( s ) = √ (cid:15) (cid:112) β ( s ) · cos ( ψ ( s ) + φ ) , (27)13here ψ describes the phase of the oscillation, φ its initial phase condition and (cid:15) is a characteristicparameter of the single particle, or considering a complete beam now, of the ensemble of particles.Indeed (cid:15) describes the space occupied by the particle in the transverse (here simplified two dimensional)phase space x, x’. And a bit more correct: The area in x, x’ space that is covered turn by turn bythe particle’s coordinates is given by A = π · (cid:15) (28)and as long as we consider conservative forces acting on the particle this area according to Liouville’stheorem is constant. Here we take these facts as given, but we would like to point out that as a directconsequence the so-called emittance (cid:15) cannot be influenced with what ever external fields. It is a propertyof the beam and we have to take it as given . . . and handle it with care.A bit more precise and following the usual treatments, we can draw the phase space ellipse ofthe particles transverse motion, like for example illustrated Fig. 21. The parameter α is used in textbooks to describe the derivative of the β − function, α = − β (cid:48) and γ = α β . While the shape andorientation is defined by these optics functions, the area covered in phase space is constant – as long asconservative forces are considered. The concept of phase space is treated in much more detail in a laterchapter of this book . . . including the fact that often we prefer to call it trace space instead. Fig. 21:
Ellipse in x-x’ phase space.
Talking a bit more about the beam as an ensemble of many (typically ) particles, and referringto Eq. (27), at a given position in the ring the beam size is defined by the emittance ε and the beta function β . Thus for a moment in time the cosine term in Eq. (27) will be one and the trajectory amplitude willreach its maximum value. Now assuming that we consider a particle at one sigma of the transversedensity distribution, and referring to the emittance of this reference particle we can calculate the size ofthe complete beam in the sense that the complete area (within one sigma) of all particles in (x, x’) phasespace is surrounded (and thus defined) by our one sigma candidate. Visualised a bit more qualitatively,we refer to Fig. 22. A small emittance means small amplitudes of the particles’ trajectories and smalldivergence. The more parallel (Laser-like) and the more dense, the better; as easy as that.In this sense the value (cid:112) (cid:15) · β ( s ) will define the one sigma beam size in the transverse plane. Asan example the values for the LHC proton beam are used: In the periodic pattern of the arc the betafunction is β = 180 m and the emittance at flat top energy is roughly ε = 5 · − rad m. The resultingtypical beam size therefore is 0.3 mm. Now, clear enough, we will not design a vacuum aperture ofthe machine based on one sigma beam size. Typically an aperture requirement corresponding to 12 σ is a good rule to guarantee sufficient aperture including tolerances from magnet misalignments, opticserrors and operational flexibility. In Figure 23 the LHC vacuum chamber is shown, including the beamscreen to protect the cold bore from synchrotron radiation. Its aperture radius corresponds to a minimumnumber of 18 σ beam size. 14 ig. 22: Schematic picture of the trajectories in a beam. Small emittance means high quality of the particleensemble, which in turn means small amplitudes and angles of the trajectories.
Fig. 23:
The LHC vacuum chamber with the beam screen to shield the super conducting magnet bore from syn-chrotron radiation.
Beyond the beam quality, expressed as "emittance" in the previous paragraph, the beam current, stored oraccelerated in our machine is subject to limitations. Keywords here are wake fields, machine impedance,and single and multi-bunch instabilities related to that. The topic is worth a book of its own (see e.g.Ref. [5]) and we only can mention a simple example here. Until now we have described the beamas an ensemble of individual particles, moving free through the accelerator. This picture however isoversimplified. In reality the beam is a highly charged dollop of particles and, as a matter of fact, talksvia its electro-magnetic field to the environment. While this is usually metallic - first of all clearlythe vacuum chamber - an image current is induced, running in parallel to our beam and creating atany moment funny shapes of field lines that act back on our primary beam. The effect is described as"impedance" and the corresponding fields can severely distort the stability of the beam. Qualitatively, asdescribed in Fig. 24, the effect is most serious whenever changes in the geometry of the environmentoccur. However, as soon as we accumulate more and more particles in our machine, we will sooner orlater hit the limit of beam stability, created by wake fields.
Fig. 24:
Schematic view of the wake fields induced due to a sudden change of the vacuum chamber geometry. P b = I πε v ln db (29)where I denotes the stored beam current, v the velocity of the bunch and d and b the vacuum chamberradius as indicated in the figure. Particle Colliders
The easiest way to do physics using particle accelerators, is to bang the accelerated beam onto a targetand analyse the resulting events. While in high energy physics we nowadays do not apply this tech-nique very often anymore it still plays an essential role in the regime of atomic and nuclear physicsexperiments. The advantage is: It is quite simple, once the accelerator has been designed and built andthe produced particles are easily separated due to the kinematics of the reaction. Schematically the situ-ation is shown in Fig. 25. The particle "A" which is produced and accelerated in the machine is directedonto the particle "B" which is at rest in the laboratory frame. The produced particles are named "C" and"D" in the example.
Fig. 25:
Schematics of a moving particle A colliding with a target particle B at rest. While the set up of such a scheme is quite simple, it is worth studying a bit the available energy,in the centre of mass system. The relativistic overall energy is given by E = p c + m c , (30)which is true for a single particle but equally valid for an ensemble of particles. Now, most important,the rest energy of the particle ensemble is constant (sometimes called invariant mass of the system ).Considering now the system of the two particles colliding, we therefore can write ( E cma + E cmb ) − ( p cma + p cmb ) c = ( E laba + E labb ) − ( p laba + p labb ) c . (31)In the frame of the centre of mass system we get by definition p cma + p cmb = 0 (32)while in the laboratory frame where particle "B" is at rest we have simply p labb = 0 . (33)The equation for the invariant mass therefore simplifies to W = ( E cma + E cmb ) = ( E laba + m b · c ) − ( p laba · c ) . (34)16nd neglecting for convenience the rest masses of the two particles, we get the quite simple expression W ≈ (cid:113) E laba · m b · c . (35)In other words the energy that is available in the centre of mass system depends on the square root ofthe energy of particle "A", which is the energy provided by the particle accelerator. A quite unsatisfactorysituation ! Fig. 26:
Schematics of the collision of two colliding particle beams with equal energies.
As a consequence, the design of modern high energy accelerators was naturally concentrated onthe development of particle colliders, where two counter rotating beams are brought into collision at oneor several interaction points (Fig. 26).Calculating again, for the case of two colliding beams of equal particles and energies the availableenergy in the centre of mass system, we get ( p cma + p cmb ) = 0 (36)and by symmetry of the situation as well ( p laba + p labb ) = 0 (37)and so the full energy delivered to the particles in the accelerator is available during the collision process. W = E laba + E labb = 2 · E laba . (38)A “typical” example of a high energy physics event in such a collider is shown in Fig. 27. Fig. 27: "Typical" event observed in a collider ring: A Higgs particle in the ATLAS detector. .2 Limit VII: The un-avoidable physics detectors While it is quite clear that a particle collider ring is a magnificent machine in the quest for higher energies,there is a small problem involved, called “particle detector”. In the arc of the storage ring usually wecan find a nice pattern of magnets that provide us with a well defined beam size, expressed as beta-function. However special care has to be taken as soon as our colleagues from the high energy physicswant to install a particle detector to analyse the events. Especially, working at the energy frontier, justlike the accelerators, these devices tend to grow considerably in size. In Figure 28 the largest particledetector installed in a storage ring is shown as impressive example: The ATLAS detector at LHC.
Fig. 28:
ATLAS detector at LHC: 46 m in length, overall weight 7000 t.
The design of the storage ring has to provide the space needed by the detector hardware and atthe same time create the smallest achievable beam spots at the collision point, which is usually right inthe centre of the detector. Unfortunately, these requirements are a bit contradictory: The equation forthe luminosity of a particle collider depends on the stored beam currents, I , I , and the transverse spotsize of the colliding beams at the Interaction Point (IP), σ ∗ x , σ ∗ y : L = 14 πe f b · I I σ ∗ x σ ∗ y . (39)The parameters f and b describe the revolution frequency of the ring and the number of stored bunches.Unfortunately, however, the beta function in a symmetric drift grows quadratically as a function ofthe distance s between the position of the beam waist and the first focusing element, namely: β ( s ) = β + s β . (40)The smaller the beam size at the IP the faster it will grow until we can apply – outside the detector region– the first quadrupole lenses. For the beam size this translates into σ ( s ) = σ (cid:115) s − s ) β , (41)where s refers to the position of the beam waist, e.g. in a collider the interaction point of the two beams.Schematically this fact is shown in Fig. 29.As a consequence this behaviour sets critical limits to the achievable quadrupole aperture - or,given the aperture, for the achievable quadrupole gradient. Therefore the focusing lenses right beforeand after the IP, are placed as close as possible to the detector, and in general they are the most critical18 ig. 29: The beam envelope in the neighborhood of a symmetric waist: the smaller the beta function at the IP,the faster the beam size is growing. and most expensive magnets in the machine: Their aperture need determines in the end the luminositythat can be delivered by the storage ring.For the experts we would like to add, that even if the bare aperture need can be fulfilled, anotherlimit is usually reached: the resulting chromaticity that is created in the mini-beta insertion and the sex-tupole strengths that we will need to correct it, usually present due to their non-linear fields serious limitsfor the stability of the particle motion; the resulting so-called “ dynamic aperture ” is the next limit thatwe will face.
The rate of physics events, generated in a particle collision process, does not only depend on the charac-teristics of the colliding beams, but first of all, on the probability to create such an event, the so-calledcross section of the process.In the case of the Higgs particle, without doubt the high light of LHC Run1, the overall crosssection is plotted in Fig. 30.
Fig. 30:
Cross section of the Higgs for different production processes (court. CMS collaboration). Σ react (cid:39) pb. (42)During the three years of the LHC Run1, namely in the years 2011-2013, an overall luminosity of (cid:90) Ldt = 25 f b − (43)has been accumulated.Combining these two numbers in the sense that the event rate of a reaction is given by R = L · Σ react we get an overall number of produced Higgs particles of "some thousand". For a Nobel prizewinning event just at the edge of a reliable statistics. As a consequence the particle colliders have tobe optimised not only for highest achievable energies but at the same time for maximum stored beamcurrents and small spot sizes at the interaction points to maximise the luminosity of the machine. Following the argumentation above, the design goal will be to prepare, accelerate and store two counterrotating particle beams to profit best from the energy of the two beams at the collision process. Still, thereis a prize to pay: unlike fixed target experiments, where the "particle" density of the target material isextremely high, in the case of two colliding beams the event rate is basically determined by the transverseparticle density that can be achieved at the interaction point.Assuming Gaussian density distributions in both transverse planes the performance of such a collider isdescribed by the luminosity formula Eq. (39).While the revolution frequency f and the number of bunches per beam b are in the end determinedby the size of the machine, the stored beam currents I and I and the beam size at the interaction point σ ∗ x , σ ∗ y have their own limitations.The most serious one is the beam-beam interaction itself. During the collision process the indi-vidual particles of counter rotating bunches feel the electro-magnetic field of the opposing bunch. Inthe case of a proton-proton collider this strong fields act like a defocusing lens, and have a strong impacton the tune of the bunches [6]. Fig. 31:
Schematic view of the beam-beam interaction during the crossing of bunch trains.
In Figure 31 the situation is shown schematically. Two bunch trains collide at the IP and duringthe collision process a direct beam-beam effect is observed. Even more, in addition to that, before andafter the actual collisions, long range forces exist between the bunches that have a non-linear componentas illustrated in Fig. 32. As a consequence [6] the tune of the beams is not only shifted with respectto the natural tune of the machine, but spread out, as different particles inside the bunches see differentcontributions from the beam beam interaction.In the tune diagram, which shows the horizontal and vertical tune in a common plot, therefore,we obtain not a single spot anymore, representing the ensemble of the particles, but a large array, thatdepends in shape, size and orientation on the particle densities, the distance of the bunches at the long-range encounters and on the single bunch intensities. The effect has been calculated for the LHC and isshown in Fig. 33. 20 ig. 32:
Beam-beam force as a function of the transverse distance of the particle to the centre of the opposingbunch.
Fig. 33:
Calculated tune shift due to the beam-beam interaction in LHC.
In linear approximation, i.e. referring to a distance of about 1-2 σ the beam-beam force in Fig. 32can be linearised and acts like a defocusing quadrupole effect. Accordingly a tune shift can be calculatedto characterise the strength of the beam-beam effect in a collider. Given the parameters described aboveand introducing the classical particle radius r p , the amplitude function at the interaction point β ∗ andthe Lorentz factor γ , we can express the tune shift due to the linearised beam-beam effect by ∆ Q y = β ∗ y · r p · N p π · γ ( σ x + σ y ) σ y . (44)In the case of LHC, with a number of particles per bunch of N p = 1 . ∗ , the design value ofthe beam-beam tune shift is ∆ Q = 0 . and in daily operation the machine is optimised to run closeto this value, which puts the ultimate limit for the achievable bunch intensities in the collider.21 Electron Synchrotrons, Lepton Colliders and Synchrotron Light Sources
While in proton or heavy ion storage rings the design can follow more or less the rules that have beendiscussed above, the case changes drastically as soon as the particles gain more and more energy. Benton a circular path, especially electrons will emit an intense light, so-called synchrotron radiation, thatwill have a strong influence on the beam parameters as well as on the design of the machine.Summarising here only briefly the situation, the power loss due to this radiation depends on the bendingradius and the energy of the particle beam: P s = 23 α (cid:126) c γ ρ , (45)where α describes the fine structure constant, ρ the bending radius in the dipole magnets of the ringand γ is the relativistic Lorentz factor. As a consequence the particles will lose energy turn by turn. Tocompensate these losses, RF power has to be supplied to the beam at any moment. An example thatvisualises nicely the problem is shown in Fig. 34. It shows the horizontal orbit of the former LEP storagering. The electrons, traveling from right to left in the plot lose a considerable amount of energy in eacharc and as a consequence are deviating from the ideal orbit towards the inner side of the ring. The effecton the orbit is large: Up to 5 mm orbit deviations are observed. In order to compensate for these lossesfour RF stations are installed in the straight sections of the ring to provide the necessary power. Fig. 34:
Measured horizontal orbit of the LEP electron beam. Due to synchrotron radiation losses the particle orbitis shifted towards the inner side of the ring in each arc.
The strong dependence of the synchrotron light losses on the relativistic γ factor sets severe limitsto the beam energy that can be carried in a storage ring of given size. And pushing for even higherenergies means either the design of storage rings that are even larger than the LEP or, in order to avoidsynchrotron radiation, the design of linear accelerating structures. To make it even more clear: Fora given maximum power that can be re-supplied to the beam (and paid by the laboratory), a factor of twoincrease of the energy of the stored electrons , means an increase of a factor of four in the size of thestorage ring to be build.At present the next generation particle colliders are being studied [7] and the ring design of thisFuture Circular Collider (FCC) foresees a 100 km ring to carry electrons (and positrons) of up to 175GeV beam energy. The dimension of this storage ring is far beyond of what has been designed up tonow. The impressive sketch of the machine layout is shown in Fig. 35 where the dashed circle refers tothe foreseen geometry of the 100 km ring and the solid one represents the little LHC machine.For the maximum foreseen electron energy of E = 175 GeV, the synchrotron radiation would leadto an energy loss of 8.6 GeV per turn or an overall power of the radiated light of 47 MW at full beamintensity. Along with the power loss of the beam due to radiation goes a severe impact on the beamemittance. In hadron machines (in general, in beams where the energy loss due to synchrotron radiationis negligible) the beam emittance as defined above can be considered as constant for a given energy. Even22 ig. 35: Schematic view of a 100 km long ring design in the Geneva region for the FCC study. better, if accelerated, it will shrink inversely as a function of the energy: ε ∝ γ ( hadrons ) . (46)In lepton rings just the opposite is true: Under the influence of the emitted radiation, which is a quantumeffect, an equilibrium is obtained between radiation damping and excitation and we get ε = 5532 √ (cid:126) cm e c γ J x ρ β { D + ( β ( D (cid:48) + αD ) } ( electrons ) . (47)A kind of formidable equation; however if we focus on the essential issue here - namely the energydependency - we can simply point out that the emittance of a lepton beam in a ring is increasing quadraticas a function of the energy, ε ∝ γ which makes it more and more difficult to achieve high luminosity orhigh brightness of the emitted light in an electron ring if we get to higher energies. As far as lepton beams are concerned, ring colliders suffer from the severe limitation of synchrotronradiation and at a certain moment the construction of such a large facility does not seem reasonableanymore. In order to avoid the synchrotron radiation, therefore, linear structures that had been discussedabove and used in the infancy of particle accelerators are en vogue again. Still the advantage of circularcolliders cannot completely be neglected: Even with a modest acceleration gradient in the RF structuresthe particles will get turn by turn a certain push in energy and they will reach some when the desired flattop energy in the ring.In a linear accelerator this repetitive acceleration is by design not possible. Within a single passthrough the machine the particles will have to be accelerated to full energy. In order to keep the structurecompact highest acceleration gradients therefore will be needed. One of the most prominent examples,proposed for a possible future collider is the CLIC design [8]. Within one passage through the 25 kmlong accelerator the electron beam will get up to 3 TeV; and the same holds for the opposing positronbeam. An artist view of tis machine is shown in Fig. 36.23 ig. 36:
Proposed location of the CLIC linear collider along the Jura mountain in Geneva region.
Table 1:
The main parameters of the CLIC study.
500 GeV 3 TeV
Site Length 13 km 48 kmLoaded accel. gradient [MV/m] 12Beam power/beam [MW] 4.9 14Bunch charge [ e+/e] 6.8 3.7Hor./vert. norm. emitt. [ − / − m ] 2.4/25 0.66/20Beta Function [mm] 10/0.07Beam Size at IP hor/vert [nm] 45/1Luminosity ( cm − s − . · . · Fig. 37:
Accelerating structure of the CLIC test facility CTF3; on the right side a electron microscope photo showsdamage effects on the surface, created due to discharges in the module.
The CLIC main parameters are listed in Table 1. Especially the accelerating gradient, i.e., the en-ergy gain per meter has to be emphasised. It has been pushed to the technically feasible maximumgradient, and the limit in the end is given by the breakdown of the electrical field in the acceleratingstructure.A picture of such a CLIC type structure is shown in Fig. 37. On the right hand side a photo, takenwith an electron microscope of the surface after a voltage breakdown is shown. At the spot of the spark-ing a little crater shows the possible damage of the surface and as a consequence the deterioration ofthe achievable gradient, which has to be avoided under all circumstances [9].Being considerably higher than the typical values in circular machines, the gradient of E acc = 100 MV/m in a linear machine still leads to a design of bout 50 km overall length for a maximum achievableenergy of E max = 3 TeV. 24 ig. 38: electron beam accelerated in the wake potential of a plasma cell. Up to 4 GeV are obtained within a fewcm length only [10]. Conclusion
Summarising the facts we state, that as soon as we talk about future lepton ring colliders (or to be moreprecise electron-positron colliders) the synchrotron losses set a severe limit to the achievable beam en-ergy. And very soon the size of the machine will be beyond economical limits: For a constant givensynchrotron radiation loss the dimensions of the machine have to grow quadratic with the beam energy.Linear colliders therefore are the proposed way to go. And in this case the maximum achievable ac-celeration gradient is the key issue. New acceleration techniques, namely plasma based concepts wheregradients are observed that are much higher than the present conventional techniques, are a most promis-ing concept for these future colliders. An impressive example is shown in Fig. 38. Within a plasma cellof only some cm in length, electrons are accelerated to several GeV. The gradients achievable are by or-ders of magnitude higher than in any conventional machine (see e.g., Ref. [10]). Still there are problemsto solve, like overall efficiency, beam quality (mainly the energy spread of the beam) and the achievablerepetition rate. But we are convinced that it is worth studying this promising field. And this is what thisschool was organised for.