LONGITUDINAL DYNAMICS OF PARTICLES IN ACCELERATORS. Author: Urša Rojec Mentor: Simon Širca. Ljubljana, November 2012

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1 Seminar I a, četrti letnik, stari program LONGITUDINAL DYNAMICS OF PARTICLES IN ACCELERATORS Author: Urša Rojec Mentor: Simon Širca Ljubljana, November 2012 Abstract The seminar focuses on longitudinal motion of charged particles in particle accelerators. The technique of acceleration by electromagnetic waves is explored and the stability of motion under such acceleration is inspected. The seminar introduces the concept of ideal particle and develops equations that treat deviations from its motion. 1

2 Contents 1 Introduction 2 2 Acceleration methods Some comments on acceleration by static fields Acceleration by radio-frequency (RF) fields Equations of motion in phase space Path length and momentum compaction Difference equations Differential equations Small oscillation amplitudes Phase Stability Phase Space Motion Phase Space Parameters Fixed Points Momentum Acceptance Emittance, Momentum Spread and Bunch Length Acceptance Acceleration Effect of RF Voltage on Phase Space Phase Space Matching Longitudinal Gymnastics: Debunching and Bunch Rotation Conclusion 15 1 Introduction Particle accelerator physics primarily deals with interaction of charged particles with electromagnetic fields. The force that the field exerts on the charged particle is called Lorentz force and can be written as F = q [E + v B] Transverse fields 1 are used to guide particles along a prescribed path but do not contribute to their energy. Acceleration is achieved by longitudinal fields and this seminar will focus on interaction of charged particles with longitudinal fields. In the simplest case, acceleration is achieved by static electric fields. Particle that travels through a potential difference of V 0 gains qv 0 energy, if q is the particle s charge. This way of acceleration is simple but limited to 10 6 V due to voltage breakdown. It is still widely used for acceleration of low energy particles at the beginning of acceleration to higher energies. Somewhat higher voltages can be achieved by pulsed application of such fields, but for acceleration to higher energies, different methods must be exploited. Most common and efficient way for particle acceleration are high frequency electromagnetic fields in accelerating structures and this is the topic covered in the seminar. In a very general way, equations of motion will be derived and stability limits will be inspected. For a more rigorous treatment of interaction of charged particles with longitudinal fields one must turn to higher order equations and take into account losses due to interactions of particles within the beam, because of incomplete vacuum and so on, that will not be covered here. 2 Acceleration methods 2.1 Some comments on acceleration by static fields Since we are limited by maximum V 0 because of voltage breakdown and accelerator s cost goes up with every meter, the first thing that comes to mind would be to curve the particle trajectory to a circle, so that it passes the same accelerating section repetitively, as is shown in Fig. 1. The field required to bend the particle 1 Mostly magnetic, because for perpendicular orientation of the fields, this is true: F E qe and F B qvb. In accelerators we are mostly dealing with particles with velocities close to the speed of light 2

3 Figure 1 trajectory is a static magnetic field. To ensure, that the electric field is non zero only between the plates, they must extend to infinity. In reality this is not possible, and we have an electric field outside the capacitor. This fringe field will decelerate the particle when it approaches or departs the capacitor. As we anticipate according to Farraday s law E ds = B ds, C t S there will be no net acceleration, which is consistent with the conservative nature of the electrostatic field. 2.2 Acceleration by radio-frequency (RF) fields With electromagnetic waves, accelerating voltages far exceeding those obtainable by static fields can be achieved. This method of acceleration is used in linear as well as in circular accelerators. For practical reasons, specifically in circular accelerators, particle acceleration occurs in short straight accelerating sections placed along the particle path. Since a free electromagnetic wave does not have a longitudinal electrical field component, special boundary conditions must be enforced. This is done by accelerating structures, called wave-guides or resonant cavities, providing a travelling or standing EM wave respectively. In a crude approximation, a waveguide is a pipe made of conducting material and a cavity is a waveguide closed at both ends. Waveguides are used primarily in linear accelerators (linacs), while resonant cavities are used in both linacs and synchrotrons. In both cases TM modes are used, because magnetic field cannot accelerate particles. The modes are found from Maxwell s equations, with boundary conditions E = 0, B = 0 at the cavity/waveguide walls. The two most commonly used modes are represented on Fig. 2. Figure 2: A cylindrical waveguide operating in TM 01 mode (left) and a pillbox cavity operating in TM 010 mode (right) Waveguides The most commonly used strategy is to excite a waveguide at a frequency above the cut-off frequency of the lowest mode, TM 01, but below the cut-off frequency for other modes. This way only one mode is propagating trough the waveguide. In order to be able to accelerate charged particles over a reasonable distance, the wave must have the same phase velocity as the velocity of the particles. This way the particle travels along the structure with the wave and is accelerated or decelerated at a constant rate. Since phase velocity of such a wave is larger than the speed of light, waveguides are loaded with discs to make this happen Cavities The most commonly used accelerating mode is TM 010. For this mode, the electric field is directed longitudinally and has constant magnitude along z. It has no azimuthal dependence and has a maximum on the axis of the cavity, decreasing in the radial direction until it is zero at the cavity walls. Accelerating structures in a circular accelerator may be either distributed around the ring or grouped together so that the ring only has one accelerating section. In both cases, the frequency of the voltage in the accelerating structures must be an integer multiple of particle revolution frequency, where the integer is called the harmonic number and denoted by h. The harmonic number is the maximum number of bunches (groups of particles moving together in the accelerator, more on this later in the text) that can be in the accelerator at the same time. 3

4 3 Equations of motion in phase space To achieve acceleration, one must ensure constructive interaction of the particles with the wave. Because EM fields oscillate, special synchronicity conditions must be met in order to obtain the desired acceleration. Weather we have a standing or a travelling wave, its current value is determined by the phase. This means that the degree of acceleration is determined by the phase. If systematic acceleration is to be achieved, this phase must be at a specific value at the moment the particle arrives to the accelerating section. This value is called synchronous phase and denoted by ψ s. We assume that the ideal, synchronous particle arrives at each station at the same phase and thus receives the same energy boost at each station. In a circular accelerator, revolution frequency of the particles and the RF frequency must be related by ω rf = hω rev, (1) where h is the harmonic number. It represents the number of times, per particle revolution period, that the RF voltage is at the correct level to accelerate particles. In the following discussion, we will introduce the concept of a synchronous particle with the ideal energy and phase, and develop equations of motion that treat deviations from its trajectory. 3.1 Path length and momentum compaction In the case of a circular 2 accelerator we come to the problem of momentum-dependent path length. The dependence arises in bending dipoles required to keep the particles on a circular path. When a charged particle enters a homogeneous magnetic field, the field exerts the Lorentz force on it and bends the particle trajectory. The radius of the bend depends on the particle s charge, as well as its velocity. Since the dipoles are tuned to a so called ideal particle, any particle with momentum different than the momentum of the ideal particle will not follow the designed path. We will denote the deviation of the particle from its ideal path by x = D(s) p p 0, where p 0 represents the momentum of the ideal particle and D(s) represents the dispersion function. It describes the effect that bending magnets have on the particles trajectory. We need not concern ourselves with more detail, and can take it as a machine parameter. The total path length can now be written as L0 ( L = 1 + x(s) ) ds, ρ 0 where ρ is the radius of the bend. We can immediately see that for an ideal particle with p = 0 the path length is just L 0. which is the ideal design circumference of the accelerator. The deviation from the ideal path can thus be obtained by integrating the second term. L = p p 0 L0 0 D(s) ρ(s) ds, To measure the variation of the path length with momentum, we define the momentum compaction factor α c = L/L 0 = 1 D(s) ds = D(s), (2) p/p 0 L 0 ρ(s) ρ The momentum compaction factor is non-zero only in curved sections, where ρ is finite. In the case of a LINAC the curvature (κ = 1/ρ) is 0, ρ = and the momentum compaction factor vanishes. However, when dealing with phase advance from one accelerating section to the other, we are not so much interested in the deviation of the path length, but would rather know the time it takes for a particle to travel between two successive accelerating sections separated by a distance L. This time is given by the equation τ = L/v. By differentiating the logarithmic version of the equation we get: τ τ = L L v v. 2 Path length along a straight line also depends on the angle that the particle trajectory encloses with the line. This, however, is a second order correction, so it will be neglected here. 4

5 The first term on the right is just momentum dependent path length that we derived earlier. To connect the second term to momentum deviation, wee need only to differentiate the momentum (p = mγv), and we get, after some manipulation: v v = 1 p γ 2 p, where γ is the Lorentz factor. We can see that both α c and γ appear as a factor before momentum deviation. We can thus put them together to form the momentum compaction and get the final expression for time deviation η c = 1 γ 2 α c, (3) τ τ = η c p p. (4) Momentum compaction vanishes when γ t = 1 αc (5) This is the transition Lorentz factor. From special relativity we know that γ = E total E rest, and so γ t is usually referred to as transition energy. Transition energy is very important when it comes to phase focusing. 3.2 Difference equations In order to derive the equations in longitudinal phase space, we take a look at phase and energy advance between successive passes through the RF cavities. The energy deviation and phase at the entrance to the (n + 1) th cavity can be expressed as: ψ n+1 = ψ n + ω rf (τ + τ) n+1, = ψ n + ω rf τ n+1 (1 + [ ] ) τ, τ n+1 E n+1 = E n + e [V (ψ n ) V (ψ s )], where ψ s is the synchronous phase and V (ψ) is the RF waveform. Since the synchronous particle always stays in phase and ω rf τ is the phase advance of the synchronous particle, we can rewrite the phase advance in the equation (6) as [ ] τ φ n+1 = φ n + ω rf τ n+1. (7) τ n+1 Here ω rf τ is the phase advance from cavity to cavity. For a circular accelerator with only one accelerating section, it must be an integer multiple of 2π (harmonic number), so that we satisfy the synchronicity condition The final form of the equations can be obtained by using equation (4) and the relationship E/E = β 2 p/p in (7) φ n+1 = φ n ω rf τη c p n+1 p s E n+1 = E n + e [V (φ n ) V (φ s )], where p s is the momentum of the synchronous particle. Both equations are coupled. This can be seen if we replace the energy deviation with momentum deviation in (8), by noting that β cp = E. p n+1 = p n + e βc [V (φ n) V (φ s )]. (9) (6) (8) 5

6 3.3 Differential equations Typically the phase and energy change by small amounts at each pass of the accelerating section, which allows us to treat them as continuous variables. We can then approximate the difference equations by differential ones, using n as the independent variable. We can rewrite equations (8) as: dφ dn d p dn = η cω rf τ p, p s = e βc [V (φ) V (φ s)]. In most practical cases, parameters like particle velocity or its energy vary slowly during the acceleration, compared to the rate of the change in phase. We can thus consider them constant an obtain a single second order differential equation from equations in (10): (10) d 2 φ dn 2 + η cω rf τe [V (φ) V (φ s )] = 0. (11) βcp s We can not go much further without making an assumption about V (φ). Since RF fields are created in accelerating cavities, we will assume a sine function. V = V 0 sin(φ). Let us now take a look at particle movement in phase space. We will take the simplest case and say that the only way that the particle s energy can change, is trough interaction with the RF field. If we rewrite the phase as φ = ϕ + φ s and expand the trigonometric term, the equation of motion in phase space becomes: ϕ + η cω rf ev 0 τβcp s (sin φ s cos ϕ + cos φ s sin ϕ sin φ s ) = 0, (12) where we have also changed from number of turns n to time t as independent variable, by noting that 4 Small oscillation amplitudes d dn = dt d dn dt = τ d dt. (13) To get some insight into the solutions and the stability of motion, we first take a look at small oscillations about the synchronous phase. Since ϕ is small we can approximate the sine and cosine term by their Taylor expansions. Keeping only the linear terms in ϕ we obtain an equation for a harmonic oscillator: where we have defined the synchrotron oscillation frequency as For real values of Ω we have a simple solution ϕ + Ω 2 ϕ = 0, (14) Ω 2 = η cω rf ev 0 τβcp s cos φ s. (15) ϕ = ϕ 0 cos(ωt + χ i ), (16) where χ i is some general phase that we will set to zero. Since φ = φ s + ϕ and φ s is constant, it is true that φ = ϕ we can construct an equation for momentum error, from (10,13) δ = p p s = ϕ η c ω RF = Ωϕ 0 η c ω RF sin(ωt + χ i ) (17) If Ω is real, both phase and particle momentum oscillate about the ideal value with synchrotron frequency - we have stable oscillations. If we join solutions for phase and momentum error, we get an invariant of motion ( ) 2 ( ) 2 ϕ δ ± = 1 (18) ϕ 0 δ 0 It describes particle trajectories in phase space. They can be ellipses or hyperbolas, depending on Ω. Ellipses represent stable motion in case of a real Ω and hyperbolas represent unstable motion, when Ω is imaginary. 6

7 Figure 3: Synchrotron oscillations in (δ, ϕ) phase space for small deviations from the synchronous phase. Trajectories for real (left) and imaginary (right) values of synchrotron frequency In Fig. 3 on the left graph for stable oscillations we can clearly see separatrices enclosing the area of stable motion. In accelerator physics, areas where stable motion exists are called buckets. All the particles sharing a bucket are called a bunch. Maximum number of bunches that can be in a circular accelerator at the same time is determined by the harmonic number (1). 4.1 Phase Stability In the previous sections, when deriving the equation for small oscillation amplitudes, we have already established that in order to have stable oscillations, synchrotron frequency needs to be real. Taking a closer look at the expression for Ω 2 (15) we can see that besides cos φ s and η c all other quantities are non-negative. Therefore these two will determine whether the motion is stable or unstable. Momentum compaction, η c, goes to zero (3) when particles cross the energy of γ t = 1 αc. (19) When this happens, the travel time from one accelerating gap to another does not depend on the particle momentum. There is no phase stability at this energy. This is a machine dependent parameter. LINACs do not have a transition energy because they are straight. Synchronous phase must be selected according to γ of the particles to obtain stable oscillations and acceleration. If RF frequency is represented by a sine wave, we have V > 0 if 0 < φ < π, hence 0 < φ s < π 2 for γ < γ t, π 2 < φ s < π for γ > γ t. For electrons the transition energy is in the range of MeV and for protons in in the range of GeV. Crossing the transition presents us with many technical problems. Since γ t for electrons is relatively small, electrons are injected to electron synchrotrons above the transition energy, thus avoiding stability problems during acceleration. This is not the case with protons. A LINAC with proton energy of 10 GeV would be very costly, so protons are usually injected into the synchrotron below γ t. An oscillating accelerating voltage, together with a finite momentum compaction produces a stabilizing focusing force in the longitudinal degree of freedom. This is the principle of phase focusing represented in Fig. 4. We now explore the effect that going over transition energy (5) has on phase focusing. compaction η c (3) changes sign when γ = γ t : The momentum η c > 0 for γ < γ t and η c < 0 for γ > γ t Below the transition energy, the arrival time is determined by the particle s velocity. After transition, the particle has a velocity close to c and its arrival time depends more on the path length than on its speed. The key difference is this: 7

8 Figure 4: Phase focusing principle. If the particle is lagging behind the synchronous particle it will see a higher accelerating voltage. This will cause it to gain more energy and because it will travel faster, deviation of its phase with respect to synchronous particle will decrease. The same principle can be applied to particles that are too fast, except that they gain less energy. A particle with momentum higher than that of the ideal particle will arrive at the acceleration section faster than the ideal particle if we are below the transition. If we are above the transition, because η c is negative, it will arrive after the ideal particle. This can be clearly seen from equation (4). In Fig. 4 the basic idea of phase focusing is introduced. We can clearly see that in order to obtain focusing, slow particles must arrive later and faster particles sooner. If the particles were to cross the γ t and RF voltage would remain the same, the motion would no longer be stable. Slower particles would be accelerated less, and faster particles more than the ideal particles. 5 Phase Space Motion The equation of motion (12) describes the particle motion in ( p, ϕ) phase space. There are two distinct cases, one where synchronous phase is set to 0, and the other when it is not. If φ s = 0, the synchronous particle will experience the voltage of V = V 0 sin φ s = 0 when it passes the accelerating section. We call this the stationary case. For all values of φ s that are not integer multiples of π, particles will be accelerated or decelerated, depending on the phase. In order to accelerate the particles, the synchronous phase must be set to a value other than nπ. Particle accelerators consist of many elements. For example, a light source would consist of an injector linac that would accelerate particles to E i. After that, the particles are transferred to a booster synchrotron that accelerates them from E i to their final energy, E f. When E f is reached, the particles are again transferred to a storage ring. The function of the storage ring is to keep the particles orbiting at constant speed. Because of synchrotron radiation, electron storage rings also contain accelerating sections to make up for energy lost due to synchrotron radiation. Since protons are much heavier (factor of 10 3 ev) and the energy lost per turn for synchrotron radiation scales as γ 4 /ρ effect of synchrotron radiation on their energy is negligible. Still RF sections are needed to provide phase focusing. By using difference equations (8) we can make simulations to take a look at particle motion in stationary and accelerating case. The following simulations were done for the Fermilab Tevatron ring, which is a proton accelerator. On the left graph in Fig. 5 we can see an example of a stationary bucket. Particles with φ = φ s Figure 5: Phase space plot for a stationary (left) and an accelerating bucket (right). are not accelerated. The phase stable region is 2π in extent and particles that find themselves out of the bucket will undulate in energy and diverge in phase. They may stay in the ring indefinitely. 8

9 Comparing the phase space plot for an accelerating bucket to the one for the stationary bucket on Fig. 5, we can see a significant change in the shape of the separatrix as well as its origin. The center of the bucket is no longer at (0, 0) but is shifted by φ s. Particles that find themselves out of the bucket will diverge in phase as well as in energy. In contrast to the stationary bucket case, these particles will eventually gain too much energy and leave the circular accelerator. The graphs in Fig. 5 were plotted for a proton energy below the transition energy of the accelerator, γ < γ t. For γ > γ t the orientation of the buckets changes. Fig. 6 shows the bucket shape before and after the transition: Figure 6: Shape of non stationary buckets before and after transition 5.1 Phase Space Parameters The equation of motion (12) can be derived from a Hamiltonian H = ϕ2 2 Ω2 cos φ s [cos(φ s + ϕ) cos φ s + ϕ sin φ s ]. (20) For the stationary case, Hamiltonian (20) simplifies and is identical to that of a mechanic pendulum: H = ϕ2 2 Ω2 cos ϕ. (21) In Fig. 7 potential for accelerating and stationary case is shown, lines representing equipotential surfaces. Figure 7: Potential well for a stationary (left) and accelerating bucket (right). which is the result of an additional linear term in (20) Accelerating potential is tilted compared to stationary, Fixed Points In the stable phase space regions, particles oscillate about the synchronous phase and ideal momentum as can be seen from equations (16,17). Within the stable regions, we can find two fixed points, one stable and one unstable. They can be calculated from H ϕ = 0, H ϕ = 0. (22) The two fixed points correspond to minima (sfp - stable fixed point) and saddles (ufp - unstable fixed point) in the potential represented on Fig. 7. From conditions (22) we obtain coordinates for fixed points in (ϕ, ϕ). sfp is located at (φ s, 0) and ufp at (π φ s, 0). 9

10 Figure 8: Characteristic bucket and separatrix parameters Momentum Acceptance ϕ is proportional to p/p 0. Maximum momentum acceptance can thus be found by differentiation of the hamiltonian, (20), with respect to ϕ. At the extreme points, where the momentum reaches a minimum or a maximum, we have ϕ/ ϕ = 0 and the contition is ϕ = 0. From Fig. 8 we see that the maximum phase elongation occurs at the ufp where ϕ is zero. Maximum momentum acceptance can then be found by equating the values of the hamiltonian for the ufp and the derived condition. We get: Emittance, Momentum Spread and Bunch Length 1 [ ( π ] 2 ϕ2 = 2Ω 2 1 s) 2 φ tan φ s. (23) The dynamics of both the stationary and the accelerating case can be described with the help of Hamiltonian equations. Following the Liouville theorem that states that the area in phase space is conserved for a system that can be described by Hamiltonian equations (this follows directly from Hamilton equations, since v = 0), we define the longitudinal emittance as the area of phase space enclosed by the beam. In what follows, we will derive the relationships between the rms momentum and the 95% emittance of a bunch. The derivations are valid for a distribution where the emittance is much smaller than the bucket area, so that effects due to non-linearities of the RF focusing and large tails can be ignored, and where the distribution of the bunch is assumed to be Gaussian. We start by evaluating (20) for small angles and obtain ϕ 2 + Ω 2 ϕ 2 = const. Since the bunch spread is usually measured in t instead of in terms of phase deviation, we will switch to ( E, t) phase space. This is important, because Liouville theorem holds only for conjugate pairs of variables. The transformation is done by substituting ϕ = ω rf t, and from (10) we obtain ϕ = 1 τ dϕ dn = p η c ω rf = 1 E p 0 β 2 E η cω rf. We are left with the equation for particle trajectory in ( E, t) phase space ( E) 2 + β2 E s ev 0 ω 2 rf cos φ s 2πhη c ( t) 2 = const. (24) which is the equation of an ellipse. To evaluate the constant, we need to find the trajectory, that encloses 95% of the particles. Since we have assumed a Gaussian distribution, the radius that corresponds to this is approximately 6σ. Our constant is than just 6σE 2, where σ E is the rms of energy deviation. We can rewrite the equation in the form of ( ) 2 ( ) 2 E t + = 1, A B 10

11 and obtain the emittance by calculating the area it encloses from S = πab. The area of this ellipse is then called 95% longitudinal emittance, denoted by ɛ l, and has units of evs. ɛ l = π 2πhη c E 3 ( ) 2 s σe ω RF β 2. (25) ev 0 cos φ s E s The emittance can be compactly written as a product of the momentum spread and the bunch length (σ t ) as ɛ l = 6πσ E σ t = 6πβcσ p σ t. (26) Given the emittance and the RF parameters, we can now express the momentum spread and bunch length and see that they scale as σ p 4 V 0, (27) σ t 4 1 V Acceptance We must distinguish between acceptance and emittance. The acceptance is associated with the bucket (available stable phase space area) whereas the emittance is associated with the bunch (actual phase space occupied by the beam). The acceptance is the maximum allowed value of emittance and is determined by the design of the transport or accelerating lines. Emitance can be obtained from the hamiltonian (20). We know that the total energy of the system is a conserved quantity. To evaluate the constant, we use the ufp, because we know that ϕ is zero at these locations. To get the acceptance we need to integrate A = S E ω RF dϕ, (28) where the integral must be taken over the separatrix. The integral can be solved analytically only for stationary buckets with φ s = 0, π. In this case we get the stationary acceptance, denoted by ɛ sta : 2eV 0 E 0 β 2 ɛ sta = 8 πh η c ωrf 2 (29) For other values of φ s the integral can be solved numerically. From Fig. 9 we can see how acceptance varies with φ s, with the largest value for the stationary bucket. Figure 9: The emittance for a moving bucket ɛ mov with respect to the emittance for a stationary bucket ɛ sta 5.2 Acceleration We will now take a closer look at the particle motion during acceleration. In a synchrotron, particles are injected at some initial energy E i. They are then slowly, through a number of turns, accelerated to their final energy, E f. Because the acceleration should be adiabatic, synchronous phase is slowly increased. The simulation tracks a single particle and varies the synchronous phase from 0 to its final (arbitrarily chosen) value of π/6. By changing the time (number of turns) it takes the particle to reach the final acceleration, we can observe the influence of non-adiabatic effects. In Fig. 10 we can see how fast changes of phase affect the longitudinal emittance. The top left graph represents adiabatic acceleration and we can clearly see how trajectory follows φ s. We can tell that the acceleration is adiabatic, since the area of the particle s phase space ellipse remains essentially constant. If we change the number of turns it takes for the particle to reach the acceleration at φ s to a smaller value, the process becomes non-adiabatic and emittance is not preserved. The smaller the number, the bigger the final emittance. In general, if the motion is to be adiabatic, the system parameters must change more slowly than the period of motion. 11

12 Figure 10: Trajectories for a particle underging acceleration from φ s = 0 to φ s = π. Number of turns is 3000, number of turns to reach 6 final acceleration φ s is??,500,100, Effect of RF Voltage on Phase Space As can be seen from Eqs. (27), (28), (23), the RF voltage has an effect on the bunch length and the momentum acceptance of the accelerator. Even for a stationary bucket we can observe that with V 0 too high, the whole phase space gradually becomes unstable. Figure 11: Phase space plot for a stationary bucket with respect to increasing accelerating voltage, from 800 kv to 550 MV 5.4 Phase Space Matching The beam transfer from one synchrotron or a linac to another synchrotron is done bucket-to-bucket. The RF systems of both machines are phase locked and bunches are transferred directly from the bucket of one machine to the other. If we want the longitudinal emittance to stay the same, the bunch must be centred in the bucket of the final machine and both machines must be longitudinally matched, meaning that they have the samelongitudinal Twiss Parameter, β L. To obtain the β L we rewrite the equation for particle trajectories in ( E, p) phase space (24) as ( ) 2 E + A ( ) 2 t = 1 E 2 B B A + A t2 B = AB, Since we previously defined the emittance as πab we obtain the final expression for phase-space trajectories: β L ( E) ( t) 2 = ɛ l β L π (30) Figs. 12, 13, 14 track the motion of a bunch for a 100 turns after transfer. They describe the motion after the transfer for a matched case, a case when the transfer occurs with a phase error of π/3, and a transfer with a β L error of factor three. 12

13 Figure 12: First hundred turns after transferring the beam from linac to the synchrotron for a matched transfer Figure 13: First hundred turns after transferring the beam from linac to the synchrotron for a transfer with phase error of π/6 Figure 14: First hundred turns after transferring the beam from linac to the synchrotron for a transfer with β L mismatch by a factor of three 5.5 Longitudinal Gymnastics: Debunching and Bunch Rotation As was already proven in previous sections, the shape of a bucket can be manipulated by changing the RF voltage. From (27) we can see that the momentum spread scales as 4 V 0 and the bunch length as 1/ 4 V 0. Since the longitudinal emittance is preserved when motion is adiabatic, we can see that the shape of the bunch can be manipulated by changing the RF voltage. The change in voltage always results in one of the δ or ϕ spreads getting larger and the other one getting smaller - there is no way to shrink them both, which is consistent with emittance preservation. Debunching We adiabatically reduce the voltage over many synchrotron periodes, untill finally it is turned off. The beam is then distributed along the whole circumference of the accelerator. If the process is adiabatic, the momentum spread of the beam is reduced, because the phase (time) spread gets large. When a beam is debunched, no RF voltage is applied to it. This is why it is not possible to debunch an electron beam at any significant energy. Because of synchrotron radiation, electron beams always need to be accelerated in order to 13

14 compensate for radiation losses. Bunch Length Manipulation The figures from the section on mismatched transfer give a hint that a bunch shape could be manipulated. We can see that the bunch rotates if mismatches occur - these mismatches are a consequence of phase/momentum offset or different RF parameters (meaning different buckets). We are more interested in the latter, since the RF parameters are something that we can change while the particles are circling in the accelerator. Let us assume that we have a bunch with a small momentum spread and long bunch length. To change the bunch to a short one, we would increase the RF voltage in the time that is short compared to the synchrotron oscillation period. Since the bucket has changed, the bunch is essentially mismatched and starts to rotate. The process is shown in Fig. 15. The bunch starts getting shorter by transferring some of Figure 15: Plots for six turns (0th, 2nd and 5th) of bunch manipulation process. On the first plot, the bunch has its original shape. It than starts to rotate until it reaches its narrowest point represented on the right graph. its phase spread to the momentum spread. After a quarter of the synchrotron period, it reaches its narrowest point and, unless the RF voltage is increased again, it will continue to rotate and start getting longer. The bunch rotates because its boundary does not coincide with a phase space trajectory, so the second time the RF voltage is ramped up, it must increase to such a value that the particles at the edge of the bunch will follow the same phase space trajectory. From (17) we get the relation Ω δ 0 = ϕ 0. (31) ω rf η c To get the overall bunch reduction factor, we proceed as follows. Before the rotation, the relation between momentum and phase deviation is Ω 1 δ 1 = ϕ 1, ω rf η c where Ω 1 denotes the synchrotron frequency with RF voltage V 1. After rotating, the phase deviation ϕ 1 is transformed into momentum deviation Ω 2 δ 2 = ϕ 1, ω rf η c and the original momentum deviation is transformed into phase deviation ϕ 2 = ω rfη c Ω 2 δ 1. We now need to stop rotation. This is achieved if the new momentum error and the new phase error are on the same phase space trajectory. The required RF voltage can be obtained from Ω 3 δ 2 = ϕ 2. ω rf η c To get the ratio of the bunch lengths we take the quotient ϕ 2 Ω 3 ϕ 1 Ω 1 = δ 2 δ 1 = Ω 2ϕ 1 Ω 2 ϕ 2 Since l ϕ 0 and Ω V we get the overall bunch length reduction factor for this process: where V 1 is the initial and V 3 final RF voltage. l 1 l 0 = 4 V1 V 3, (32) 14

15 The bunch length manipulation described in this section is applicable only to non-radiating particles. For particles that radiate bunch manipulation is easier due to damping effects. Relation (31) still holds, but the momentum spread is independently determined by synchrotron radiation and the bunch length scales proportionally to V. 6 Conclusion The first idea to use RF cavities instead of static fields was to achieve higher energies. Throughout the seminar the stability of motion of particles interacting with such fields was inspected and it was shown that besides higher accelerating voltages, oscillating fields also provide an additional stabilizing force, that results in phase focusing. The result of oscillating fields are bunched beams, within which the particles oscillate about the ideal values of momentum and phase with the frequency known as the synchrotron frequency. References [1] Helmut Wiedermann, Particle Accelerator Physics - Basic Principles and Linear Beam Dynamics (Springer- Verlag, New York, 1993). [2] D.A. Edwards, M.J. Syphers, An Introduction to the Physics of High Energy Accelerators (Wiley-VCH, Weinheim, 2004). [3] William Bartletta, Linda Spentzouris, USPAS - U.S. Particle Accelerator School Slides, Fund.shtml ( ) [4] M. J. Syphers, Some Notes on Longitudinal Emittance, syphers/accelerators/tevpapers/longemitt.pdf ( ) 15