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Le traitement du cancer au moyen d'acclrateurs a une longue histoire avec l'utilisation de linacs, de cyclotrons et, plus rcemment, de synchrotrons. Le traitement peut se faire essentiellement de deux manires: par faisceau largi ou

  1. Introduction Hadrontherapy was first proposed by Robert R. Wilson in 1946 [1]. He understood that by exploiting the typical, energy-deposition pattern for charged hadrons (known as the Bragg peak ) that is shown in Figure 1, it would be possible to deliver a high radiation dose to deep tumours while limiting the dose received by the surrounding healthy tissue and the entry channel. Figure 1. The energy deposition for charged hadrons, is more favourable than for other particles. By virtue of the Bragg peak behaviour and the low lateral scattering of charged hadrons, it is possible to confine the radiation dose to the target volume of the tumour ( conformal treatment) to a much higher accuracy than with conventional radiotherapy (X-rays and electrons). Hadrontherapy is particularly suited to tumours of the eye, brain, salvary glands, spinal chord, for some prostate and uterus tumours and generally for all those tumours that are situated close to a critical organ, which must be spared any radiation dose. In addition to the better conformation of the dose shared by protons and light ions, the latter have the advantage of much higher linear energy transfer (LET). This is due to ionisation and scales with the square of the ion’s charge ( Z ). Whereas protons mainly cause cell damage by creating free radicals, light ions cause additional damage in the form of double strand breaks in the DNA via the ionisation electrons. A still more interesting feature is that the LET function is higher on the Bragg peak than in the entry channel. Too large Z ions cause severe damages in the entry channel as well, resulting in higher morbidity (complications), and show a larger tail due to fragmentation . Thus the charge Z 3

  2. Introduction ________________________________________________________________________________________________________________________________________________________________________________________________________ of the ion has to be chosen carefully. Radiobiologists and therapists have agreed that the optimum ion is carbon ( Z = 6), which delivers a radio-biological effect (RBE) that is several times that of protons in the Bragg peak region, while still being close to protons in the input channel. This makes carbon ions better suited to oxygen-poor and radio-resistant tumours. Light ions also exhibit fragmentation, which increases with Z . The negative aspect is the creation of an exit dose, while the positive aspect is that for certain ions it creates positron emitting isotopes that can be used with a PET camera to visualise the dose distribution directly after irradiation. Carbon is again the optimum ion that has a weak exit dose, but still creates the useful positron emitters carbon 10 and 11. Since the Bragg peak is rather narrow, a spread-out Bragg peak (SOBP) is necessary to treat thick tumours (see Figure 2). The SOBP is obtained by the superposition of many single Bragg peaks. The range (depth) of the peak can be changed in steps by inserting material in the beam path ( degrader ) or by the active use of the accelerator producing the beam. To limit the number of steps, a small energy spread is created in the beam by a ridge filter , since accelerators are, relatively speaking, mono-energetic. A variable degrader may also be used to create the full energy spread directly. In all cases, the ratio of the entry channel dose to the peak is less favourable than for the single Bragg peak, but it is still better than for conventional therapy. Figure 2 also shows the relative importance of the exit dose due to fragmentation for different ions and protons Figure 2 The Spread Out Bragg Peak needed for treating thick tumours is still preferable to conventional radiation. The transverse spreading of the beam can be obtained either with a passive system using scatterers and collimators or by actively scanning a pencil beam into the tumour volume. 4

  3. Introduction ________________________________________________________________________________________________________________________________________________________________________________________________________ The latter allows a better conformation of the dose and does not require patient-specific hardware. The two systems are schematically shown in Figure 3. Patient scatterer collimator Patient Scanning magnets Figure 3. Active and passive spreading systems. Therapeutic beams can be produced with linacs, cyclotrons and synchrotrons, but when carbon ions are also to be accelerated, the synchrotron is the only practical choice. The first design of a dedicated carbon ion synchrotron was the EULIMA study [26]. To reach a depth of, say, 27 cm, carbon ions have to be accelerated to a total energy of 4800 MeV (400 MeV/u). Such a high energy makes cyclotrons and linacs unsuitable. One of the main clinical specifications is the uniformity of dose that has to be distributed over the treatment volume with a maximum tolerable error of r 2.5%. This implies that the dose has to be carefully measured during irradiation and that the flux of particles at the patient has to be low enough to allow such an accurate measurement. This calls for a slow extraction and for a very constant spill to avoid overdose due to fluctuations. With active scanning, the beam is distributed in small volume units named voxels (short for volume pixels). The large number of voxels needed to cover the whole tumour means a short time per voxel and imposes very tough conditions on the spill uniformity. Passive beam delivery systems are less prone to fluctuations but, on the other hand, require patient- specific hardware and the dose conformation is less accurate. The clinical specifications and the consequent physical specifications for the beam are, of course, similar in all the existing and planned facilities. The Proton-Ion Medical Machine Study (PIMMS) at the PS-Division in CERN was set up following an agreement between TERA, MED-AUSTRON, CERN and GSI. The agreed aim of the study was to investigate and design a synchrotron-based medical facility that would allow the direct clinical comparison of protons and ions for cancer treatment. The machine was to be designed primarily for high-precision active beam scanning for both protons and carbon ions, but was also to be capable of delivering proton beams by passive scattering. The main beam specifications agreed for PIMMS are resumed in Table 1. 5

  4. Introduction ________________________________________________________________________________________________________________________________________________________________________________________________________ PIMMS performance parameters Active scanning Passive scanning (Pencil beam) (large area beam) Extraction energies for carbon ions 120-400 MeV/u - Extraction energies for protons* 60-220 MeV 60-250 MeV Nominal treatments 60 spills of 1 s +1s to 120 spills of 0.25s +1s to ramp up and down = ramp up and down = 2 minute 2.5 minute Nominal dose delivered 2 Gray in 2 litre 2 Gray in 7.5 litre Number of protons in one spill at 10 10 2 u 10 10 patient Number of carbon ions in one spill 4 u 10 8 at patient - Spot sizes variation at all energies - (full-width half-height). 4-10 mm Field size 20 cm u 20 cm 20 cm u 20 cm Table 1. PIMMS main performance parameters Within the framework of PIMMS, the aim of this thesis is to study in detail the resonant slow extraction and its consequences on the extracted beam in order to choose the most suitable extraction scheme. The study of the transfer lines to the treatment rooms is also included. A new approach to beam transport has been used to exploit the particular transverse characteristics of the extracted beam. Chapters 1 to 3 act as an introduction, summarize and discuss the state of the art and define the main quantities, Chapters 4 to 7 describe the original work of this thesis. 6

  5. Chapter 1 BASIC ACCELERATOR PHYSICS The guiding and focusing of a charged particle beam in a circular accelerator rely on a series of magnetic elements, separated by field-free drift spaces, that form the accelerator lattice. The design of the lattice is one of the first tasks for the accelerator designer. One normally starts with a simplified structure, containing only ideal magnetic dipoles and quadrupoles. This basic structure is referred to as the linear lattice and once it has been determined the character of the machine is more or less fixed. In a separated function synchrotron, dipole magnets bend the particles onto circular trajectories and quadrupole magnets are needed for the focusing of particles with small deviations compared to the ideal trajectory. The equilibrium orbit can be defined as the orbit of a particle with the design momentum p 0 that closes upon itself after one turn and is stable. The motion of a particle in an accelerator is then conveniently described by the deviations of its trajectory with respect to this equilibrium orbit. 1.1 Coordinate system Generally in a synchrotron, there is bending in only one plane referred to as the horizontal plane which contains the equilibrium orbit. For a description of particle motion a right- handed curvilinear coordinate system ( x , s , z ) following the equilibrium orbit is used. The azimuthal coordinate s is directed along the tangent of the orbit. The radial transverse direction x is defined as normal to the orbit in the horizontal plane and the vertical ˆ and s ˆ . The transverse direction is given by the cross product of the unit vectors x ˆ and z ˆ then define the vertical plane. The local radius of curvature U 0 and the vectors s bending angle s / U are defined as positive for anticlockwise rotation when viewed from positive z . The choice of the coordinate system is shown in Figure 1.1. particle trajectory x z equilibrium s orbit ˆ z ˆ s ˆ x U 0 horizontal plane local centre of curvature 0 Figure 1.1. Curvilinear coordinate system. 7

  6. Chapter 1 Basic accelerator physics ________________________________________________________________________________________________________________________________________________________________________________________________________ 1.2 Accelerator magnets For this study, the magnetic elements are considered as purely transverse, two- dimensional fields in the curvilinear coordinate system that follows the equilibrium orbit. It is assumed that each element can be decomposed into a harmonic series that extends only as far as the sextupole component and that each element can be represented by a ‘block’ of field that is discontinuous in the axial direction. This is the so-called hard-edge approximation . For a final design, end-field corrections will have to be applied to improve the accuracy of the hard-edged model . The equilibrium orbit corresponds to a particle with the nominal momentum and starting conditions. Particles of the same momentum, but with small spatial deviations will oscillate about this orbit. The equilibrium orbit is variously known as the central orbit , the reference orbit and the closed orbit . In the hard-edged model, this orbit is a series of straight lines connecting circular arcs of cyclotron motion (see Figure 1.2). v Am av v 2 / U B z nev B z U Figure 1.2. Cyclotron motion for a particle of mass Am av with charge ne . The force experienced by a charged particle, moving in a magnetic field is given by the Lorenz equation as d p F q v B , (1.1) � u d t where v is the particle velocity, q its charge and B the magnetic field. Since the Lorenz force always acts perpendicular to the particle velocity, only the direction of movement changes while the magnitude of the velocity is constant. The nominal trajectory in a circular accelerator is defined by the main bending magnets which provide a dipolar field, ˆ B . (1.2) B z 0 Equating the magnetic deflection to the centrifugal force (scalar values) gives, 2 Am v Am v av av B nev B , (1.3) � U z z ne U where A is the atomic mass number, m av is the average relativistic mass per nucleon, n is the charge state of the particle, e is the elementary charge, and U is the radius of the cyclotron motion. The reluctance of the particle to be deviated is characterised by B U � which is known as the magnetic rigidity . 8

  7. Basic accelerator physics Chapter 1 ________________________________________________________________________________________________________________________________________________________________________________________________________ The application of a sign convention for the magnetic rigidity is usually avoided and an ‘engineering’ formula is quoted in which the momentum is always taken as positive. � � 1 A � � Magnetic rigidity: B z [ T ] [ m ] p [GeV/c ] (1.4) U av 0 . 299792458 � n � where p av is the average momentum per nucleon. From (1.4) it can be seen that the higher the particle momentum , the higher the magnetic field needed to keep the particle on the nominal trajectory. Therefore, the magnetic fields in an accelerator have to be continuously adjusted, according to the actual particle momentum during acceleration. In order to avoid the momentum dependency when characterising magnetic elements in an accelerator, the fields are normalised w.r.t. the magnetic rigidity (particle momentum). For a dipole magnet the normalised bending strength h , is given by 1 h . (1.5) U The absolute field level required to keep a particle with a certain momentum on the nominal trajectory is then simply found by multiplying (1.5) with the magnetic rigidity (1.4). The equilibrium orbit in a synchrotron is defined by the main dipole magnets. In general particle trajectories will deviate slightly from this ideal design orbit. To maintain stability about the design orbit, focusing forces are needed. These are provided by quadrupole magnets. A quadrupole magnet has four poles with a hyperbolic contour and the magnetic field in the current-free region of the magnet gap can be derived from a scalar potential [2], ( x , z ) G x z , (1.6) ) � � � where G is known as the quadrupole gradient, being defined as: d B � � z G . (1.7) � � d x � � 0 The equipotential lines are the hyperbolae x · z = const and the field lines are perpendicular to them. The horizontal and vertical magnetic fields in a quadrupole magnet are linear in the deviation from the magnet centre: B G z and B G x . (1.8) � � x z The transverse forces acting on a particle with a deviation ( x,z ) from the equilibrium orbit are obtained with (1.1) as F qvB ( x , z ) qvGx (1.9) x z and F qvB ( x , z ) qvGz . (1.10) � � z x 9

  8. Chapter 1 Basic accelerator physics ________________________________________________________________________________________________________________________________________________________________________________________________________ Depending on the sign of G , the force will be focusing in the horizontal direction and defocusing in the vertical direction or vice versa. A quadrupole that focuses horizontally ( G < 0), is called a focusing quadrupole and a quadrupole that defocuses horizontally ( G > 0), is called a defocusing quadrupole . By alternating focusing and defocusing quadrupoles an overall focusing effect can be obtained in both the horizontal and the vertical plane ( alternating gradient focusing ). An important property of a quadrupole magnet is that the horizontal force depends only on the horizontal and not on the vertical position of the particle trajectory. Similarly, the vertical component of the Lorentz force depends only on the vertical position. The consequence is that in a linear machine, containing only dipolar and quadrupolar fields, the horizontal and vertical motions are completely uncoupled. In analogy to the normalised bending strength of a dipole magnet (1.5), the quadrupole gradient can be normalised w.r.t. the magnetic rigidity, thus defining the normalised quadrupole strength, G k . (1.11) B U 1.3 Transverse optics When analysing the transverse motion in an accelerator, it is practical to use the distance s measured along the trajectory, instead of the time as independent variable, d d d s d v . (1.12) d t d s d t d s It is usual to consider only dipole and quadrupole fields (linear lattice), which leads to uncoupled motions in the two transverse planes. The equations of motion are obtained from (1.1) while retaining only first order terms in the transverse coordinates [3]. The horizontal and vertical motions can be represented by one simple expression valid for both planes, y c c ( s ) K ( s ) y ( s ) 0 (1.13) � where, for the horizontal motion, 2 y x and K ( s ) K ( s ) h ( s ) k ( s ) (1.14) { � x and, for the vertical motion, y { z and K ( s ) K ( s ) k ( s ) . (1.15) z For the hard-edged model, the focusing function K y ( s ) is piecewise constant along the trajectory and for a circular machine, it has the periodicity of the lattice, K ( s L ) K ( s ) , (1.16) � y y where the accelerator is composed of N identical sections or cells, with C = N·L and C being the circumference of the machine. 10

  9. Basic accelerator physics Chapter 1 ________________________________________________________________________________________________________________________________________________________________________________________________________ The motion of mono-energetic particles about their equilibrium orbit is known as the betatron motion and it is usual to parameterise the motion so that its pseudo-harmonic behaviour is brought into evidence. The first stage is to express the motion in a form developed by Courant and Snyder [4], s � � d V � � � Betatron motion: y ( s ) A s cos B , (1.17) � E � � � � E V � � 0 where y ( s ) represents either transverse coordinate as a function of the distance s along the equilibrium orbit, A and B are constants depending on the starting conditions and E ( s ) is the betatron amplitude function (dimension length). The phase P ( s ) of the pseudo oscillation is given as s d V � � � s . (1.18) P � � E V 0 To complete this description, the derivative of E ( s ) is added in the relations, 1 d E � � s (1.19) D � 2 d s and 2 1 � D � � s . (1.20) J E The expressions (1.17)-(1.20) all depend on a knowledge of E ( s ). Although the analytic solution for E ( s ) is more complicated than the original motion equation, it is reasonably easy to evaluate this function numerically and to tabulate it for any lattice. The parameters D ( s ), E ( s ) and J ( s ) are collectively known as the Courant and Snyder parameters , or more usually the Twiss parameters . The above parameterisation is now so commonplace that it is the starting point for nearly all lattice design. The pseudo-harmonic motion can be further transformed into a simple harmonic motion by returning to equation (1.17), introducing P and differentiating, to obtain, � � y ( s ) A E cos P � B (1.21) D 1 � � � � � � y c s � A cos P � B � A sin P � B . (1.22) E E The phase terms can be extracted and used to define new coordinates Y ( P ) and Y c ( P ) that are known as normalised coordinates, 1 � � � � � � Y P A cos P B y s (1.23) � E D � � � � � � � � Y c A sin B y c s y s . (1.24) P � P � E � E 11

  10. Chapter 1 Basic accelerator physics ________________________________________________________________________________________________________________________________________________________________________________________________________ It is useful to represent real-space coordinates by lower case symbols and normalised coordinates by upper case symbols ( x o X , etc.). Normalised coordinates use the phase advance as independent variable and real-space coordinates use the distance. The transformations between the two systems are conveniently expressed in matrix form as, � � � � � � Y 1 / 0 y E � � � � � � Real to normalised: (1.25) � � � � � � d Y / d d y / d s � P � / � � D E E � � � � � � � � y Y E 0 � � � � � � Normalised to real: . (1.26) � � � � � � � d y / d s d Y / d � � / 1 / � P � D E E � � The elimination of the phase advance from equations (1.23) and (1.24) yields an invariant of the motion, c c c 2 2 � 2 � 2 � � � � 2 A Y Y J y 2 D y y E y ( constant ) . (1.27) Equation (1.27) is in fact the equation of an ellipse in ( y , y c ) phase space. The constant A 2 equals the ( area/ S ) of the ellipse described by the betatron motion in either the normalised ( Y , Y c ) or the real ( y , y c ) phase spaces (see Figure 1.3). When referring to a single particle, this area is sometimes called the single-particle emittance . When referring to a beam it is known as the emittance . The situation with a beam is complicated by the definition used for deciding the limiting ellipse that defines the area. This may be related to a number of standard deviations of the beam distribution, or the overall maximum. It is useful to note that the emittances in real and normalised phase spaces are equal and to re- express (1.27) as, E E � � � � E S y y y ˆ ˆ 2 2 Emittance: ˆ ˆ and , (1.28) E y Y y Y S � y E S S y ˆ are the maximum excursions in real and normalised phase spaces that ˆ and Y where y define the beam-size at the observer’s position. y c [rad] 0.0015 0.0015 Y c [m 1/2 ] � ( E y J y / S ) � ( Y 2 +Y c 2 ) - D y � ( E y / S E y ) 0 0 y [m] Y [m 1/2 ] -0.0015 0 0.0015 -0.0015 0 0.0015 ˆ � ( E y E y / S ) = y ˆ � ( E y / S ) = Y -0.0015 -0.0015 Figure 1.3. Phase-space ellipses in real and normalised phase spaces. 12

  11. Basic accelerator physics Chapter 1 ________________________________________________________________________________________________________________________________________________________________________________________________________ It should be noted that during acceleration the emittance decreases and therefore the phase space density changes. This is not against Liouville’s theorem, but due to the definition of the transverse phase space. The y c -coordinate, describing the divergence of a particle with respect to the nominal orbit, v d y 1 d y y y c , (1.29) d s v d t v is not a generalised momentum as defined in classical mechanics. As long as the momentum is constant, the difference is just a scaling factor but during acceleration this factor changes and so does the emittance. The physical reason is the increase of the longitudinal momentum during acceleration, whereas the transverse momentum is constant. Therefore the divergence v y / v decreases and the beam shrinks, which is known as adiabatic damping . Normalising the emittance w.r.t. the particle momentum gives a constant of the motion, the so called normalised emittance , E E , (1.30) E J n , y rel rel y where E rel and J rel are the relativistic parameters. 1.4 Transfer matrix formalism Another basic expression that is needed for the present study is the general transfer matrix . This can be derived by expanding (1.21) into two terms, � � y s A cos B sin (1.31) E P � E P where A and B are new constants. Differentiation of (1.31) with respect to s gives, A B � � � � � � y s cos sin cos sin . (1.32) c � � � � D P P P D P E E The constants A and B can be replaced using the initial conditions at s = s 1 , choosing P 1 = 0 y D 1 1 A and B y y , (1.33) c � E 1 1 1 E E 1 1 to give the general transfer matrix from position s 1 to position s 2 . The phase advance from s 1 to s 2 is written as ' P , � o � General transfer matrix : s s M 1 2 � � � E � � ' � ' � ' 2 cos sin sin P D P E E P 1 1 2 � � . (1.34) E 1 � � 1 � E � > @ � � � � ' � � � � ' 1 � ' � ' � � 1 sin cos cos sin D D P D D P P D P � 1 2 2 1 2 E � � E E 2 1 2 13

  12. Chapter 1 Basic accelerator physics ________________________________________________________________________________________________________________________________________________________________________________________________________ The phase-space coordinates at position s 2 are then given by � � � � y y � 2 � � 1 � o ( s s ) . (1.35) M � � � � c 1 2 c y y � � � � 2 1 When equation (1.35) is applied to a full turn in a ring, the input conditions equal the output conditions ( D = D 1 = D 2 , E = E 1 = E 2 , ' P = �S Q ), so that � � � � cos 2 Q sin 2 Q sin 2 Q � S D S E S � � M , (1.36) � � � 1 turn � � � sin 2 Q cos 2 Q sin 2 Q � � J S S D S where Q is known as the betatron tune and is equal to the number of transverse oscillations the particle makes during one turn in the machine. Expression (1.36) describes the evolution of the phase-space coordinates of a particle at a certain position in the machine. A plot of the coordinates for a large number of turns gives a phase-space trajectory . In a linear machine phase-space trajectories are always of elliptical shape; the orientation of the ellipses at any position s in the machine is determined by the local Twiss parameters and the beam size is found with the emittance according to Figure 1.3. The general transfer matrix for normalised coordinates is simply a 2x2 rotation matrix describing a clockwise rotation by the phase advance ' P between position s 1 and s 2, ' ' � � cos sin P P o � � ( s s ) . (1.37) M � � N 1 2 � ' ' sin cos � � P P The single-turn matrix for normalised coordinates is � � cos 2 Q sin 2 Q S S � � . (1.38) M � � N,1 turn � � sin 2 Q cos 2 Q � S S The 2x2 transfer matrix formalism is commonly known as Twiss-matrix formalism . 1.5 Off-momentum particles and dispersion function A simple extension of (1.17) allows the motion of particles with different momenta to be described, s d V G p � � � y ( s ) A E ( s ) cos B D ( s ) (1.39) � � � � E ( s ) p � � � 0 � � � � � � � � � � � � � � � � � Dispersion motion Betatron motion where D ( s ) is known as the dispersion function and G p / p = ( p part. - p 0 )/ p 0 is the relative momentum deviation of the particle. The dispersion is created by the momentum dependency of the bending radius in dipole magnets and appears therefore only in the plane of bending (generally the horizontal plane). The equilibrium orbit for an off-momentum particle is to first order displaced from the central orbit by the product of the dispersion function and the momentum deviation, p p G G y D and y D . (1.40) c c EQ.O y EQ.O y p p 14

  13. Basic accelerator physics Chapter 1 ________________________________________________________________________________________________________________________________________________________________________________________________________ An analytic derivation of the dispersion function is given in [4], but it is again common practice to rely on lattice programs to supply numerical listings of D ( s ) and its derivative with distance, as for the betatron amplitude function. The momentum deviation G p/p is treated as a quasi-variable and particles are transferred through the lattice with 3 u 3 matrices of the form, y m m m y � � � � � � 11 12 13 � � � � � � y c m m m y c (1.41) � � � � � � 21 22 23 � � � � � � p / p 0 0 1 p / p G G � � � � � � 2 1 where m 11 , m 12 , m 21 and m 22 are the coefficients of the general transfer matrix (1.34) and m 13 , m 23 are additional dispersion coefficients (see Section 3.10). The dispersion vector in the form ( D , D c , G p/p =1) also propagates through the lattice according to (1.41). The normalised form of the dispersion function ( D n , D c n ), similar to (1.25), will also be frequently used, D � � D 1 / E 0 � � � � n � � . (1.42) � � � � � � � � � � d D / d d D / d s P / D E E � � � � n � � It should be mentioned that (1.41) is strictly applicable to only small values of G p/p and to linear lattices. For trajectories with more than a few per mil momentum deviation, or for trajectories that pass through non-linear magnetic lenses, it is advisable to perform a numerical tracking if the orbit position is required to a high precision. Exactly this situation arises when calculating the position and angle of the separatrices for a resonant extraction [5] and it is useful to be able to incorporate the more exact tracking method into the general transfer matrix as, > @ � c � � � � � m m y m y m y / G p / p � 11 12 EQ, 2 11 EQ, 1 12 EQ, 1 � > @ � � � c � � c � M s s � m m y m y m y / p / p � (1.43) G o 1 2 21 22 EQ, 2 21 EQ, 1 22 EQ, 1 � � 0 0 1 � � where y EQ and y c EQ are the position and angle of the tracked off-momentum equilibrium orbit. A more detailed and complete introduction to accelerator physics can be found in References [3,6,7]. 15

  14. Chapter 2 THEORY OF THE THIRD-INTEGER RESONANCE The third-integer resonance can be used to extract particles from a synchrotron during a large number of turns [8]. In this case, the slowly extracted beam is known as the spill . In a medical machine, it is essential to extend the extraction time (spill time) to about one second (~10 6 turns), in order to facilitate the measurement of the radiation dose delivered to the patient. This is done by making particles of the circulating beam unstable in a controlled way. For this, the betatron tune in the extraction plane (in general the horizontal plane is used) has to be close to the resonance condition, 3 Q h = integer. The resonance is excited by sextupole magnets. 2.1 Sextupole magnets In the current-free region of a magnet gap, the field can always be derived from a scalar potential ) . Assuming that the magnetic field has only transverse components, the scalar potential of a magnet with 2 m poles is given by [2], m m A Re( x iz ) B Im( x iz ) . (2.1) ) � � � m m � � � � � � � � � � � � � � Skew magnets Normal magnets The field components are obtained by differentiation according to w ) w ) B x and B z . (2.2) � � w x w z The transverse fields in a normal sextupole magnet ( m = 3) are then found to be 2 . 2 � � � � B x z , � 6 B xz B x z , 3 B x ( z ) (2.3) � � x 3 z 3 The relationship between the coefficient B 3 and the magnetic field is found by comparing (2.3) to the Taylor expansion of the magnetic field in the horizontal plane, 2 d B d B 1 1 � � � � 2 B x z ( , 0 ) B z x z x ... (2.4) � � � � � z 0 � � 2 1 ! d x 2 ! d x � � � � 0 0 so that 2 d B 1 � � z B � . (2.5) � � 3 2 6 dx � � 0 The vertical and horizontal fields in a sextupole can then be written as 2 2 d B d B 1 � � � � � � 2 2 z z B x z and B xz . (2.6) � � � � � z x 2 2 2 d x d x � � � � 0 0 17

  15. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ The effect of a sextupole on a particle trajectory can be described in a simple way by considering the magnet as a thin lens. The integrated magnetic field seen by the particle is given by the product of the magnet length " S and the field at the average transverse particle position. As the lens is considered to be infinitely thin, the particle position remains unchanged during the passage, i.e. ' x = ' z = 0. The sextupole gives a kick to the particle that changes the direction of the trajectory. With the magnetic rigidity | B U | of the particle the thin-lens kicks given by the sextupole are 2 � d � B " 1 " B 1 � � � � z S S z 2 2 2 2 ' x c � � x � z " k c x � z (2.7) � � S 2 B 2 B d 2 x U U � � 0 and d B " � � S z z c xz k c xz , (2.8) ' � � " � � S 2 B d x � � U 0 where k c is the normalised sextupole gradient. 2.1.1 Sign conventions for sextupole magnets To define a sign convention for sextupole magnets, the trajectory of a particle in the horizontal plane (i.e. z = 0) is considered. Figure 2.1 illustrates the sign convention and shows the corresponding magnet geometry. x If the sextupole focuses for positive x , then it can be said to be an F-sextupole. x If the sextupole defocuses for positive x , then it can be said to be a D-sextupole. F-Sextupole ( B z negative, k c negative) z z S s N N B B z S S x N B z D-Sextupole ( B z positive, k c positive) z z N s B z S S B B z N N x S Figure 2.1. Sign convention for sextupole magnets. 18

  16. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.1.2 Normalised sextupole strength In order to describe the effect of a thin-lens sextupole in normalised coordinates, the kick introduced by the sextupole and the particle position have to be expressed in terms of normalised coordinates. From the above, the effect of a thin lens sextupole in real space is 2 B 1 � d B � " " � � 2 2 z S S z ' x 0 and x c � � x z . (2.9) ' � � � 2 2 B U B U d x � � 0 Many problems are simplified by a transformation into normalised coordinates that converts the basic betatron motion into a simple harmonic motion. The normalisation matrix and its inverse are quoted below, � � 1 � � � � 0 E 0 � � y � � � � � � E Y y y � � � � � � � 1 D M � � ( M ) 1 and M . (2.10) � � � � � y N, y D N, y N, y � � c c Y y � � � � � � y � � E E E � � � � y y y E � � y When applied to (2.9) the relations between normalised and unnormalised coordinates are found to be, 1 ' c ' c x X x X z Z . (2.11) E E x z E x Thus the effect of the sextupole in normalised coordinates appears as, � � � � � � 2 1 d B E E " c 3 / 2 � � ' X 0 ' � 2 � z 2 � � 2 � z 2 � S z X E X Z S X Z (2.12) � � � � � � x 2 2 B U d x � E � � E � � � x x 0 where S is the normalised sextupole strength , � � 2 1 d B 1 " 3 / 2 � � 3 / 2 c S z S k . (2.13) E E " � � x x S 2 2 B U d x 2 � � 0 The expressions describing the effect of the thin-lens sextupole on the vertical trajectory are obtained in the same way as � � � � 2 1 d B E E " � � � � ' Z 0 ' c � 3 / 2 z � z S z Z 2 E XZ 2 S XZ . (2.14) � � � � x 2 B 2 U d x E E � � � � x x 0 From the above expressions, it can be seen that a sextupole couples the horizontal and vertical motion of a particle. The strength of the coupling is proportional to the ratio of the vertical and horizontal beta functions ( E z / E x ) at the position of the sextupole. For a horizontal extraction, choosing a small E z / E x ratio and provided the vertical tune does not satisfy a resonance condition, the influence of the vertical motion can be neglected to first order. For this reason only the horizontal motion is considered in the following analysis. 19

  17. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.2 Basic theory for third-integer resonance A simple method that gives an immediate insight into the physics of resonant extraction was developed by Kobayashi [9,10]. The method describes the effect of the sextupole as a perturbation to the linear machine. The general transfer matrix M k for normalised coordinates, describing k turns in the machine is given by: � � cos 2 ( kQ ) sin 2 ( kQ ) S S � h h � M . (2.15) � k sin 2 ( kQ ) cos 2 ( kQ ) � � S S h h Consider a particle with a horizontal betatron tune close to a third-integer, Q h = n r 1/3 + G Q, ( n is integer, | G Q| <<1/3), where G Q is defined as the tune distance of the particle from the resonance, � G Q Q Q , (2.16) particle resonance the explicit transfer matrices for one, two and three turns in the unperturbed machine can be written as: > @ > @ � � � � 2 1 3 2 1 3 cos n / Q sin n / Q 1 2 / 3 2 / � S G S G � r � r � � � � r M 1 (2.17) � � � # > @ > @ � � � � � � 3 2 / 1 2 / � sin 2 n 1 3 / Q cos 2 n 1 3 / Q # S G S G � � r � r � � � � � > @ > @ � � � � cos 4 n 1 3 / Q sin 4 n 1 3 / Q 1 2 / 3 2 / � S G S G � r � r � # � � � (2.18) M 2 � � � # > @ > @ � � � � � � 3 2 / 1 2 / � sin 4 n 1 3 / Q cos 4 n 1 3 / Q S G S G r � � r � r � � � � � > @ > @ � � � � cos 6 n 1 3 / Q sin 6 n 1 3 / Q 1 6 Q � S G S G � r � r � SG � � . (2.19) M 3 � � � # � � � > @ > @ � � � � 6 Q 1 � sin 6 n 1 3 / Q cos 6 n 1 3 / Q SG � � S G S G � r � r � � � For the following calculations, the tune distance G Q is kept only in the transfer matrix for the full three turns. The coordinates of the particle after one turn are then found as: X � � X 1 2 / 3 2 / � � � r � � (2.20) � � # � � � � X X c c � � 3 2 / 1 2 / � � � # � � 1 0 and similarly for 2 turns X � � X 1 2 / 3 2 / � � � # � � � � # � � . (2.21) � � X X c c � � 3 2 / 1 2 / � � � r � � 2 0 By defining the small quantity H = 6 SG Q , which will be referred to as the modified tune distance , it can be seen that a particle with exactly resonant tune (i.e. H = 0) will return to its initial position every three turns. X 1 X � � � �� � � H (2.22) � � # � � � � X ' 1 X ' � � � � � � H 3 0 20

  18. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ The effect of the sextupole during three turns in the machine is now calculated as a perturbation by linearly adding: (A) The effect of 3 turns with a sextupole placed after the 3rd turn, M Sextupole 3 � (B) the effect of 3 turns with a sextupole placed after the 2nd turn, M Sextupole M � � 2 1 (C) the effect of 3 turns with a sextupole placed after the 1st turn. M � Sextupole � M 1 2 as shown schematically in Figure 2.2 below. (A) (B) Add (C) Figure 2.2. Perturbative treatment of the sextupolar effect during three turns. With the effect of the sextupole as derived earlier, SX 2 , ' X 0 ' c X (2.23) the calculation of the three terms gives: (A) 3 turns + sextupole � � X X X � c H 3 0 0 (2.24) � 2 � X c � X � X c � S X � X c H H 3 0 0 0 (B) 2 turns + sextupole+1 turn 2 � � � � � � 1 1 3 3 3 1 1 3 � � X X X X X S X X � � # c � r r � c � � # c � � � 3 0 � 0 0 0 0 � 2 2 2 2 2 2 2 2 � � � � � � (2.25) 2 � � � � � � 3 1 3 1 3 1 1 3 � � X X X X X S X X c # � # c � � r � c � � # c � � � 3 0 � 0 0 0 0 � 2 2 2 2 2 2 2 2 � � � � � � (C) 1 turn + sextupole+2 turns 2 � � 1 � 1 3 � 3 3 1 � 1 3 � X X X � X X S X X � � � r c � c � � r c # # � � � � 3 0 0 0 0 0 � � 2 2 2 2 2 2 2 2 � � � � � � (2.26) 2 � � � � � � 3 1 3 1 3 1 1 3 � � X X X X X S X X c r � r c � � # � c � � r c � � � 3 0 0 0 0 0 � � 2 2 2 2 2 2 2 2 � � � � � � 21

  19. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ Subtracting the initial coordinates ( X 0 , X c 0 ) and adding the three terms (A), (B) and (C) while keeping only first-order correction terms in H gives 2 2 � � � � 3 1 3 3 1 3 X X S X X S X X ' c r � c � r c # # H � � � � 3 0 0 0 0 0 2 � 2 2 � 2 � 2 2 � (2.27) 2 2 1 � 1 3 � 1 � 1 3 � 2 ' X c � X � SX � S � X X � � S � X r X . # c c � � � H 3 0 0 0 0 0 0 2 2 2 2 2 2 � � � � The cancellation of signs shows that there is no fundamental difference between the 1/3rd and 2/3rd resonances. 2 2 � � � � 3 1 3 3 1 3 X X S X X S X X ' c � � � c � � � � c � � � H 3 0 0 0 0 0 2 � 2 2 � 2 � 2 2 � (2.28) 2 2 � � � � 1 1 3 1 1 3 2 X X SX S X X S X X ' c � � � � � c � � � � c H � � � 3 0 0 0 0 0 0 2 � 2 2 � 2 � 2 2 � Finally, the following expressions for the change of position and divergence of the particle within three revolutions in the machine including the sextupolar effect as a perturbation are obtained: 3 X X SX X ' c � c H 3 0 0 0 2 (2.29) 3 � � 2 2 X X S X X ' c � � � c H 3 0 0 0 4 2.3 Kobayashi Hamiltonian The time needed for three revolutions in the machine is short compared to the spill time and can be safely used as the basic time unit. The elementary changes occurring in this time are also the smallest that need to be resolved to understand the physics of the extraction. Thus the subscripts are no longer needed and (2.29) can be treated as a continuous function that is derived from a Hamiltonian H , such that 3 ' X H � � w ' X � X c � SX X c � � H 3 t X c 2 ' � � w t 1 (3 turn) ' (2.30) X c H 3 � ' � � � w 2 2 ' ' X � X S X X c � � � � � � H 3 t X 4 ' � � w t 1 (3 turn) ' The Hamiltonian is then found by integrating the above partial differentials, S � � � � H 2 2 2 3 H X X 4 3 XX X . (2.31) � c � c � 2 It should be noted that in this formulation time is dimensionless. 22

  20. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.3.1 Properties of the Hamiltonian The Hamiltonian is time independent and therefore a constant of the motion. The plot of this function, for constant values of H, creates maps of particle trajectories in normalised phase space at the sextupole. The above Hamiltonian consists of two parts. The term � � H 2 2 X X (2.32) � c 2 describes particle motion in the linear, unperturbed machine (i.e. S = 0). These trajectories are circles with the radius � (2 H / H ) in normalised phase space. The perturbation term S � � 2 3 4 3 XX X (2.33) c � is proportional to the sextupole strength and distorts the circles. The resulting trajectories are symmetric with respect to the X -axis since X c appears only in a quadratic form. A change in sign of the modified tune distance H corresponds to a reflection about the X c axis and a change in sign of the normalised sextupole strength S is equivalent to a rotation of the phase-space trajectories by 180º as illustrated in Figure 2.3. All the properties of the system can be derived from the Hamiltonian. In particular, when H has the value [(2 H /3) 3 / S 2 ], it factorises into three straight lines, S X 4 4 � � � � � � H H H 3 X X 3 X X 0 . (2.34) � � � � c � � � � c � � � � 4 6 � � 3 S � � 3 S � These three lines are referred to as the separatrices that define the boundaries between stable and unstable regions in phase space. The size of the stable region is determined by the ratio | H / S| . For a particle that has exactly the resonance tune, there are no stable trajectories. Figure 2.4 shows phase-space maps at the sextupole illustrating the dependency of the stable region on the particle tune, represented by H . 1 1 X’ X’ 0.5 0.5 0 0 0.5 0.5 1 1 1 0.5 0 0.5 X 1 1 0.5 0 0.5 X 1 Norm. phase space H � / S 0 > 0 Norm. phase space H � / S 0 < 0 Figure 2.3. Particle trajectories at the sextupole. 23

  21. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ 1 1 X’ X’ 0.5 0.5 0 0 0.5 0.5 1 1 X 1 0.5 0 0.5 1 1 0.5 0 0.5 X 1 Norm. phase space H = 0; S 0 Norm. phase space 0.5( H � / S 0 ) > 0 Figure 2.4. Particle trajectories at the sextupole. 2.3.2 Stable triangle geometry The Hamiltonian yields the equations of the three separatrices from (2.34) that divide up the phase space and hence the geometry of the phase space as shown in Figure 2.5. The equations of the separatrices denoted by A, B and C are explicitly given by: � 2 H B : � 3 1 2 H C : 3 1 2 H . A : X X X X X (2.35) c � c � S 3 2 2 3 S 2 2 3 S A X’ P 3 h P 1 X P 0 stable region C Unstable region surrounding stable P 2 B triangle Figure 2.5. Geometry of separatrices and stable triangle for a third-integer resonance. 24

  22. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ Fixed points in phase space are derived from the Hamiltonian by solving the system H H w w c 0. (2.36) X X w w The phase-space map for a third-integer resonance contains four fixed points. The three unstable fixed points are the crossing points of the separatrices, the stable fixed point P 0 is given by the equilibrium orbit. The coordinates of the unstable fixed points are: P 1 = 0 4 � 2 2 � � 2 2 � � � H H H H H , � P 2 = � , P 3 = � , . (2.37) � � � � � � � � 3 S � 3 S � � 3 S � 3 S 3 S The geometry of the stable triangle is conveniently described by introducing the apothem h of the stable triangle, 2 4 H S G h Q . (2.38) 3 S S x For a positive value of h , the upright separatrix corresponds to a negative value of X . x From the equations of the separatrices and the coordinates of the fixed points it can be seen that a change in sign of h (i.e. either S or H changes sign) is equivalent to a 180 q rotation of the stable triangle around the origin or a reflection about the X c axis. x The area of the stable triangle is given by: 48 3 2 S G 2 � � Area of triangle Stable emittance = 3 3 h Q (2.39) S 2 S The trajectory of a particle in normalised phase space in the unperturbed machine (i.e. S =0) is a circle. The area of this circle is the so called single particle emittance . When the sextupole is raised to the nominal strength in an adiabatic way (i.e. during a number of turns much larger than the revolution period in phase space), a particle with a certain G Q will remain stable if its emittance in the unperturbed machine was not bigger than the area of the stable triangle calculated for the nominal sextupole strength. Therefore the area of the stable triangle is sometimes referred to as the stable emittance . 2.4 Transverse particle dynamics Up to now, only the time-independent geometry of the phase space for a third-integer resonance has been considered, but it is also necessary to analyse the dynamic aspects and to describe particle movement in phase space. 2.4.1 Movement on a separatrix The change of the particle coordinates within three turns in the machine is derived from the Hamiltonian: 3 w H � � ' X c H X c � SXX c 3 2 w X (2.40) 3 w H � � 2 2 � � X ' X S X X ' � � H � � c 3 4 w X 25

  23. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ All separatrices are essentially equal (see Section 2.6) and there is no loss in generality if only the upright separatrix ( A ) from Figure 2.5 is considered. By inserting the equation of the separatrix � 2 H X (2.41) 3 S into (2.40), the phase-space velocity for particles moving on the separatrix is obtained as, H 2 3 2 ' c X � SX and ' X 0 . (2.42) c S 4 It can be seen that there are two contributions to the phase-space velocity in (2.42); the constant term H 2 / S and the quadratic term dependent on the square of X c . The contributions of the two terms are of opposite sign summing to zero at the fixed points P 2 and P 3 . The velocity therefore changes sign when crossing a fixed point. This means that one fixed point acts as an attracting pole with particles on either side moving towards it, while it is the contrary for the other. Thus the separatrix is divided into three sections: the incoming section, the side of the stable triangle and the outgoing section. This is shown schematically in Figure 2.6, the arrows indicate the direction of particle movement. velocity ' X’ X’ 0 fixed fixed point point separatrix incoming outgoing side of section section stable triangle Figure 2.6. Particle movement on a separatrix The separatrices divide the phase space into stable and unstable regions. Trajectories within the stable triangle are closed and correspond to stable particles whereas trajectories outside are not closed and belong to unstable particles. However, the movement of a particle on any phase-space trajectory follows the same direction as the movement on the closest separatrix. Particle movement on stable trajectories in the sense of clockwise or anticlockwise rotation is determined only by the sign of the modified tune distance H and is independent of the sign of the sextupole strength S . This follows immediately from (2.41) and (2.42) as a change of the sign of S reverses the velocity on the separatrix but at the same time rotates the phase-space trajectories by 180 q , resulting in an unchanged direction of rotation of the particles. 26

  24. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.4.2 Phase space maps at the sextupole The geometry of the phase-space map for a third-integer resonance is determined by the normalised sextupole strength S and the modified tune distance H of the particles. Figure 2.7 summarises the possible geometry of the separatrices at the sextupole, the arrows indicate the direction of particle movement. x The size of the stable triangle is proportional to �H� S � � . x The orientation of the stable triangle at the sextupole is determined by the sign of H / S . A positive value corresponds to Figure 2.7 (a),(d), with the upright separatrix at a negative value of X . Changing the sign rotates the phase-space map by 180 q . x The sign of H determines absolutely the direction of rotation in phase space within the stable triangle. A positive value means that the operational tune is above resonance and particles within the stable triangle will perform a clockwise rotation. x Finally, the direction of the outgoing section of the upright separatrix is always downwards for positive values of S and upwards for negative values. H = 6 SG SG Q > 0 H = 6 SG SG Q < 0 H H tune above resonance tune below resonance (a) (b) X’ X’ S > 0 X X (c) (d) X’ X’ S < 0 X X Figure 2.7. Overview of stable triangle geometry at the resonance sextupole. 27

  25. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.5 General first-order Hamiltonian In the Kobayashi theory as derived above, it is assumed that the equilibrium orbit is coincident with the centre of the sextupole. However, the equilibrium orbit of particles with a momentum deviation compared to the design momentum of the accelerator is displaced to first order from the origin by the product of the relative momentum deviation ( G p/p ) and the dispersion function ( D,D c ) in the machine. In normalised coordinates, the equilibrium orbit for an off momentum particle is given by: p p G G X D X D . (2.43) c c EQ.O n EQ.O n p p where ( D n , D c n ) is the normalised dispersion function. When going round the machine, particles will perform betatron oscillations around the orbit defined by the dispersion function. For the derivation of a general Hamiltonian including the effect of an off-centred equilibrium orbit at the sextupole, it is convenient to introduce a second coordinate system ( X E , X c E ) with its origin on the off-centred orbit. The particle coordinates are then split into a constant term, given by the dispersion function, and a betatron part as shown in Figure 2.8. X c X c E p p G G (2.44) X X D X X D � � � E n E n p p X E X X E G p G p (2.45) X X D X X D c c � c � c c � c E n E n p p D n G p/p X Figure 2.8. Coordinates of the betatron motion. The general Hamiltonian is derived following exactly the mathematics of the Kobayashi theory. Particle motion in the unperturbed machine is, as before, described by transfer matrices. For k revolutions in the machine, X X � � � � E E . (2.46) � M � � � X k X c c � � � � E E k 0 The main difference appears when calculating the kick of the sextupole on the off-centred equilibrium orbit: 2 Sextupole in dispersion-free region: ' X 0 ' X c SX (2.47) E E E 2 � p � G 2 Sextupole in dispersion region: ' X 0 ' X c SX S X � D (2.48) � � n E E E p � � 28

  26. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ This yields the more general Hamiltonian: � p � S 3 � � � � G � � H 2 2 2 3 2 2 H X � X � 4 3 X X c � X � S D X � X (2.49) c � � c n E E E E E E E 2 2 p � � It can be seen that the dispersion-dependent term affects (apart from the shift of origin) only the circular part of the trajectories. Reordering (2.49) shows that this can be considered as a change in tune of the particle. � � � � p 1 S G � � � � 2 2 2 3 (2.50) H � � 3 S D � X � X c � 4 3 X X c � X � � H n E E E E E 2 p � � � � The initial tune distance of the particle was given by Q 1 . (2.51) G S H 6 Assuming that the dispersion-dependent term is of the same form leads to the following expression for an additional change in tune, ~ � � p 1 G Q � 6 3 S D . (2.52) � � G n p � � S The above expression contains the normalised sextupole strength and dispersion. Rewriting the tune change in terms of unnormalised coordinates with (2.13) gives 2 ~ 1 � d B � 1 " � p � p G G S z Q � � D k D (2.53) � � � � " c G E E � � x � � x S 2 4 B d x p 4 p S U � � S � � 0 with k c being the normalised sextupole gradient � 2 � 1 d B c � � z k (2.54) � � 2 B dx � � U 0 By introducing the chromaticity Q c as the linear change of the betatron tune with momentum, the chromatic effect of a sextupole can be expressed as ~ � � d Q 1 c � � c � Q k D , (2.55) " � � E S dp / p 4 � � S which corresponds exactly to the well-known linear form for the tune shift introduced by a sextupole. Although the above derivation is restricted to the region in tune close to the third-order resonance the result is in fact generally valid. 29

  27. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ From the above theory it can be deduced immediately that the resonance sextupole is best positioned in a dispersion-free region, otherwise any change of the sextupole strength will also alter the tune distance of the particles and therefore the extraction phase space. The general Hamiltonian describes the physics of a third-integer resonance to first order correctly without any further restrictions on sextupole locations or particle momenta. Comparing (2.50) with the Kobayashi Hamiltonian (2.31) shows that both expressions are of the same qualitative shape although they are defined in different coordinate systems. This means that the considerations about phase-space geometry and dynamics derived in the previous sections can be adopted by simply substituting � � p G o � � � 6 Q 6 Q 3 SD (2.56) H SG S G n � � p G p o � X X D (2.57) n p p G c o c � c X X D (2.58) n p in all the relevant expressions. Equation (2.56) represents the chromatic effect of the sextupole, (2.57) and (2.58) are the coordinate transformations from the system ( X E , X c E ) back to the reference system with its origin at the sextupole centre. The general expressions for the separatrices are given below. Figure 2.9 shows the phase-space geometry at the sextupole for positive values of h , D n , D c n and G p/p. A X’ G p (2.59) A : X D h � � n p h � � � � 3 G p 1 G p X B : � (2.60) X D X D h c � c � � � � � � n n 2 p 2 p � � � � 0 C equilibrium orbit � � � � G p G p B 3 1 C : (2.61) X D X D h c � c � � � � shifted by dispersion � � n n 2 p 2 p � � � � � � ' X D n G / p p � � n G / ' c X D c p p Figure 2.9. Geometry of the separatrices at the sextupole for an off-momentum beam. Finally the change of the phase-space map when the sextupole is located in a non-zero- dispersion region can be summarised: x All phase-space trajectories are shifted by the offset of the equilibrium orbit. x The size of the stable triangle is changed due to the tune change caused by the chromatic action of the sextupole. 30

  28. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ Figure 2.10 shows phase-space maps for an off-momentum particle ( G p / p > 0) with a modified tune distance H 0 in the unperturbed machine. To illustrate the effect of the dispersion, the sextupole position is changed from a zero dispersion region to a region with finite dispersion while keeping the normalised sextupole strength S 0 constant. 1 1 X’ X ’ 0.5 0.5 0 0 0.5 0.5 1 1 1 0.5 0 0.5 X 1 1 0.5 0 0.5 X 1 norm. phase space H � / S � > 0 D n = D n ’ = 0 norm. phase space H 0 / S 0 > 0 D n = D ’ n < 0 Figure 2.10. Change of phase-space map for off-momentum beam due to dispersion at the sextupole. 2.5.1 Closed-orbit distortions In a real machine, it has to be expected that there is a distortion of the equilibrium orbit due to magnet imperfections and misalignments. This will give an effect similar to the one caused by a finite dispersion function. The fundamental difference is that the closed orbit distortion will be to first order independent of the particle momentum. It is straight forward to also include orbit distortions at the sextupole in the Hamiltonian. The coordinates of the particle at the sextupole can be split into three parts: x Contribution of the dispersion function x Closed orbit distortion at the sextupole x Betatron oscillation of the particle The Hamiltonian is derived following exactly the same scheme as above: 1 � p � S � G � � � � � 2 2 2 3 � 3 S D X � X X c 3 X X c X . (2.62) H H � � � ' � � � � � � � n CO. � E E E E E 2 4 p � � � � The geometry and dynamics of the phase-space can be analysed as in the previous section. The shift of phase-space trajectories consists of the momentum-dependent dispersion part and the constant offset given by the closed orbit distortion. The chromatic term shows the same dependency. The total change in tune will be: ~ ~ ~ 1 p 1 � G � � � � � � � CO. Q Q Q k c D k c x (2.63) G G � G � E " � � � E " ' � � Disp. CO. x S x S 4 4 S p S � � 31

  29. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.6 Phase-space maps along the machine To draw the phase-space map at any position s in the machine, it is sufficient to describe the evolution of the separatrices around the machine. All the relevant physics of the slow extraction process can be obtained from this. The following considerations are based on the general Hamiltonian (2.50), derived earlier. It is assumed that there is only one sextupole � , defining the reference position (betatron-phase P = 0) in the machine. 2.6.1 General description of separatrices The description of particle motion in the linear machine (outside the sextupole) is done by using the coordinate systems introduced in Figure 2.8. There are two contributions to the evolution of the separatrices and the stable triangle when tracking around the machine. x The momentum-dependent equilibrium orbit describes the centre of the stable triangle at any position s in the machine G p G , p � � � � � � � � X s D s X s D s (2.64) c c EQ.O n EQ.O n p p x The phase advance ' P from the sextupole to a position s determines the orientation of the stable triangle, in fact the separatrices are simply rotated clockwise around the equilibrium orbit by ' P . This behaviour is implied by the transfer matrix, describing the betatron motion in normalised phase space, which has the form of a rotation matrix, � � � � cos sin P P ' ' � � (2.65) M R � � ' ' P � � � � P � sin cos � P P � ' ' � � The size of the stable triangle is determined absolutely by the normalised sextupole strength S and the modified tune distance H (including chromatic effects) and remains unchanged in normalised phase space. A schematic example is given in Figure 2.11. A X c X c B A h h D n ( s ) G p/p X X 0 0 C D n,Sext G p/p B phase advance ' P = 90° C clockwise rotation by 90° Normalised phase space Normalised phase space at sextupole (a) at element with ' P = 90° (b) Figure 2.11. Normalised phase-space maps separated by ' P = 90°. � This is not a restriction on the generality of the theory at the level of approximation being used. It will be shown later that from the standpoint of resonance excitation (described on a time scale of 3 turns) many sextupoles can be replaced by a single virtual sextupole. 32

  30. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ From the considerations above, it follows that a general description of the separatrices has to contain the dispersion function D n ( s ) and the phase advance from the sextupole ' P ( s ) as parameters. However, the earlier derived expressions (2.59)-(2.61), describing the separatrices at the sextupole, contain only the dispersion function. The constant dispersion function at the sextupole can be simply replaced by D n ( s ). The dependency on the betatron phase advance is implemented via geometrical considerations. A standard form for the description of a straight line is shown in Figure 2.12. y x cos y sin h (2.66) � D D h D The angle D is measured anticlockwise from the x-axis to the perpendicular h . x Figure 2.12. Perpendicular form of a straight line. Rewriting the expressions of the separatrices in the form of (2.66) gives exactly the required angular dependency. With h being the distance from the side of the stable triangle to its centre, the equations of the separatrix will have the form: � p � � p � G G � � � � � X D s � cos � X c D c s � sin h . (2.67) � � � D D � � � � n n p p � � � � By inspection of Figure 2.11.(a), the values for D at the sextupole are: ( A ) D � = 180° anticlockwise ( B ) D � = 300° anticlockwise ( C ) D � = 420° anticlockwise The separatrices rotate clockwise with the betatron phase advance measured from the sextupole. Thus, the effective angle at an element with a phase advance ' P from the sextupole will be ( D � �' P ). For convenience, it is easier to think in terms of ' P appearing as a positive term, which gives: � p � � p � ' ' � � � � � � � � ( A ) X D s cos X D s sin h (2.68) � � ' � c � c ' � � � � P P n n p p � � � � � p � � p � ' ' � � � � � � � � ( B ) X D s cos 120 X D s sin 120 h (2.69) � � ' � q � c � c ' � q � � � � P P n n p p � � � � � � � � p p ' ' � � � � � � � � ( C ) X D s cos 120 X D s sin 120 h (2.70) � � ' � q � c � c ' � q � � � � P P n n p p � � � � Equations (2.68)-(2.70) describe the separatrices and therefore the stable triangle at any position s with a phase advance ' P from the sextupole. It was mentioned earlier that all the separatrices are essentially equal, which is obvious now by inserting the phase advance for one turn, ' P � turn # 2 S n/3 = ±120° mod 360°, in the above equations. This means that a particle will appear to jump from one separatrix to the next on consecutive turns. 33

  31. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.7 Resonance excitation by sextupoles The theory derived so far shows how a single sextupole magnet excites the third-integer resonance. In general, in a real machine there will be several sextupole magnets (e.g. for the correction of chromaticity). Apart from these magnets there will also be some unwanted sextupole field components introduced mainly by the end fields of the main dipole magnets. The common effect of all these sextupole fields on the resonance is described by the so-called driving term N [11 @� For the third-order resonance 3 Q h = n , the driving term is, � � C � � � � 2 � 1 1 d B � � � 3 / 2 z ^ ` s exp 3 i d . (2.71) � � � � N E P x x 2 � B � 24 C d x � � � U � S 0 0 The integral is made around the full machine circumference and includes all sextupole fields. For short sextupoles, the above can be rewritten as a sum, using the normalised sextupole strength S , as defined earlier, 1 � � � � S n exp 3 i (2.72) N P x,n 12 C S n An equivalent sextupole can be found by evaluating the above driving term and equating to a single ‘virtual’ sextupole. � � � � � S exp 3 i S exp 3 i . (2.73) P P virt x,virt n x,n n By separating real and imaginary parts, � � � � � � � � � � S cos 3 S cos 3 S sin 3 S sin 3 , (2.74) P P P P virt x,virt n x,n virt x,virt n x,n n n the betatron phase of the equivalent sextupole is found from � � � S sin 3 P n x,n � � n tan 3 , (2.75) P � � x,virt � S cos 3 P n x,n n and the equivalent sextupole strength is given by 2 2 � � � � � � � � � � 2 � � S S cos 3 S sin 3 . (2.76) P P � � � virt n x,n n x,n � � � � n n The virtual sextupole describes the resonance excitation of all sextupole fields in the machine. The stable triangle will have its characteristic shape with one upright separatrix at the position with the betatron phase P virt of the virtual sextupole. The orientation of the separatrices at an element with a betatron phase P can now be evaluated by inserting ' P = ( P - P virt ) into equations (2.68)-(2.70). 34

  32. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ However, the size of the stable triangle depends on two parameters: x The strength of the sextupoles driving the resonance which is represented by S virt . x The modified tune distance H of the particles given by the initial value H 0 in the linear machine (without sextupoles) and the chromatic effects of all the sextupoles. For the determination of the size of the stable triangle it is therefore necessary to evaluate, separately from the driving term, the chromatic effect of all the sextupoles in dispersion regions which is done in Section 2.8 below. 2.7.1 Combination of sextupoles In a series of an even number of sextupoles with identical normalised strength, S 0 , it can be seen from (2.76) that a separation in phase of ' P = S /3 leads to cancellation of the resonance driving term, � � � � ^ ` � � 2 2 S S cos 3 S cos 3 60 ... � � q � � P P Virtual 0 x,1 0 x,1 . (2.77) � � � � ^ ` � � 2 sin 3 sin 3 60 ... S S � � q � P P 0 x,1 0 x,1 Similarly, a phase separation of ' P = 2 S /3 leads to a reinforcement of the driving term, � � � � ^ ` � � 2 2 cos 3 cos 3 120 ... S S S � � q � � P P Virtual 0 x,1 0 x,1 . (2.78) � � � � ^ ` � � 2 S sin 3 S sin 3 120 ... � � q � P P 0 x,1 0 x,1 2.8 Chromatic effects of sextupoles The chromatic effect of a thin-lens sextupole was derived earlier (2.53), (2.55). In general the effect of sextupoles (in a region of finite dispersion) on the horizontal and vertical chromaticities of the linear machine, d Q d Q � � � � x z Q Q (2.79) c c � � � � x , 0 z , 0 � � � � d / p p d p / p � � � � 0 0 is given by, C C 1 1 � c � � � � � � � c � � � � � � Q k s s D s d s Q k s s D s d s . (2.80) c c ' � E ' E x x x z z x 4 4 S S 0 0 For short sextupoles, using the thin-lens approximation, the above can be rewritten as 1 1 � c � � � c � � Q c k E D Q c k E D . (2.81) ' � " ' " x i i x x x i i z x i i 4 4 S S i i Expressions (2.80) and (2.81) describe the chromatic effect of all sextupoles in the machine. The modified tune distance H of a particle with a momentum deviation G p/p is now obtained as p G 6 Q x c . (2.82) H H � S ' 0 p 35

  33. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ Finally, the concepts for the description of the phase space of a third-integer resonance, derived earlier for a single sextupole magnet, can now be generalised for any sextupole distribution in the machine by using: x Expression (2.82) for the modified tune distance of the particles. x The virtual sextupole strength (2.76) as normalised sextupole strength. x The phase of the virtual sextupole (2.75) as the reference position. 2.8.1 Schemes for chromaticity correction Chromatic effects in accelerators are caused by the momentum dependency of the focal properties of lattice elements. Sextupole magnets, placed in dispersion regions, are used to correct such effects and to adjust the chromaticity of the machine, e.g. to ensure the transverse stability of the beam [12], or to meet the Hardt -condition [13] for resonant extraction (see Section 3.8). For an independent adjustment of the horizontal and vertical chromaticities at least two magnets are needed. It can be seen from (2.80) that ' Q c x and ' Q c z are of opposite signs and therefore focusing ( k c SF ) and defocusing ( k c SD ) sextupoles, as defined in Section 2.1, are used. In general, chromaticity correction is done not with one F- and one D- sextupole, but with groups consisting of several magnets, each group powered by a single power converter. Such sets of magnets with identical fields form so called magnet families . A simple scheme for chromaticity control is given below. Consider two sextupole families, one focusing with strength k c SF , the other one defocusing with k c SD . Assuming that all the magnets are of the same length and using the thin-lens approximation, gives two expressions for the change of the chromaticities, NF ND � � " � � � � � � S ' c � Q k D � c k D (2.83) c � E E � x SF x x n SD x x n 4 S � � n 1 n 1 NF ND � � " � � � � � � S Q k D k D , (2.84) ' c c � c � E E � z SF z x n SD z x n 4 S � � n 1 n 1 with NF and ND being the numbers of magnets within the focusing and defocusing families. To get some degree of orthogonality for the correction, the positions of the F-sextupoles should be in ‘large E x -small E z ’ regions and the D-sextupoles should be in ‘small E x -large E z ’ regions. The unwanted effect of a chromaticity correction is that in general the resonance driving term will also be changed. It should be noted that spacing the chromaticity sextupoles at intervals of 60 0 in betatron phase as shown in (2.77), is not sufficient for a cancellation of the resonance driving term. It can be seen that (2.83) and (2.84) contain the normalised sextupole gradient k c , but for a cancellation the normalised sextupole strengths have to be identical. This means it is also essential that the horizontal beta functions are identical at all sextupole locations. 36

  34. Theory of the third-integer resonance Chapter 2 ________________________________________________________________________________________________________________________________________________________________________________________________________ 2.9 Sextupole families for small machines Equation (2.71) for the driving term of the third-integer resonance shows that in principle all sextupole magnets in a machine contribute to the resonance excitation. Thus, any correction of the chromaticity affects not only the modified tune distance H for an off- momentum particle, but at the same time changes the strength of the resonance excitation. Ideally one would like to have completely independent handles for the control of the resonance strength and the horizontal and vertical chromaticities, especially in a machine using a third-integer resonance as these parameters determine the extraction set-up. In small medical machines, it is not always possible to have several sextupole families, mainly for the lack of space and for the cost of the additional power supplies. As mentioned above there are three parameters to control, x the resonance excitation represented by S virt , x the horizontal chromaticity Q c x , x the vertical chromaticity Q c z . Therefore three independent power supplies will be needed and the minimum number of sextupole magnets will be three. From (2.80) it can be seen that sextupoles in dispersion-free regions do not change the chromaticity, for this reason sextupoles that excite or control the resonance are best placed in a zero dispersion region. Unfortunately, for the chromaticity sextupoles no similar situation exists, but by taking advantage of the rather special third-integer tune, an approximate solution that leaves the resonance excitation unchanged, can be found when the tune is near an even number (i.e. Q h | n r 1/3, with n even). The scheme is based on a lattice with a periodicity of two, which means the ring can be split in two halves with identical sequences of magnetic elements. Due to the symmetry, the dispersion and beta functions on opposite sides of the ring will be equal. Consider now two sextupoles placed on opposite sides of the ring. If the betatron phase at the first sextupole is taken as the origin, the second sextupole will then be at P x = Q S | ( n r 1/3) S� The effect on the resonance and the chromaticities is now evaluated with (2.76), (2.83) and (2.84). Resonance excitation: 2 2 2 � � � � � � � � S S cos 0 S cos 3 Q S sin 0 S sin 3 Q (2.85) � � � S S virt 1 2 1 2 S S S (2.86) r virt 1 2 Since the tune is very close to ( n r 1/3) S� the sine terms are zero and the cosine terms are unity. The upper sign is for n odd and the lower sign for n even. Chromaticity: " " � � � � S S Q c D k c k c ; Q c D k c k c (2.87) ' � � ' � E E x x x 1 2 z z x 1 2 4 4 S S 37

  35. Chapter 2 Theory of the third-integer resonance ________________________________________________________________________________________________________________________________________________________________________________________________________ Because of the equal lattice functions at the sextupoles the relation between normalised and unnormalised sextupole strength (2.13) can be written as c � c and c � c , with k S k S 1 1 2 2 c = constant. Now (2.87) can be rewritten as " " � � � � S S Q c D c S S ; Q c D c S S (2.88) ' � E � ' E � x x x 1 2 z z x 1 2 4 4 S S From (2.86) and (2.88) it can be seen that for n even, the excitation of the resonance is proportional to ( S 1 - S 2 ) whereas ( S 1 + S 2 ) changes the chromaticity. Consider now a family of two sextupoles in series on one power converter. When connected with the same polarity (i.e. S 1 = S 2 ) the sextupoles change only the chromaticities without affecting the resonance, connecting them with opposite polarity (i.e. S 1 = - S 2 ) has the converse effect. For n odd, there is no such fortuitous situation. 38

  36. Chapter 3 RESONANT SLOW EXTRACTION The use of a resonant slow extraction scheme allows the continuous extraction of particles from a synchrotron during a large number of turns. Slow extraction is a dynamic process that can be described by combining the different phase-space properties that characterise particle behaviour in a synchrotron with a horizontal betatron tune close to a third-integer, as derived in the previous chapter. For clarity and simplicity it is assumed that there is only one sextupole for the resonance excitation, located in a zero-dispersion region. The chromaticities of the machine and the tunes are therefore independent of the resonance strength. It is also assumed that the resonance is positioned in the machine centre, i.e. the resonance tune ( G Q = 0) corresponds to the central orbit momentum ( G p = 0). 3.1 Set-up of the extraction process First, the linear machine is considered. During injection and acceleration of particles, the presence of resonances has to be carefully avoided, as this leads to immediate particle loss. This is done by adjusting the tune values such that they are sufficiently far from dangerous resonances and by compensating for non-linear field errors that could drive resonances. At the end of the acceleration process the so-called flat top is reached (i.e. the fields in the main magnets are kept constant). The resonance sextupole is still switched off and the particle trajectories are circles in normalised phase space. The radius of each circle is determined by the particle position and divergence, ( X 0 , X c 0 ), and is referred to as normalised amplitude A , with 2 . 2 A X X (3.1) � c 0 0 The circle area is the so-called single particle emittance , � � S 2 2 2 E A X X . (3.2) � c S 0 0 Before starting the extraction process, the circulating beam has to be positioned with respect to the resonance. The horizontal tune is moved closer to the resonance by adjusting the currents in the quadrupole magnets and the resonance sextupole is slowly raised to the nominal strength S in a quasi-adiabatic way. As shown earlier, the circular trajectories are deformed and the phase space is split into stable and unstable regions by the separatrices that define the stable triangle. A particle with a certain tune distance G Q remains stable if its single particle emittance in the unperturbed machine was not larger than the area of the corresponding stable triangle. Figure 3.1 shows the trajectory of a stable particle at the position of the sextupole in normalised phase space. The particle starts on a circle and ends on an almost exactly triangular shaped trajectory but remains stable. 39

  37. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ Figure 3.1. Particle trajectory during adiabatic raise of sextupole, normalised phase space. 3.2 Resonance stopband The resonance sextupole strength is one of the main parameters for the slow extraction. For a nominal value of S , the size of the stable triangle for a certain tune distance G Q , is fixed. The stability of a particle with G Q depends on its initial amplitude in the unperturbed machine. A criterion is found by comparing (3.2) and expression (2.39) for the size of the stable triangle, as 48 3 S 2 2 � � S E A Q . (3.3) d S G stable stable 2 S A particle will be stable if its emittance is smaller than or equal to the area of the stable triangle. Therefore the area of the stable triangle is often called stable emittance . Inverting the inequality in (3.3) and solving with respect to G Q gives the tune interval for which particles with a certain amplitude will be unstable. This interval is referred to as the stopband of the third-integer resonance, 1 Q A S . (3.4) � G 48 3 S Equation (3.4) can be rewritten in terms of absolute tune with G Q Q Q as � particle resonance 1 1 . (3.5) Q � S A � Q � Q � S A res part res 48 3 48 3 S S The above expressions are best illustrated graphically in the Steinbach diagram as shown in Figure 3.2. 40

  38. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ Amplitude A , resonance line � emittance, � ( unstable region stopband for A 0 particles with stable stable amplitude A 0 region region Q res -| G Q ( A 0 )| Q res +| G Q ( A 0 )| Q res tune, Q Figure 3.2. Steinbach diagram for third order resonance stopband, amplitude-tune space. From (3.4) it can be seen that for every tune distance G Q , there is a maximum stable amplitude A ( G Q) . Particles will be unstable if their amplitude is bigger than A ( G Q ) and stable if their amplitude is smaller. There are no stable amplitudes for particles on resonance, i.e. all particles with G Q = 0 are unstable. The range of stable amplitudes increases when the tune is moved away from the resonance. Replacing the inequality in (3.4) by an equality yields the equation of the boundary between stable and unstable regions in Figure 3.2, sometimes referred to as resonance line , Q G A 48 3 . (3.6) S S Particles that exactly fulfil (3.6) are on the limit of stability, which means they are moving on a stable triangle in phase space. 3.3 Overview of extraction methods The width of the third-integer stopband is symmetric with respect to the resonance, therefore there is no loss in generality when considering only particles with G Q < 0. It is assumed that the particles are stable after the machine has been set to extraction conditions. In order to extract particles from the stable region in Figure 3.2, they either have to be moved into the unstable region or the size of the unstable region has to be increased. Inspection of (3.6) shows that this can be done by: x Widening the stopband by increasing the resonance sextupole strength S . x Moving the particles into the stopband by changing their tune values. x Increasing the particle amplitudes until the critical value A ( G Q ) is reached. The different methods are explained in more detail below. Extraction by changing the resonance strength For the description of the extraction process two particles with a tune distance G Q 0 and amplitudes A 1 and A 2 are considered. Figure 3.3 (a) shows the situation before the start of extraction. Both particles are stable, their amplitudes are smaller than the critical 41

  39. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ amplitude A ( S 0 ) given by (3.6) and their phase space trajectories are deformed circles inside the stable triangle defined by S 0 and G Q 0 . A , � E A , � E G Q 0 G Q 0 A ( S 0 ) A 1 = A 1 (a) (b) A ( S 1 ) A 2 A 2 tune Q tune Q Q res - G Q 0 Q res - G Q 0 Q res Q res Figure 3.3. Extraction via resonance sextupole change. When increasing the sextupole strength, the unstable region widens, the critical amplitude A ( S ) is decreasing which also means, the size of the corresponding stable triangle decreases. Figure 3.3 (b) shows the extraction process at the moment when the critical amplitude has reached the value of the amplitude A 1 . This means the particle is moving on a stable triangle and will start going outwards on one of the separatrices, finally being extracted. Basically, the same applies to the particle with the smaller amplitude A 2 , but later on, when the sextupole strength has been further increased. The main difference is that this particle will be extracted from a smaller stable triangle. Inspection of equation (3.6) shows that zero-amplitude particles, with a finite tune distance, can never be extracted as this would require an infinite sextupole strength. Extraction by changing the tune distance The substantial difference from the above method is that, instead of increasing the resonance strength, the tune of the particles is slowly moved towards resonance tune. In Figure 3.4 (a) both particles are stable, their tune distance is larger than the width of the resonance stopband. A , � E A , � E G Q 0 G Q 1 A ( G Q 0 ) A 1 A 1 (b) (a) A ( G Q 1 ) A 2 A 2 Q res - G Q 0 tune Q Q res - G Q 1 tune Q Q res Q res Figure 3.4. Extraction via tune change. It can be seen that every amplitude is linked to a critical tune value at which the particle becomes unstable. The smaller the particle’s amplitude, the further away it’s tune from the corresponding critical tune. Therefore, when the tune is moved towards the resonance, the particle with the larger amplitude will be extracted earlier, as indicated in Figure 3.4 (b). As in the extraction method described above, particles that reach the limit of stability are moving on stable triangles. The size of these triangles is again determined by the initial particle amplitudes. 42

  40. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ Extraction by increasing the particle amplitude The third possibility for extraction is to increase the transverse amplitudes of the particles. The main difference compared to the extraction techniques described earlier is that particles with the same tune distance will be extracted from stable triangles with equal size, independent of the initial particle amplitudes. 3.4 Spiral step After a particle has become unstable, it starts moving on the outgoing part of one of the separatrices. In Section 2.6, it was shown that all three separatrices are essentially equal, they simply exchange their position from turn to turn. However, when analysing the extraction process, it is more convenient to consider the separatrices as geometrically fixed in phase space. Then, an unstable particle moves out along a separatrix, one turn later it is on the next and after every third turn it returns to its initial separatrix. Earlier (2.40), the change of the particle coordinates within three revolutions in the machine was derived as 3 ' X X c � SX X c H 3 0 0 0 2 (3.7) 3 � � 2 2 X c X S X X c ' � � � H 3 0 0 0 4 By inserting (3.7) into (3.1) the amplitude increase during three turns, for a particle that is just on resonance (i.e. H = 6 SG Q = 0) is obtained as 3 2 A A A S A . (3.8) ' � 3 0 0 4 From (3.8) it can be seen that the growth increases rapidly as the particle progresses along the unstable section of the separatrix. After a certain number of turns, the particle amplitude has increased so much, that the particle ‘jumps’ into the electrostatic septum and is extracted. Depending on the phase advance from the sextupole, ' P ES , the electrostatic septum is positioned in the outer or inner side of the vacuum chamber. The phase advance has to be chosen such that only one separatrix is intercepted by the electrostatic septum. Figure 3.5 gives a schematic view of the normalised phase space at the resonance sextupole and the electrostatic septum for a phase advance of ' P ES = 225°. extraction X’ X’ separatrix 3 ' R c 0 2 ' R 2 1 T X ES X X phase advance ' P ES = 225° 1 clockwise rotation by 225° 0 electrostatic extraction septum 3 separatrix Figure 3.5. Amplitude increase during last three turns before extraction. 43

  41. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ The maximum possible amplitude, A last , of a particle, which is not extracted, can be calculated from the normalised position of the electrostatic septum, X ES , and the angle T between the extraction separatrix and the horizontal axis as 1 A X . (3.9) last ES cos( - ) Three turns later the particle will be extracted and by inserting (3.9) into (3.8), the increase of amplitude during these last three turns in the machine is found to be 2 3 3 X � � 2 ES A S A S . (3.10) ' � � last 4 4 cos - � � From (3.10) it can be calculated how far the particle jumps into the septum, the so-called spiral step ' R , and the spiral pitch ' R c (see Figure 3.5), 3 1 2 R A cos - S X , (3.11) ' ' last ES 4 cos - and 3 tan - 2 R c A sin S X . (3.12) ' ' - last ES 4 cos - The spiral step and pitch define the size of the extracted beam. It is important to note that the slow-extracted beam differs markedly from the circulating beam. In the plane of extraction, the phase-space footprint is a narrow rectangle that corresponds to the segment of the outgoing separatrix that is cut and deflected by the electrostatic septum and the emittance is visibly smaller then the circulating beam’s one. In the orthogonal plane, the phase space distribution and the emittance are to first order the same as those of the circulating beam [14]. This asymmetry of the extracted beam requires special techniques for the design of the beam delivery system towards the patient, discussed in Chapter 7. It should be noted that (3.11) and (3.12) are only valid for particles which are exactly on resonance ( G Q = H = 0). The spiral step and pitch for particles with a finite tune distance are in general different and can be evaluated by transforming the coordinates of a particle, that just misses the electrostatic septum, to the position of the resonance sextupole and then applying (3.7). Equation (3.11) gives the maximum distance that a particle can jump into the septum. The space on the extraction separatrix from the position of the electrostatic septum to this point will be filled by particles with different starting conditions, some of the particles will hit the septum directly and will be lost. Therefore the extraction efficiency depends critically on the thickness of the septum. This is the main reason for the use of an electrostatic septum, which is in generally built as a wire septum with a wire thickness of the order of 0.1 mm. Electrical fields up to 100 kV/cm can be obtained in such septa. The drawback is that these fields are often not sufficient to kick the beam directly out of the machine. The small deflection provided by the electrostatic septum translates into a physical gap between the circulating beam and the extracted particles further downstream, where one or more magnetic septa are positioned to finally extract the beam. Ideally the extraction channel has to be designed such that the only particle loss occurs at the electrostatic septum, which is unavoidable due to its finite thickness. A detailed discussion of the layout of the extraction channel is given in Sections 3.9 and 3.10. 44

  42. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ 3.5 Effect of the electrostatic septum The electrostatic septum cuts off particles from the separatrices. Particles that have jumped into the septum are deflected by the electrostatic field, corresponding to an angle M . This kick transforms into a physical gap between the extracted and the circulating part of the separatrix further downstream, where the magnetic septum is positioned. X’ X’ A kick M Magnetic Septum B X X Electrostatic Septum A B phase advance P gap Figure 3.6. Kick of electrostatic septum in normalised phase space. The gap size is calculated by comparing the movement of two on-momentum ( G p = 0) particles from the electrostatic to the magnetic septum; particle A starts just inside the electrostatic septum, B just outside, as shown in Figure 3.6. The thickness of the electrostatic septum (typically 0.1 mm) is neglected, both particles are assumed to start from the radial position of the septum x ES , and with the same angle x c ES but only particle A receives the kick M . With the 2x2 Twiss-matrix (real phase space), the positions and angles of the particles at the magnetic septum are obtained as A: x m � x � m � x c � m � M 11 12 12 MS ES ES x c m � x � m � x c � m � M MS 21 ES 22 ES 22 and (3.13) B: x m x m x c � � � MS 11 ES 12 ES , x m x m x c � � � c MS 21 ES 22 ES where the suffices ES and MS denote the positions of the electrostatic and magnetic septa. Thus, the effect of the kick appears at the magnetic septum as a difference in position and angle of the particles, ' x m (3.14) MS 12 M � and x MS m , (3.15) ' c � M 22 where ' x MS is the gap, available for the thicker magnetic septum, which is explicitly given by ' x MS MS sin . (3.16) M � E � E � P ES From (3.16) it follows that to relax on the kick (voltage), that has to be provided by the electrostatic septum to create the space for the magnetic septum, the lattice functions of the machine and the positions for the septa have to be chosen such that: x The phase advance between the septa is close to 90° + n � 180 q . x The beta functions at the septa have reasonably large values. 45

  43. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ 3.6 Separatrix geometry at electrostatic septum So far, in all the figures of the normalised phase space at the electrostatic septum, the extraction separatrix has been shown in the first quadrant, where X and X c > 0, with a positive kick, M > 0. In fact, only the first and third quadrants are useable. This follows from purely geometrical considerations as shown in Figure 3.7. For the first and fourth quadrant X is always positive, which means that the kick of the electrostatic septum must also be positive, otherwise the extracted particles are driven back into the septum wires. For the first quadrant, a positive kick opens a gap for the magnetic septum further downstream, whereas it is the contrary in the fourth quadrant, as a positive kick drives the extracted beam segment closer to the machine centre. The same considerations apply when comparing second and third quadrant operation. X’ NO O.K. Kick is positive at a Kick is positive at a positive position and negative position and creates a gap for the drives beam back to I magnetic septum II centre and wires X Kick is negative at a Kick is negative at a III IV positive position and negative position and drives beam back to creates a gap for the centre and into wires magnetic septum NO O.K. Figure 3.7. Possible working quadrants for the electrostatic septum. A further limitation for the separatrix geometry at the electrostatic septum is introduced by the angular spacing of 120° between adjacent separatrices. From equation (3.16) it was found that, for an optimum usage of the electrostatic septum’s kick, the phase advance to the magnetic septum should be 90° + n ·180°. Figure 3.8 shows a logical layout for first quadrant operation with the extraction separatrix at 45° at the electrostatic septum and a phase advance of 90° to the magnetic septum. X c X c safety kick margin phase advance ��q MS ES 45° -45° X X safety margin Figure 3.8. Ideal separatrix geometry at electrostatic and magnetic septa for first quadrant operation. 46

  44. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ The extraction separatrix could be rotated by a maximum of 15° anticlockwise before the safety margin in Figure 3.8 is lost and the preceding separatrix hits the electrostatic septum as shown in Figure 3.9 (a), or by 15° clockwise before the following separatrix hits the magnetic septum, Figure 3.9 (b). This limits the possible angles for the extraction separatrix at the electrostatic septum to 45° ± 15° for first quadrant operation and to 225° ± 15° for third quadrant operation. X c X c (a) Limited at kick electrostatic phase advance ��q septum. MS ES 60° -30° X X no safety margin (b) X c X c Limited at no safety magnetic margin septum. kick MS ES 30° -60° X X phase advance ��q Figure 3.9. Limitations of separatrix geometry by electrostatic and magnetic septa. The alternatives to the extraction layout with a phase advance of 90° between the septa, as shown above, are limited. x The first is to accept a much smaller phase advance, which might be imposed by lack of space, but this requires a stronger electrostatic septum which may have implications for the reliability. x The second is to use 270° phase advance, but this has two unfavourable aspects. Firstly, the extracted beam has to be transported for a longer distance in the machine, which for a small synchrotron means that crossing of non-linear magnetic elements (e.g. chromaticity or resonance sextupoles) is almost unavoidable. Any change in these elements would result in a change of the extraction geometry. Secondly, a phase advance of 270° (more generally 270° + n ·360°) means that the electrostatic and magnetic septa are on opposite sides of the vacuum chamber. This may have the drawback that aperture is lost for the circulating beam by ‘encasing’ it between the septa. 47

  45. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ 3.7 Phase-space representation of beam and resonance Up to now only single particles have been considered, but the beam in an accelerator always has a certain momentum spread G p / p and, due to the chromaticity in the machine, this momentum spread translates into a tune spread according to G p G Q Q c . (3.17) p The size of the beam in an accelerator is characterised by the emittance , which is defined as an area in phase space that contains a certain percentage of the beam particles. In general, a beam will contain particles with amplitudes between zero and a maximum amplitude, corresponding to a maximum single particle emittance, which is numerically equal to the total emittance of the beam. At a given position s in the machine, the beam can be represented in phase space by a series of ellipses (circles in normalised phase space), centred around the dispersion vector D ( s )· G p/p , as shown in Figure 3.10. x c Real phase space Areas are emittance of betatron motion for each momentum. x ( G p/p ) 2 ( G p/p ) 1 ( G p/p ) 0 X c Normalised phase space Areas in normalised phase space are still equal to the emittance. X Figure 3.10. Representation of a beam in phase space and normalised phase space. The circles that represent the beam emittance in normalised phase space become triangles under the influence of the resonance. The resonance for particles of each momentum is represented by a triangle, corresponding to the last stable orbit, and the extensions along the outward separatrices. According to (3.17) each momentum corresponds to a certain tune distance from the resonance and the corresponding triangles are of different sizes as shown in Figure 3.11. X’ Normalised phase space Size of the stable triangles depends on the tune distance from the resonance. Orientation of the triangles X depends on phase advance from resonance sextupole. ( G p/p ) 2 ( G p/p ) 1 ( G p/p ) res Figure 3.11. Normalised phase space representation of a beam under the influence of the resonance. 48

  46. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ When considering the interactions of the beam and the resonance, it is very convenient to use the Steinbach diagram that represents the stopband of the third-integer resonance. With (3.17) the expression for the stopband (3.6) can be rewritten with the momentum spread and the chromaticity as Q c p G A 48 3 . (3.18) S � S p It is sometimes more convenient to use the momentum spread rather than the tune as abscissa, since the momentum spread is an independent beam parameter (changing the chromaticity will change the tune but not the momentum). Figure 3.12 (a) shows the circulating beam before extraction, corresponding to the phase space representation in Figure 3.10. Figure 3.12 (b) shows the beam during the extraction process, corresponding to the phase-space diagram 3.11. A , � E A , � E �G p/p) inst P 2 band of A max A max unstable particles stack stack P 0 �G p/p) stack �G p/p) res ' p/p �G p/p) 2 �G p/p) res = �G p/p) 0 ' p/p (b) (a) Figure 3.12. Steinbach diagram of the beam before and during extraction. It can be seen that during the extraction a quasi-static situation is reached and particles with all amplitudes in the beam enter the resonance. These particles have different momenta and, due to the chromaticity, different tunes, which means the corresponding stable triangles are of different sizes. Figure 3.13 shows the separatrix geometry at the resonance sextupole and the electrostatic septum in normalised phase space. X’ X’ Extraction separatrices M X ES X X phase advance ' P ES = 225° clockwise rotation by 225° Electrostatic septum Extraction separatrices Figure 3.13. Separatrices at the resonance sextupole and the electrostatic septum. The dotted lines are the separatrices corresponding to zero amplitude particles P 0 in Figure 3.12 (b), the full lines are the separatrices of particles with the maximum amplitude, P 2 . 49

  47. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ The separatrices of all other particles are found in between these two extreme cases. The instantaneous momentum spread of the extracted beam can be calculated from (3.18) with the maximum amplitude A max as p 1 S G A . (3.19) max p Q c 48 3 S inst. 3.8 Hardt condition During the extraction process, there will in general be particles with all amplitudes becoming unstable at any given instant and being extracted along separatrices from stable triangles with different sizes. As shown in Figure 3.13, the separatrices at the electrostatic septum are in general not superimposed, which means that particles, moving outwards along different separatrices, reach the septum with different angles. Therefore, not only particles that hit the septum cross-section directly will be lost, but due to the finite septum length, also particles that are moving close to the septum wall with a certain divergence. x X’ x > x ES E 0 X c ES x ES x < x ES particle trajectories X X ES s Septum length L ES (a) (b) Figure 3.14. Particle losses at the electrostatic septum due to angular spread of separatrices. Figure 3.14 (a) shows the trajectories of two particles in real space, one starting just short of the electrostatic septum at x < x ES , and the other just beyond the septum wires at x > x ES . From geometrical considerations it follows that particles starting from x < x ES will be lost along the septum wall if their relative divergence, compared to the septum angle, is larger than ( x ES - x )/ L ES . A similar situation exists for particles inside the septum with x > x ES , except that the electric field curves their orbits and modifies the forbidden angles. Figure 3.14 (b) shows the so called ‘shadow’ regions (grey shaded) in normalised phase space. All the particles on the segments of the separatrices that cross the shadow regions are lost along the septum wall. It was mentioned earlier that stable triangles of different sizes correspond to different momenta. At a position in the machine where the dispersion is non-zero, the triangles will be shifted according to their momentum (see Section 2.6). With a finite dispersion at the electrostatic septum, this effect can be used to superimpose the extraction separatrices in order to avoid particle losses in the shadow regions. It should be noted that in Figures 3.13 and 3.14 (b) all the triangles are centred which means the dispersion was assumed to be zero at the electrostatic septum. The condition, to arrange the separatrix geometry and to set the optics at the electrostatic septum such that all separatrices are superimposed, is known as the ‘ Hardt condition’ [13]. The requirements on the lattice functions can be derived with a 50

  48. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ purely mathematical approach from the general expressions for the separatrices (2.67)- (2.70), given in Section 2.6, � p � � p � G G � � � � � � � � cos sin . (3.20) � X D s � � X c D c s � h � � ' � � � ' D P D P � n � � n � 0 0 p p � � � � The angle D � describes the orientation of the separatrices at the sextupole, ' P is the phase advance from the sextupole to the considered lattice element and h is the distance from the side of the stable triangle to its centre, 2 4 H S G h Q . (3.21) 3 S S With (3.17) the above expression can be rewritten as 2 4 p H S G h Q c . (3.22) 3 S S p Inserting (3.22) into (3.20) gives the general expression for the separatrices as a function of particle momentum and chromaticity of the machine, p p 4 p � � � � G G S G � � � � � � � � X D s cos X D s sin Q . (3.23) � � � � ' � � c � c � � ' c D P D P � n � 0 � n � 0 p p S p � � � � It can be seen that the size of the stable triangles and the displacement of the separatrices due to the dispersion is determined by the momentum of the corresponding particles. In order to superimpose the extraction separatrices the momentum dependency has to be removed from the equation of the separatrices, giving the Hardt condition, 4 S � � � � D cos D sin Q . (3.24) � ' � c � ' � c D P D P n 0 n 0 S The above equation may appear very flexible, but there are some boundary conditions that restrict the parameters. x The choice of the angle ( D � - ' P ) is restrained by the geometry of the extraction. For optimised operation in the first or third quadrant, as discussed earlier, it can be shown that the phase ( D � - ' P ) is either 135° for particles with tune values below resonance ( G Q < 0), or 315° for particles above resonance ( G Q > 0). The different possibilities for the extraction layout are summarised in Figures 3.17 and 3.18. x The sextupole strength cannot be used as a variable since it determines the spiral step and spiral pitch and therefore the horizontal size of the extracted beam. x The horizontal chromaticity Q c is the main variable, but for small, low-energy machines which are working below transition, the chromaticity should be negative to ensure the stability of the coasting beam [12]. x The normalised dispersion function depends on the lattice. The task of the designer is to foresee a position for the electrostatic septum with suitable values of D n and D c n . 51

  49. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ The left-hand side (LHS) of (3.24) can be seen as a scalar product of two vectors, � ' � � � � D cos( ) D P � n � � 0 � � LHS = . (3.25) � � � � c � ' D sin( ) � � � � D P n 0 where the (cos,sin) vector is the unit vector, perpendicular to the extraction separatrices as was shown in Section 2.6. The value of the LHS is therefore fixed by the extraction geometry and the dispersion vector at the position of the electrostatic septum. In order to superimpose the separatrices and to fulfil the Hardt condition, the right hand side of (3.24) is then adjusted by changing the horizontal chromaticity of the machine. For a more physical insight, the expressions for the stopband of the resonance (3.18), the momentum spread of the extracted beam (3.19) and the Hardt condition (3.24) have to be considered together. A change of the chromaticity changes the width of the stopband (the slope of the resonance line) and therefore the extracted momentum spread. This means that the momenta corresponding to different-size triangles change and the triangles are shifted along the dispersion vector, according to their new momenta. In general, the extraction should be arranged such, that the momentum spread of the extracted beam is small for two main reasons: x The optics of the beam delivery system towards the patient (Chapter 7) is rather complicated. In a gantry it is likely that the dispersion takes large values which, together with a large momentum spread, leads to enlarged magnet apertures and therefore bigger and more expensive magnets. x The transfer between electrostatic and magnetic septum is in general chromatic (see Section 3.9). These chromatic effects can be partially corrected with a higher voltage of the electrostatic septum, but are less important for a small momentum spread. Inspection of (3.19) shows that the extracted momentum spread will be small for large absolute values of chromaticity and therefore a large absolute right-hand side in (3.24). The LHS (3.25) is zero for perpendicular vectors, which is the case when the normalised dispersion is parallel to the extraction separatrix. For this configuration it is obvious that the separatrices cannot be superimposed for any finite value of chromaticity. (The zero chromaticity extraction is considered in Section 3.11.) The absolute value of the LHS will have a maximum for parallel vectors in (3.25), this means, the normalised dispersion is directed at right angle to the extraction separatrices and the shift of triangles with different momenta, relative to each other, is most effective. The following Figures 3.15 and 3.16 show the separatrix geometry at the electrostatic septum for first quadrant operation. The full lines represent the extraction separatrices corresponding to zero amplitude (zero-size triangle) and maximum amplitude (maximum-size triangle) particles. The dotted lines indicate the position of the maximum- size triangles if the dispersion function was zero. In 3.15 the normalised dispersion vector is almost parallel to the separatrices, thus the absolute value of the LHS is small, whereas in 3.16 the normalised dispersion is perpendicular to the separatrices and the LHS is large. In 3.15 (a) and 3.16 (a) the chromaticity is not adjusted to fulfil the Hardt condition, the RHS of (3.24) is equal for both diagrams. In 3.15 (b) and 3.16 (b) the chromaticity has been adjusted in order to superimpose the separatrices. In 3.15 (b) the dispersion vector is not well suited, the final absolute chromaticity is small and the extracted momentum spread is large, as can be seen from the Steinbach diagram. In 3.16 (b) the normalised dispersion vector is perpendicular to the separatrices that are superimposed for a small momentum spread of the extracted beam and a large absolute chromaticity. 52

  50. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ (a) X c A , � E shift band of D n ( G p/p) unstable D c n particles D n stack ' p/p X, D n �G p/p) � D n ( G p/p) extracted momentum spread (b) X c A , � E shift band of unstable D n ( G p/p) D c n particles D n stack ' p/p X, D n �G p/p) � large extracted D n ( G p/p) momentum spread Figure 3.15. Hardt condition for small chromaticity and large momentum spread of extracted beam. (a) X c A , � E band of unstable D c n particles D n ( G p/p) stack ' p/p �G p/p) X, D n � extracted D n shift momentum spread D n ( G p/p) X c (b) A , � E band of unstable D c n particles D n ( G p/p) stack ' p/p �G p/p) X, D n � small extracted D n shift momentum spread D n ( G p/p) Figure 3.16. Hardt condition for large chromaticity and small momentum spread of extracted beam. 53

  51. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ Figures 3.17 and 3.18 summarise all the possible extraction layouts for optimised first and third quadrant operation (derived according to Section 2.6). The arrows indicate the required direction of the normalised dispersion vector to fulfil the Hardt condition with the smallest possible momentum spread of the extracted beam. The dotted lines are the extraction separatrices for the zero-amplitude particles which are exactly on resonance ( G Q = 0, G p = 0), the full lines correspond to maximum amplitude particles with a momentum deviation G p and a tune deviation G Q = Q cG p / p . c > 0 c < 0 Q c Q c (a) G Q > 0 (b) G Q < 0 X’ X’ � D n D c � c n � � G p/p > 0 G X X � � D n D c c n � � ES ES (c) G Q < 0 (d) G Q > 0 X’ X’ � � D n D c c n � � G p/p < 0 G X X � D n D c � c n � � ES ES Figure 3.17. Separatrix geometry for first quadrant operation, ideal normalised dispersion vector. c > 0 c < 0 Q c Q c (a) G Q > 0 (b) G Q < 0 X’ X’ � � D n D c c n � � ES ES G G p/p > 0 X X � � D n D c c n � � (c) (d) G Q > 0 G Q < 0 X’ X’ � D n D c � c n � � ES ES G G p/p < 0 X X � D n D c � c n � � Figure 3.18. Separatrix geometry for third quadrant operation, ideal normalised dispersion vector. 54

  52. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ It was mentioned earlier that for a small, low-energy machine that works below transition the chromaticity should be negative in order to ensure the transverse stability of the beam [12]. This constraint leaves only four possibilities for the extraction geometry at the electrostatic septum. In Figures 3.17 (b) and (d) the electrostatic septum is on the outside of the vacuum chamber, the dispersion required for the Hardt condition is D n > 0 and D c n < 0. In Figures 3.18 (b) and (d) the electrostatic septum is on the inside of the vacuum chamber, the dispersion required for the Hardt condition is D n < 0 and D c n > 0. In general, in small machines the dispersion D n is positive. This fact leaves only the two layouts in Figures 3.17 (b) and (d), for the extraction. Figure 3.19 shows the qualitative shape of the normalised dispersion for a dispersion bump, an arc with regular distributed dipoles and a regular cell structure. The favoured position for the electrostatic septum is on the downward slope of the dispersion (shaded areas), where D n > 0 and D c n < 0. D n Correct orientation for ( D n ,D n ’ ) in a dispersion bump P D n Correct orientation for ( D n ,D n ’ ) in an arc with spread out dipoles P D n Correct orientation for ( D n ,D n ’ ) in a regular cell lattice P Figure 3.19. Qualitative shape of the normalised dispersion for typical lattice structures. In Figures 3.17 (b) and (d) the electrostatic septum is in the outer half of the chamber. Earlier, in (3.16) it was shown that the ideal phase separation between electrostatic and magnetic septa is P = 90° + n ·180°. An odd value of n (i.e. P = 270°, 630° etc.) means that the septa are on opposite sides of the chamber and, with the electrostatic septum being on the outside, the magnetic septum has to go to the inside. One feels intuitively that the extraction should not be done to the inside of the machine because of the high magnetic rigidity of the beam. Therefore, the phase separation should be around 90° to have both septa on the outside of the chamber. A larger phase advance of 90°+ n ·360°is less convenient as the extracted beam has to be transported for a longer distance in the machine. The final decision for the extraction layout of Figure 3.17 (d) is based on aperture considerations. The resonance should be positioned in the centre of the chamber, (i.e. the resonance tune should correspond to the central orbit momentum) to ensure a balanced growth of the separatrices. During the extraction set-up the beam has to be kept sufficiently far from the resonance. With both septa being on the outside of the vacuum chamber it is natural to position the beam in the inner half of the chamber to avoid aperture limitations. In this case the beam is below the resonance in momentum and, due to the negative chromaticity, the resonance is reached from above in tune. 55

  53. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ 3.9 Transfer between extraction septa Generally, during the extraction process, particles with different momenta are extracted at the same time. When the Hardt Condition is fulfilled, all the extraction separatrices are superimposed at the electrostatic septum and all particles reach the septum on the same separatrix, independent of their momentum. In Section 3.5, the transfer of on-momentum particles between electrostatic and magnetic septa was considered. However, the extracted beam has a certain momentum spread, and in general, the trajectories of particles with a momentum deviation are different to those of on-momentum particles. This may cause a reduction of the space available for the magnetic septum, as calculated in (3.16). It is therefore also necessary to analyse the transfer of particles with the maximum momentum deviation. This is done in a similar way to above, by considering the trajectories of two particles, C and D, starting from the radial position of the electrostatic septum, x ES with the septum angle x c ES , but only particle C receives the kick M of the septum. In a linear lattice, the movement of particles with a momentum deviation can be described with the 3x3 transfer matrix formalism. The horizontal coordinates of the particles at the magnetic septum are derived as � � C: , x m x m x m M � m G p / p c � � MS ES ES 11 12 12 13 � � , x m x m x m m p / p M � G c � c � MS 21 ES 22 ES 22 23 and (3.26) � � D: x m x m x m p / p , G c � � MS 11 ES 12 ES 13 � � . x m x m x m G p / p c � c � MS 21 ES 22 ES 23 Comparison of (3.26) and (3.13) shows that the separation between circulating and extracted particles ' x MS is the same as calculated for on-momentum particles, but the gap appears at a different position and angle. The shift in position reduces the effective gap width for the magnetic septum. Figure 3.20 shows the transfer between the septa for particles representing the full momentum spread of the extracted beam. The grey shaded areas represent particles with momenta in between these limits. X’ X’ A, C Magnetic kick M Septum phase advance P B, D X X C Electrostatic A Septum D B gap eff C A ' div extr gap eff D B gap for G p = 0 ' " extr Figure 3.20. Transfer from electrostatic to magnetic septum. 56

  54. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ The effects of the chromatic terms, m 13 and m 23 in the transfer matrix, on the phase space geometry at the magnetic septum and the extracted beam are described below. x A non-zero m 13 causes a loss of space for the magnetic septum and has to be corrected with a stronger kick of the electrostatic septum. The effective gap width is � � gap m m G / p p . (3.27) M eff � � � 12 13 x The horizontal width of the extracted beam is increased, and, to avoid losses inside the magnetic septum, the horizontal aperture has to be enlarged. � � ' l m 13 G / p p (3.28) extr � x A non-zero m 23 is leading to a larger overall divergence of the extracted beam at the magnetic septum and also requires an enlarged horizontal aperture to avoid losses. � � ' div m 23 G / p p (3.29) extr � It should be mentioned that at the electrostatic septum any angle error of the extraction separatrices will increase particle losses. At the magnetic septum it is usual to foresee a small clearance of say a few millimetres between beam and septum and therefore angular spreads up to 1mrad (approx.) will not lead to losses. For this reason, only the m 13 term will be considered in the further discussion. 3.9.1 Minimisation of chromatic effects One way of reducing the above mentioned disadvantages is to arrange the extraction such, that the momentum spread of the extracted beam is small. However, the momentum spread is not a free parameter as it is fixed when applying the Hardt condition to avoid losses at the electrostatic septum. It is therefore of prime importance to optimise the lattice in such a way that the superposition of the extraction separatrices results in a small momentum spread of the extracted beam. The explicit form of the element m 13 , expressed in terms of normalised dispersion, � � m D D cos D c sin , (3.30) � � � E P P 13 MS n , MS n , ES n , ES shows that the loss of space for the magnetic septum is proportional to � E MS . Decreasing E MS reduces the influence of m 13 , but the gap created by the electrostatic septum (3.16) is also proportional to � E MS and becomes smaller. The effective gap (3.27) for the magnetic septum is reduced by decreasing E MS . � � p p G � P G � . (3.31) gap m � m sin � D � D cos � D sin c � � M E E P P eff n n n 12 13 MS ES , MS , ES , ES p p � � The only effective approach to reduce chromatic effects is to minimise m 13 directly. 57

  55. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ 3.10 Achromatic transfer between extraction septa A momentum-independent transfer between electrostatic and magnetic septa requires the chromatic term m 13 in the general 3x3 transfer matrix to vanish. Inspection of (3.30) shows that the size of m 13 is determined by the normalised dispersion functions at the septa. In order to reduce m 13 , the lattice needs to provide suitable values of the dispersion at the positions of the septa. The shape of the dispersion function in a lattice is determined by the distribution of the dipole magnets. In the analysis below, the effect of the dipoles is approximated as a point kick. In this model, every passage of a dipole adds a kick D c n,0 to the actual value of the normalised dispersion function, according to � � � � D 0 D � � � � � n n � � � . (3.32) � c c � c D � D � � D D � c � � n n ,0 n n ,0 , � � � � � � � � entrance po kick exit of int to dipole of dipole dipole In a bending-free region, the dispersion function acts like the betatron oscillation of a particle and can be described with the 2x2 Twiss-matrix formalism. The transformation of the dispersion function between two lattice elements, denoted by the suffices 1 and 2, without crossing dipoles is given by � � � � � � D m m D � � � 11 12 � � � � . (3.33) � � c c D m m D � � � � � � 21 22 2 1 In normalised phase space, the normalised dispersion vector simply rotates by the phase advance ' P between the elements, according to � � � ' ' � � � cos sin D D P P � n � � � � � n � . (3.34) � � � � c c � ' ' D sin cos D � � � � � � P P n n 2 1 Combining (3.34) and (3.32) allows an approximate analysis of the transfer between the extraction septa. Some particular cases are demonstrated below. Both septa in a bending-free dispersion region D c n D n ( D n , D c n ) ES kick (b) (a) dipole1 D n ES MS kick phase 90° dipole2 ( D n , D c n ) MS 180° Figure 3.21. 180°dispersion bump. (a) Normalised coordinates ( P , D n ), (b) normalised phase space ( D n , D c n ). Figure 3.21 shows a dispersion bump as created by two dipole point kicks of identical strength, spaced by 180° in phase. The electrostatic septum is positioned in the first half of the bump, the magnetic septum 90° later, in the second half. 58

  56. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ To prove that in such a case the transfer is always achromatic, the closed orbit for particles with a momentum deviation G p is considered. The closed orbit at any position s in the machine is to first order defined by the product of dispersion function and momentum deviation, � � � � � � � � � � � � x s , p D s � p / p and x c s , p D c s � p / p . (3.35) G G G G EQ.O. EQ.O. The 3x3 transfer matrix links the orbit coordinates between different positions (1 and 2) in the machine according to x ( p ) m m m x ( p ) � � � � � � G G 2 11 12 13 1 � � � � � � x c ( p ) m m m x c ( p ) . (3.36) � G G � � � � � � 2 21 22 23 1 � � � � � � p / p 0 0 1 p / p � G � � � � G � By inserting (3.35) into (3.36), general expressions for m 13 and m 23 in terms of the dispersion functions and the 2x2 Twiss-matrix elements are found, m D m D m D c and m D c m D m D c . (3.37) � � � � � � � � 13 2 11 1 12 1 23 2 21 1 22 1 In regions without bending the dispersion function transforms according to (3.33) and by inserting this into (3.37) it follows directly that m 13 and m 23 are zero and therefore: x The transfer via a dispersion region without crossing bending magnets is always achromatic with respect to position and angle. It should be noted that this result is exact and valid not only inside a dispersion bump, but in any lattice section without bending. A typical structure where this result can be applied is the so called ‘square’ lattice. The dipole magnets are grouped in the corners, to create dispersion bumps on two opposite sides and dispersion-free straight sections on the remaining two sides. The fact of the fully achromatic transfer between the septa may give the impression that such a structure is best suited for a slow extraction scheme, but there are two major problems with a square lattice: x The natural position for the electrostatic septum (i.e. the first half of the dispersion bump) is the worst place to put it when adjusting for the Hardt condition. It was shown in Section 3.8, that for ideal operation D n > 0 and D c n < 0, as in the second half of a bump is required, whereas in the first half D n > 0 and D c n > 0, as can be seen from Figure 3.21. x The second problem is that in general there is too little useable phase advance in the dispersion straight-section. An important fraction of the 180° is lost because of the length of the corner dipoles. To get a reasonable phase separation between the extraction septa, they have to be positioned close to the corners on either side of the straight section. Such a layout requires a very strong kick from the magnetic septum in order to clear the dipole magnets that close the dispersion bump. Moving the magnetic septum closer to the centre of the drift space reduces the phase separation from the electrostatic septum. In this case, the electrostatic septum has to kick very hard, to create the space needed for the magnetic septum. 59

  57. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ Both septa in regions with dispersion and bending D c n D n P kick kick dipole1 D n MS (b) dipole2 (a) ES ( D n , D c n ) ES kick phase dipole3 ( D n , D c n ) MS >180° Figure 3.22. Extended dispersion bump, (a) normalised coordinates ( P , D n ), (b) normalised phase space ( D n , D c n ). Figure 3.22 shows a so-called ‘extended-dispersion-bump’. The free space within the dispersion bump is increased by adding a central dipole in order to overcome the problems of the ‘square’ lattice. The disadvantage of such a structure is, that in general, the transfer between the extraction septa is chromatic, as a bending magnet has to be crossed. Some design guidelines can be found for an extraction layout with optimised phase advances of P = 90 q + n � 180 q between the septa. Using expression (3.30) for the chromatic term, � � m D D cos D sin , (3.38) � � � c E P P 13 MS n, MS n, ES n, ES and assuming a phase advance P = 90 q + n � 360° between electrostatic and magnetic septa gives � � m D D . (3.39) E � � c 13 MS n, MS n, ES To make the transfer achromatic ( m 13 = 0), D D c (3.40) n, MS n, ES is required, but as shown earlier, one needs to work with a negative D c n,ES for adjusting the Hardt condition, therefore m 13 can only be made zero by having negative dispersion at the magnetic septum. For a phase advance of P = 270 q + n � 360 q (septa on opposite sides of the vacuum chamber) it follows � � m D D c (3.41) E � � 13 MS n, MS n, ES and therefore to make the transfer achromatic, D D c (3.42) � n, MS n, ES is required. In this case, m 13 can be made zero by having a positive D n,MS and a negative D c n,ES just as required for the Hardt Condition. A disadvantage of this solution might be that the particles which are extracted, have to be transported for a longer distance in the machine (e.g. crossing of sextupoles between the septa would be more difficult to avoid). � � If a sextupole is crossed (either resonance or chromaticity) between the ES and the MS, then there is a variable optical element in the extraction channel. Any change in the Q c or resonance-strength alters the extraction geometry. 60

  58. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ Electrostatic septum in dispersion region and magnetic septum in zero-dispersion region D c n D n P T kick dipole1 ES MS D n (a) (b) kick phase dipole2 180° ( D n , D c n ) ES Figure 3.23. 180°dispersion bump. (a) Normalised coordinates ( P , D n ), (b) normalised phase space ( D n , D c n ). Figure 3.23 shows a 180° dispersion bump; contrary to the extraction layout in Figure 3.21, the electrostatic septum is positioned in the second half of the bump where D n > 0 and D c n < 0, as required for the Hardt condition. The magnetic septum is positioned in the dispersion-free straight section after the dipoles that close the bump. As for the extended dispersion bump, the transfer between the septa is in general chromatic due to the passage of dipoles. Assuming the dispersion bump is created by dipole point kicks, the normalised dispersion function can be described with (3.32) and (3.34) as � � � � D D sin and D D cos , (3.43) - c � - - c � - n n , 0 n n , 0 where - is the phase advance counted from the first dipole kick and D c n,0 is the strength of the kick. With D n,MS = 0 in (3.30), the chromatic term m 13 can be written as � � m D cos D sin , (3.44) � � � � c � E P P 13 MS n, ES n, ES and by inserting (3.43) into (3.44) a simple expression for m 13 is obtained, � � m � � D � sin � . (3.45) E - P 13 MS n ,0 It follows immediately that the transfer between the septa will be achromatic( m ) for 0 13 � � - P n 180 (3.46) � � Obviously it is impossible to use exactly n = 1, since this gives the position of the dipole which is closing the dispersion bump. To keep m 13 small, the magnetic septum has to be positioned as close to the dipole as possible � . This solution has been adopted for the lattice of the medical synchrotron. The electrostatic septum is positioned in the second half of the dispersion bump, which allows the Hardt condition to be fulfilled with negative chromaticity and a small extracted momentum spread while keeping the chromatic effects small. The only resulting compromise is a reduced phase advance of less than 90° between the septa. � For larger n , there is again the problem of transporting the extracted part of the beam through a large distance in the machine. 61

  59. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ Transfer for un-fulfilled Hardt Condition Adjusting for the Hardt Condition fixes the chromaticity and therefore the slope of the resonance line (3.18) and the momentum spread of the extracted beam (3.19). Instead of superimposing the extraction separatrices at the electrostatic septum, the chromaticity can be used to arrange the extraction geometry such that particles with different momenta arrive at the septum with different angles. With a proper choice of these starting conditions a reduction of the space available for the magnetic septum can be avoided even though the transfer between the septa is chromatic. This method is used in the present CERN-PS slow extraction scheme [15]. Transfers between zero dispersion regions By inspection of the general expression (3.30) for m 13 , it follows immediately that transfers between zero-dispersion regions are always fully achromatic. However, with the electrostatic septum located in a zero-dispersion region, the separatrices can be superimposed only for zero chromaticity, which will be discussed later. 3.11 Comparison of extraction techniques In order to slowly extract a beam, either the particles have to be moved progressively into the resonance stopband or the stopband has to be moved onto the beam. This means that some physical quantities have to be changed. The varying parameter, the way of changing it and the initial geometry of the beam in the amplitude-momentum (amplitude-tune) space distinguish the extraction method. In Section 3.3, a general overview of the different extraction techniques was given. 3.11.1 Varying the sextupole strength Extraction by changing the sextupole strength is generally not used. The main argument against a change of the resonance strength is found by considering expressions (3.11) and (3.12) for the spiral step and pitch that determine the size of the extracted beam. As both terms are proportional to the resonance strength S , any change of S during the extraction leads to a change of the beam size and therefore, (in the case of a medical machine), the beam spot at the patient, which is not acceptable. 3.11.2 Varying the tune distance The most common extraction method is to slowly change the tune distance G Q , of the circulating particles. There are two fundamentally different approaches. x To move the resonance through the beam by changing the betatron tune of the machine with quadrupole magnets. x To move the beam into the stationary resonance by acceleration (deceleration) of the circulating particles. The change in momentum translates via the chromaticity into a change of the tune (3.17). The main problem, when moving the resonance through the beam, is that the Hardt condition is ineffective. The extraction separatrices can be superimposed at any time but during the extraction they change their position. This means that the divergence of particles that arrive at the electrostatic septum changes from the beginning to the end of the extraction process. This variation in angles also affects the transfer between the extraction septa and reduces the space available for the magnetic septum. For the 62

  60. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ compensation of these effects, e.g. a dipole correction bump, that changes the orbit around the electrostatic septum, can be used. However, such a dynamic correction complicates the operation of the machine and should be avoided, especially for a medical machine where simplicity and reliability of operation are of major concern. Figure 3.24 (a) shows the principle of this method for a beam with a large horizontal emittance E x and a small momentum spread. The first particles that are extracted have large amplitudes and a large tune distance ( G Q = Q cG p / p ), the small amplitude particles are extracted at the end of the spill with a small G Q . The variation of the mean tune distance of the extracted particles results in the problem of a changing beam size during extraction as can be seen from equations (3.7). The extracted momentum spread is always equal to the total momentum spread of the beam. Figure 3.24 (b) shows the same extraction technique for a beam with a small transverse emittance and a large momentum spread. During the whole spill particles of all amplitudes are extracted, the beam size remains therefore unchanged. The instantaneous momentum spread of the extracted beam is small but the mean momentum varies. This has two undesirable features. x Firstly, it could require dynamic changes of the septa strengths and the magnets in the extraction line within a single spill. x Secondly, for a medical machine, a change of the mean momentum of the beam during the spill causes a variation in the depth of the Bragg-peak in the target volume. Extraction via tune change by moving the resonance Small horizontal emittance Large horizontal emittance (a) (b) Large momentum spread Small momentum spread Amplitude Amplitude ' p / p ' p / p unstable unstable Amplitude Amplitude particles particles ' p / p ' p / p unstable unstable Amplitude Amplitude particles particles ' p / p ' p / p x Momentum of extracted beam varies x Size of extracted beam varies x Size of extracted beam is constant x Momentum of extracted beam is constant Figure 3.24. Slow extraction via changing the tune by moving the resonance. 63

  61. Chapter 3 Resonant slow extraction ________________________________________________________________________________________________________________________________________________________________________________________________________ For the second method, the particles are accelerated and their tune changes due to the chromaticity according to (3.17). In this case the resonance is stationary, the Hardt condition can be applied, the transfer between the septa is unchanged during the extraction. A further advantage is that all lattice parameters are kept constant during the extraction and all lattice power supplies have constant settings. Figure 3.25 (a) shows the extraction process for a beam with a large horizontal emittance and a small momentum spread. As in Figure 3.24 (a) the first particles that are extracted have large amplitudes, the main problem is that the stationary resonance transforms the horizontal emittance into a momentum spread. It can be seen that the mean momentum of the extracted beam varies during a single spill, resulting in the same difficulties as discussed for Figure 3.24 (b). The most preferable situation is shown in Figure 3.25 (b). The beam has a large momentum spread and a small horizontal emittance. During the whole spill time particles of all amplitudes are extracted. The momentum spread of the extracted beam is small and the mean momentum is constant. With such a geometry, all machine parameters can be kept constant and no dynamic correction systems are needed. Extraction via tune change by acceleration (deceleration ) of the beam Small horizontal emittance Large horizontal emittance (a) (b) Large momentum spread Small momentum spread Amplitude Amplitude ' p / p ' p / p Amplitude Amplitude unstable unstable particles particles ' p / p ' p / p Amplitude Amplitude unstable particles unstable particles ' p / p ' p / p x Size of extracted beam is constant x Size of extracted beam varies x Momentum of extracted beam varies x Momentum of extracted beam is constant Figure 3.25. Slow extraction via changing the tune by acceleration (deceleration) of the beam. 64

  62. Resonant slow extraction Chapter 3 ________________________________________________________________________________________________________________________________________________________________________________________________________ Acceleration (deceleration) of the beam into the resonance There are a number of different methods for the acceleration of the beam into the stationary resonance. The best known are listed below, this is a partial list and a complete discussion of the different techniques would go beyond the scope of this thesis. x Acceleration of the beam with the rf cavity. x Use of longitudinal stochastic noise [16]. x Use of a betatron core, inductive acceleration [17,18]. 3.11.3 Varying the amplitude (zero chromaticity extraction) The third method for slow extraction is to increase the particle amplitudes until they enter the resonance stopband. This method is particularly well suited to an extraction set-up based on a machine with zero chromaticity. In Section 3.1 it was shown that the stability of a particle, for a given sextupole strength, is determined by its tune distance G Q and its amplitude A . From (3.17) it follows with Q c = 0 that all particles have the same tune distance and their stability is therefore only a function of the particle amplitude. This in turn means that all particles that become unstable will be extracted from stable triangles of equal size, independent of their momentum. From the expression for the Hardt condition (3.24), it can be seen that the right hand side will be zero for Q c = 0. Inspection of (3.25) shows that in this case, the separatrices can be superimposed by simply positioning the electrostatic septum in a zero- dispersion region. In order to further optimise the extraction geometry, the transfer between electrostatic and magnetic septa should be achromatic. In Section 3.10, it was shown that transfers between zero-dispersion regions are always fully achromatic. Therefore the magnetic septum should also be positioned in a dispersion-free region, 90° downstream from the electrostatic septum. Finally, the extraction process is performed by blowing up the beam with transverse rf excitation. Rf kicks applied at the betatron frequency (product of fractional horizontal tune and revolution frequency) cause the betatron amplitudes to grow and particles to enter the stopband. Alternatively, transverse stochastic noise can be used to blow-up the beam. Amplitude unstable Amplitude Resonance particles Stopband ' p / p ' p / p Figure 3.26. Zero chromaticity extraction by transverse beam blow up. Figure 3.26 shows the amplitude momentum space and the resonance stopband for the zero-chromaticity extraction. During the extraction all machine parameters are kept constant; as for the extraction technique with the stationary resonance, no dynamic correction systems are needed. The extracted beam has the momentum spread of the circulating beam, the mean momentum and the beam size are constant during the extraction. The only disadvantage of this extraction method is the lack of intrinsic transverse stability of the coasting beam, due to the zero-chromaticity. 65

  63. Chapter 4 TRANSIT TIME The precise control of the slow-extracted beam is a key point in designing a synchrotron for hadrontherapy. The flux of extracted particles, in particular, should be as smooth as possible in order to avoid overdosing. A more detailed study of the dynamics of the extraction is thus needed to get a deeper understanding and to choose the extraction scheme that suits the need for a constant spill best. The first step is to evaluate the time that a particle spends in the accelerator after becoming unstable and before being extracted, which will be referred hereafter as transit time [19]. 4.1 Translated hamiltonian The motion of the particles is described by the Kobayashi hamiltonian (2.31). This is an approximate hamiltonian, chosen because it can be manipulated relatively easily and because numerical simulations show that the final expressions describe the physics to an accuracy of a few percent. In Figure 4.1 the geometry of phase space for positive H and S is shown with the direction of motion of a particle just out of the stable triangle illustrated by dashed arrows. The equivalent pictures when H and/or S change sign can be determined by mirroring/rotating Figure 4.1 as shown in Chapter 2. Furthermore the position in which one of the separatrices intercepts the electrostatic septum downstream in the lattice, is shown. Electrostatic Septum Intercept A X’ P 3 h P 1 X P 0 stable region C Unstable region surrounding stable P 2 B triangle Figure 4.1. Separatrices geometry at the sextupole. Since, as already mentioned in Chapter 2, after each revolution the separatrices exchange their position and since the infinitesimal time considered is three revolution periods, all the separatrices are equivalent and there is no loss of generality in choosing a particular one. 67

  64. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ When making an extraction, the stable area is shrunk over a period of the order of 1 s, whereas the revolution time in the machine is typically of the order of 1 P s and the transit time of the order of some hundreds of revolutions. Thus, particles that cross the limiting separatrix of the stable area stay very close to the separatrices during their extraction. In the following, the particles that become unstable along separatrix A in Figure 4.1 will be considered. The proximity of the extraction trajectories to the separatrices A and C means that the influence of the stable point P 3 will be strong and the particles will spend most of their time in the vicinity of this point. It is convenient therefore to change the coordinate system and translate the origin to P 3 , as illustrated in figure 4.2. X ' Electrostatic Septum Position A X P 3 h P 1 P 0 Stable region C Unstable region surrounding stable P 2 B triangle Figure 4.2. Translation of the reference system to stable point P 3 . The Kobayashi hamiltonian H [m], in the new coordinates, reads S 3 2 2 3 H ( 4 h 6 h X 6 3 h X X ' 3 X X ' X ) (4.1) � � � � 4 2 H [m 1/2 ] is the apothem of the triangle and S [m -1/2 ] is the normalised where h 3 S sextupole strength. To simplify the notation, from now on we will omit the bar over H , X and X ’. The Kobayashi hamiltonian has been derived by approximating the phase-space displacement of a particle every third revolution of the accelerator, and considering the time of 3 turns as the infinitesimal increment for time. Thus, in the following, the unit of measurement of the time is 3 revolution periods of the machine, dt { 3 T REV but is dimensionless, as already mentioned in Chapter 2. The times derived from the following formulae, multiplied by 3, will thus give the number of turns necessary to reach the electrostatic septum. 68

  65. Transit Time Chapter 4 ________________________________________________________________________________________________________________________________________________________________________________________________________ The equations of motion in the new reference frame are given by: dX w H S � � 6 3 hX 6 XX ' ( a ) � dt w Y 4 (4.2) H dX ' w S � � 2 2 12 hX 6 3 hX ' 3 X ' 3 X ( b ). � � � � � dt X 4 w The transit time for a particle will be worked out integrating the equations of motion between the starting position and the electrostatic septum position. 4.2 Extraction time with constant parameters If H and S are constant during the extraction time, the particle will follow a trajectory with constant H . Thus it is possible to work out the trajectory equation by equating the general hamiltonian to the value that it assumes with the initial conditions ( X 0 , X c 0 ). From (4.1) S 3 2 2 3 ( 4 h 6 hX 6 3 hX X ' 3 X X ' X ) � � � � 4 (4.3) S 3 2 2 3 ( 4 h 6 hX 6 3 hX X ' 3 X X ' X ). � � � � 0 0 0 0 0 0 4 The total transit time is calculated in the following by evaluating and summing the time needed by the particle to travel from P 3 to the electrostatic septum and the time from the position where the particle jumps across the separatrix to P 3 . As an example the expressions derived, will be evaluated for the following static parameters: x T REV = 0.5 P s x Spill length = 0.5 s = 10 6 T REV x G Q = 3.782 10 -3 x H = 6 SG Q = 7.13 10 -2 x O † = 15 10 -6 x S = 27.7897 m -1/2 x h = (2/3)( H / S ) = 1.71 10 -3 m 1/2 x n ‡ = 4.79 4.2.1 Time from nearest fixed point In the vicinity of the fixed point P 3 , that is | X |, | X c | << h , the third order terms in X and X c can be neglected. Simplifying, the trajectory equation (4.3) results in 2 2 X 3 X X ' X 0 � 0 0 � X ' (4.4) 3 X † The value of O is chosen to correspond to the distance moved by the separatrix during 3 turns in the example in next sections, when acceleration is taken into account. Its meaning will be explained later. ‡ The value of n is a measurement of the ES position. Its definition is given later in this paragraph. 69

  66. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ where X 0 and X c 0 are the coordinates of the starting point. The substitution of this trajectory into the equation of motion for X , equation (4.2) ( a ), yields: dX S 6 6 � 2 2 � 6 3 hX X 6 X X ' X . (4.5) � � � � � 0 0 0 dt 4 � � 3 3 � � This equation can be integrated by variable separation. By definition, the trajectory equation (4.4) is valid only close to P 3 , but as the particle approaches asymptotically the 1 separatrix the third order terms in the hamiltonian cancel out. With this X ' X � 3 consideration they can also be neglected far from P 3 along the outgoing separatrix and integration can be extended to the electrostatic septum. Using the standard integral formula: 2 1 1 2 ax b b 4 ac � � � � dx ln (4.6) 2 ax bx c 2 2 b 4 ac 2 ax b b 4 ac � � � � � � valid for b 2 > 4 ac , the transit time T c,s can be written as t T X � 0 c 2 ES 1 � � T dt dX c , s 2 2 3 S X 3 hX X 3 X X ' � � � � 0 0 0 t X 0 0 X ES � � 2 2 � X h h X X X � 2 3 9 4 3 ' 2 1 � � � � � 0 0 0 ln . � � � � � � S 3 2 2 � 2 2 � h X X X X h h X X X 9 4 3 ' 2 3 9 4 3 ' � � � � � � � 0 0 0 0 0 0 � � X 0 Since | X 0 |, | X c 0 | << h , we can expand the square roots to first order: X ES 2 X 3 X X ' � 2 � � � � 0 0 0 2 X 3 h 3 h 1 � � � � � � � � 9 2 h 2 � � � � T ln � � | c , s � � 3 3 Sh 2 X 3 X X ' 2 � � � � 0 0 0 � 2 X 3 h 3 h 1 � � � � � � 9 2 � h � � � � � � � X � � 0 (4.7) X ES 2 X 3 X X ' � � � 0 0 0 X � � � � 2 3 h ln . � � | 2 3 3 Sh X 3 X X ' � � � 0 0 0 X 3 h � � � � � 3 h � � X 0 It is useful to express all the distances in units of h, thus we define 70

  67. Transit Time Chapter 4 ________________________________________________________________________________________________________________________________________________________________________________________________________ X ES - nh , X 0 - O h , . X ' 0 2 3 h � / These definitions are illustrated in Figure 4.3 X ' X ES = - nh ES A X P 3 X c 0 = -2 � 3 / h h P 1 P 0 X 0 = - O h C P 2 B Figure 4.3. Definition of O , / and n . For O = 0, / = 0 corresponds to the stable point P 3 and / = 1 to stable point P 2 . Using the new parameters O , / and n in (4.7) gives: 1 1 � � � � 2 2 nh 3 h h 3 h 3 h O O O O O � � / � � � / 2 3 3 T s ln | � c , 1 1 � � � � 3 3 Sh 2 2 nh 3 h 3 h h 3 h O O O O O � � � / � � / 3 3 2 n 3 ln . | � n 3 O 3 3 Sh � 2 Inside the logarithm, the term � � h 2 has been neglected with respect to nh in the O 3 O � / 3 first fraction and with respect to O h in the denominator of the second fraction (since / is small near to P 3 ). In the numerator of the second fraction only the term 3 h has been kept. 2 H Remembering that , it results that the transit time from the nearest fixed point in h 3 S the static case is 71

  68. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ 1 n 3 T s ln . (4.8) | c , n 3 3 O H � Note that, provided / << 1, the time needed for a particle to be extracted is independent of the initial X c coordinate. For the numerical data given, T c,s = 95 which means that the particle needs 285 revolutions to reach the ES. A numerical simulation of the extraction process, yields 312 revolutions for / = 0.1 and 306 revolutions for / = 0. The simulation just advances the particle, applying the displacement that it undergoes each third revolution, that is the Kobayashi equations (2.29), until it reaches the ES. The number of iterations multiplied by 3 gives the number of revolutions. This simplified tracking is necessary in order for the used quantities h , O etc to be meaningful. An exact tracking, as will be seen better in Chapter 6, yields to a distortion of the separatrices that destroys the simple parameterisation used herein. The validity of this simple description will be confirmed by the results described in Chapter 5, which are checked against an exact tracking. 4.2.2 Time spent along the side of the stable triangle For a particle starting somewhere along the side of the stable triangle, far from the unstable fixed point P 3 , the extraction time is given by the sum of the time needed to travel to P 3 and the time needed to go from P 3 to the ES. The motion along the side of the triangle is determined by the second equation of motion, (4.2) ( b ). The trajectory of a particle starting near the side of the stable triangle is not a straight line but it remains close to the side for most of its length. Until | X| << | X c | it is possible to approximate equation (4.2) ( b ), by neglecting the terms in X and X 2 , so that ' dX S � � 2 6 3 hX ' 3 X ' (4.9) | � � dt 4 which, similar to the previous case, can be integrated by variable separation, yielding: 1 2 3 A / � T ln (4.10) | s 1 , s 1 A 3 � / H where the integration has been done between the starting point X ' 0 2 3 h and � / � a point near the corner where the curve will be joined to the trajectory X F ' Ah � calculated in the previous Section 4.2.1. As the junction point has to be near P 3 to allow the use of (4.8), A has to be smaller or of the order of 0.1 � 2 3 , which corresponds to / = 0.1. 72

  69. Transit Time Chapter 4 ________________________________________________________________________________________________________________________________________________________________________________________________________ 4.2.3 Time for particles starting along the side of the stable triangle To correctly join the two trajectories, it is necessary to evaluate the position of the start of the second. To do this we can evaluate the hamiltonian (4.1) at the starting point � � and work out the X F coordinate at X c F = - Ah . From h , 2 3 h � � / O S 2 3 2 � � � � 3 ( 4 h 6 hX 6 3 hX Ah 3 X Ah X ) � � � � � � F F F F 4 S � � � � � � 2 � � 2 � � � � ) 3 3 ( 4 h 6 h O h 6 3 h O h 2 3 h 3 O h 2 3 h O h � � � � � / � � � / � � 4 neglecting the higher order terms in O and X F , X F is obtained � � � � 12 1 12 1 / � / / � / X h h h . (4.11) O O O � � � | � { � F F A 2 3 A 2 A 3 � Starting from this point it is possible to evaluate the total time needed to reach the electrostatic septum: 1 2 3 A n 3 1 2 3 n 3 / � / T T T ln ln . � | | s , s s 1 , s c , s 1 A n 3 O 1 A n 3 O 3 3 H H � / � F � / � F And finally, by substituting the definition of O F (4.11), the transit time for a particle starting along the side of the stable triangle in the static case can be evaluated as 1 1 n 3 T s ln (4.12) | s , � � 2 n 3 O 3 1 H � � / which does not depend on A , that is on the exact position of the junction point between the two trajectories. This expression is valid until (1- / ) >> O , such that the velocity in X c does not depend on X . A comparison of (4.12) with numerical simulations is summarised hereafter Formula (4.12) Numerical / (revolutions) Simulation (revolutions) 0.1 285 312 0.5 312 339 0.9 393 423 0.999 615 675 4.3 Extraction time with varying parameters Resonant slow extraction can be “activated” in many ways, but the most common is to vary the tune Q of the machine, by varying the focusing quadrupoles, to bring the beam into resonance. Alternatively it is possible to vary the momentum of the particles, as G Q = Q c ' p / p . Varying G Q causes the triangle to shrink, in fact H 6 Q is directly SG 73

  70. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ proportional to the apothem h of the stable triangle. Reducing the size of the triangle causes some particles to pass from the stable region inside the triangle to the unstable region outside and thus to be extracted. In this Section a linear variation of the tune with � = constant, will be considered. time, that is Q The main approximations used in the following are: x the relative variation in the size of the triangle during the transit time will be small, that is ' h << h ; x instead of considering the movement of the separatrices as the triangle shrinks, the relative movement of the separatrix to the particle, will be considered as an additional contribution to the particle velocity and the triangle will be considered fixed during the extraction time. This approximation is illustrated in Figure 4.4, where the situation is sketched at two different times t 1 and t 2 , with t 2 > t 1 . Stable region at t 2 Particle Particle position position at t 2 at t 2 Particle Particle Stable region Stable region position position at t 1 & t 2 at t 1 at t 1 at t 1 ' h ' h Real situation Approximation Figure 4.4. Relative motion of particle and stable triangle. For numerical checks few more data are needed beyond the ones used in the static case. It will be assumed: x 'G Q per turn = -1.891 10 -8 Q = 'G Q in 3 turns = -5.673 10 -8 � x x O = - ' h in 3 turns/ h = - � Q /( G Q) 0 = 15 10 -6 4.3.1 Time from nearest fixed point It was shown in the static case, that in this region the extraction time does not depend on the initial value / . Following this hint, / is set to zero and the particle is assumed to move 1 on the outgoing separatrix X ' X . With this assumption, equation (4.2) ( a ) becomes � 3 74

  71. Transit Time Chapter 4 ________________________________________________________________________________________________________________________________________________________________________________________________________ dX 3 S � � 2 . 3 hX X � dt 2 When considering acceleration, it is necessary to add the velocity of the separatrix as it recedes from the particle. It is easy to evaluate it in the non-translated frame and to note that the velocity has to be the same in both frames. In this frame, the separatrix equation is given by X = - h . Thus: dh 4 dQ S . (4.13) � dt S dt As the separatrix moves to the right, the particle moves to the left with respect to it. Thus, the relative velocity of the particle with respect to the separatrix, is given by: dX 3 S 4 � � S 2 � � 3 hX X Q . (4.14) � dt 2 S This equation can again be integrated by separation of variables and the transit time results: nh � 2 � � 3 3 3 8 3 � SX Q 1 H H S � � � � T ln � � c , d 2 2 � � 3 8 3 Q 3 SX 3 3 8 3 Q � � H S H H S � � � � � h � � O � nh � � 1 8 3 Q � � � � S � 3 SX 3 3 1 H H � � � � � � � 2 2 3 1 � � H � � ln � � | � � 3 � 1 8 3 Q H � S � 3 3 3 1 � SX � H H � � � � � � 2 2 3 � � � � H � � � � O h � � � 2 Q 2 Q � � � � n 4 2 3 4 S O S � � � � � � � 2 2 1 � 3 � � 3 � H H ln � � � � � � 3 2 Q 2 Q H � � � � n 2 3 4 4 S O S � � � � � � � 2 2 � 3 � � 3 � H H � � � � whence the transit time from the nearest fixed point in the dynamic case is 1 n 2 3 1 n 3 | T d ln ln (4.15) c , � � � � � � � � � � n 3 n 3 � 1 3 2 4 Q 3 � H S H H � � � � � � � � � O � O 2 2 � 3 � � 3 � H H where the assumption ' h << h which translates into: 75

  72. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ � 4 Q 1 2 1 S H 2 � Q �� � �� H S 3 S 3 2 3 H S � Q has been used and O and have been neglected with respect to unity. In principle, 2 H � is also negligible with respect to � H H H 2 . This meams that particles start within one O H step of the separatrix and the separatrix moves many times during extraction. The term in � O has not been suppressed in (4.15) in order to recover (4.8) when Q 0 . In the numerical example the transit time results T c,d = 231 machine revolutions, to be compared with the numerical simulation result of 249 turns for / = 0, and 252 turns for / = 0.1 4.3.2 Time spent along the side of the stable triangle The next step is to evaluate the time needed to travel along the side of the stable triangle. The same approximations will be used as before, with the only difference that the velocity 2 , to obtain the relevant component of the of the separatrix has now to be multiplied by 3 motion of the extraction separatrix C in Figure 4.2. The side of the stable triangle is also moving, but the motion in X c is quasi-independent of O . Separatrix C moves with the same speed as separatrix A in the previous section, but only the X c component of its motion is of interest. Equation (4.2) ( b ), provided that X remains negligible during the particle movement, becomes dX ' S 8 � � S 2 � 6 3 hX ' 3 X ' Q (4.16) � � � dt 4 3 S which, integrated between X ' 0 2 3 h and X ' Ah � gives: � / � F , d 1 2 3 A / � T ln (4.17) | s 1 , d 1 A 3 H � / which is exactly the same as in the case of the static situation, equation (4.10). This is reasonable considering the approximation used, in which the velocity in X c does not depend on X and in which the variation in the length of the triangle side is negligible in the time considered. 4.3.3 Time for particles starting along the side of the stable triangle It only remains to evaluate the point where the second integral begins. The assumption is made that this point can be calculated by adding the position calculated under stationary conditions using (4.11) and the movement of the separatrix, evaluated as the product of the time needed to move along the side of the triangle times the velocity at which the separatrix moves. The starting point for the second integral is then: 76

  73. Transit Time Chapter 4 ________________________________________________________________________________________________________________________________________________________________________________________________________ � � 12 1 1 2 3 4 A S / � / / � � ln X O h Q O h � � F , d � � � F , d 1 A S (4.18) A 2 3 A 3 H � / � ' X Ah F d � , The total time to reach the ES for a particle starting from � � H , 2 3 H with / greater � O � / or of the order of 0.1, taking into account the shrinking of the stable triangle, is given by the sum of the time to move along the side and the time to reach the ES from the junction point near P 3 , that is: 2 4 S � 2 3 Q O � � F , d 2 1 2 3 A 1 n 3 H / � T T T ln ln � | � s , d s 1 , d c , d 2 4 1 A n 3 3 3 S H H � � / � Q O F d � , 2 3 H And finally the transit time for a particle starting along the side of the stable triangle in the dynamic case can be evaluated to 1 n 2 3 A 3 � / ln T s , d | n 3 A 1 3 2 3 H � � / S � Q O F , d � 2 H (4.19) 1 n 2 3 A 3 � / ln . 1 � n 3 A 1 3 H H � � / O F , d � 2 3 H This time, A does not fortuitously disappear, as in the static case, thus a value must be given for it. As the two regions (near and far from P 3 ) have been previously divided at / = 0.1, it seems reasonable to choose A such that the junction point corresponds to / = 0.1. This means A 01 2 3 . . Evaluating (4.18) and (4.19) with the numerical data � used earlier, and comparing with the simulation, yields: Formula (4.19) Numerical / (revolutions) Simulation (revolutions) 0.1 231 252 0.5 258 267 0.9 300 306 0.999 399 432 For an initial tune distance G Q = 3.782 10 -4 , which corresponds to an amplitude 10 times smaller and thus a single particle emittance 100 times smaller, one would have the following transit times: 77

  74. Chapter 4 Transit Time ________________________________________________________________________________________________________________________________________________________________________________________________________ Formula (4.19) Numerical / (revolutions) Simulation (revolutions) 0.1 1323 1431 0.5 1575 1584 0.9 1986 2061 0.95 2136 2349 Particles with / = 0.999, as discussed in the next chapter, are extracted along the following separatrix and are thus not reported. 78

  75. Chapter 5 TIME PROFILE OF THE EXTRACTED BEAM In Chapter 4 the time needed for a particle to be extracted after becoming unstable has been evaluated. The transit time formulae can be used to estimate the temporal structure of the extracted beam, that is the number of particles reaching the electrostatic septum as a function of time, when the beam is slowly brought into the resonance. This will be expressed as the density in time P ( t ), defined such that P ( t ) d t is the number of particles that reach the ES between t and t +d t. This function depends on the extraction method and can thus be used to compare different extraction techniques. 5.1 “Strip” profile During the extraction process, the stable region is shrunk slowly, such that a narrow strip * of particles becomes unstable and is thus extracted. This situation is represented in Figure 5.1. X ' Electrostatic Septum Position A X P 3 P 1 P 0 C P 2 B Figure 5.1. Shrinking the stable region leaves a narrow strip of particles unstable. In the figure only one of the three sides of the triangle is considered and the motion of the particles is shown by the arrows. The first particle will reach the ES at t = T c,d and the last one at t = T s,d ( / = / F ). Note that the maximum time does not correspond to / = 1. This is due to the fact that the particles that start very near to the stable point P 2 will be overtaken by the inward movement of separatrix B. These particles will be extracted along the following separatrix. This is * The name “strip” will be reserved for the narrow region of monoenergetic particles on the side of a stable triangle. The name “band” will be introduced later for the series of strips in a beam with a momentum spread. 79

  76. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ equivalent to considering them as starting near P 3 (they are in fact replaced by particles from the separatrix C that are overtaken by the movement of separatrix A). The delay with which they appear near P 3 will not be considered, for the moment. The value of / F will be given later in this Chapter. As noticed in Chapter 4, all the particles starting with 0< / <0.1, reach the electrostatic septum at virtually the same moment, so the spill will start with a spike at t = T c,d . If U ( / ) is the linear probability density of particles in the strip, and N strip is the total number of particles contained in the strip, the spike will contain N spike particles, where: 0 . 1 1 � � N N d N d ( ) ( ) (5.1) / / � / / U U spike strip strip 0 / F � � � � � � � � � � � � � � Particles in the Particles overtaken vicinity of P by the separatrix 3 1 � . Thus the initial spike can be approximately described by: where ( ) d 1 / / U 0 N strip P spike ( t )d t = N spike G ( t - T c,d )d t. (5.2) After the initial spike, the spill shape can be evaluated by noticing that the particles coming out between T ( / ) and T ( / )+d t are the ones which started between / and / +d / . This means: d / N P ( t ) dt N P ( T ( )) dt N ( ) d N ( ) dt (5.3) / / / / U U strip tail strip tail strip strip dt for t 0 < t < t F , which corresponds to 0.1 < / < / F . The time profile for the elementary strip is then given by the sum of P spike and P tail . P strip ( t ) dt = P spike ( t ) dt + P tail ( t ) dt. (5.4) Thus to evaluate the time profile of the elementary strip P strip , / F , U ( / ) and / ( t ) have to be determined. The shape of this elementary spill is the key to calculating the spill profile for the whole beam and for including the influence of ripple (an explanation of ripple is given later in Section 5.8). 5.2 Linear probability density Assuming that the initial beam is smoothly distributed in phase space and that the resonance is applied adiabatically, then the density probability U ( / ) in the strip will be proportional to 1/ v ( / ), where v ( / ) = d X c /d t is the velocity in normalised phase space of the particles. With the Kobayashi Hamiltonian translated to P 3 , v ( / ) is given by (4.2) ( b ) dX ' H S w � � 2 3 2 12 hX 6 3 hX ' 3 X ' X . (4.2) ( b ) � � � � � dt w X 4 80

  77. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ Restricting (4.2) ( b ) to trajectories close to separatrix A, substituting the definitions of O and / neglecting the term in O 2 , yields dX ' � � 2 2 v ( ) 3 Sh 3 3 / � � O � / � / dt which yields 1 1 ( ) U k k / � � 2 2 v ( ) / 3 Sh 3 3 � � O � / � / where k is a normalisation constant defined by 1 � . ( ) d 1 U / / 0 Thus 2 9 Sh � k . O 2 ln 3 Finally, the line density is given by 3 ( ) . (5.5) U / O � � 2 2 3 3 ln / � / � O 3 5.3 Inverting the Transit Time The second element that is necessary for the evaluation of P tail , is d / /d t . The time needed for a particle to reach the electrostatic septum starting from X = - O h and X ' 2 3 h , near � / separatrix A, is: 1 n 2 3 � A / 3 T ln . (4.19) s , d 1 n 3 A 1 � � � / 3 H H � O F , d 2 3 H Re-arrangement of (4.19) to solve for / , and the use of t instead of T s,d , to stress that it is now the independent variable, yields n 2 3 A 3 � / 3 t e H n � 3 A 1 � / � 1 � � H � � � O � � F , d 3 � H � H 81

  78. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ and, using (4.18), � � / n 3 2 3 A 1 3 t e . (5.6) H � / � � 3 n A 1 / � � / � � / 12 1 1 2 3 A � � � H � � 1 ln O � � � � � / � A 1 A 2 3 A 3 H � � H � H As O is of the order of , and H << 1, the first term in the denominator of the last � H fraction in (5.6) can be neglected with respect to the second. Equation (5.6) can then be rewritten as R K K ln R (5.7) � where 2 3 A n 3 � � / � H 3 t R ; K e H . (5.8) � 1 2 3 A n 3 � / H In order to invert T ( / ), it is necessary to solve (5.7) with respect to R . This equation has real solutions only when K t 1, which corresponds to t t T c,d . This is to be expected since no particle is extracted prior to T c,d . Let R = R ( K ) be the solution of equation (5.7), then: g ( K ) (5.9) / 1 g ( K ) � A where g ( K ) R ( K ) . Referring back to equation (5.3), the aim is to evaluate 2 3 A � d 3 d 3 d / / / � � . (5.10) U / | O O dt dt dt � � � � 2 2 2 3 3 ln 2 3 3 ln O / � / � / � / 3 3 Substitution of expression (5.9), yields: 2 � � � � � � � � � � � d 1 g K g K 1 g K 1 R K � � � � / � U (5.11) / � � � � � � � 2 dt O � � � � O g K O R K 1 g K � � 2 g K ln 2 ln 2 ln � 3 3 3 which no longer depends on A . Note that neglecting O with respect to ( / 2 - /� in equation (5.10), means that / >> O , which is always true for / > 0.1, and (1 - / ) >> O . This will be shown later to be also true for / < / F . Derivation of (5.7) with respect to time, gives 82

  79. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ � R � � � R K K ln( R ) K � � R 1 ln( R ) � � � R K K 1 � R which, substituted into (5.11), gives � d 1 1 ln( R ( K )) 1 1 ln( R ( K )) Q n 3 / � � � t S � � � 3 K e H . (5.12) U / � dt R ( K ) K R ( K ) K n O � O � H 2 ln ln 3 3 Note that neglecting the derivatives of n and H was already included in the model when it was assumed that the stable region does not change during the extraction time. A plot of expression (5.12) is shown in Figure 5.2 compared with a numerical simulation. 50 P tail ( t ) 40 30 20 10 Turns 100 200 300 400 500 600 Figure 5.2. Strip Profile (blue line) shown with simulation results (points). When K >> 1, then R >> K and K can be neglected in the denominator. From (5.7), 1 ln( R ) 1 � . Thus, for K >> 1, the asymptotic value of the spill density is R K � d 1 K 3 H / ( ) (5.13) U / | � � O O dt K 2 ln 2 ln t o f 3 3 5.4 Spill length The end of the spill (and thus the spill length) can be evaluated by noticing that separatrix B in its movement overtakes some particles with / sufficiently close to unity that their velocity is slower than the velocity of the separatrix itself. Those particles will then be extracted along the following separatrix. The effect is that some particle near P 2 will 83

  80. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ disappear to reappear near P 3 . The spill therefore ends with the particle whose velocity is equal to the velocity of the separatrix. The X c coordinate of this particle is given by dX ' S 8 � � S 2 2 � 12 hX 6 3 hX ' 3 X ' 3 X Q . (5.14) � � � � � dt 4 3 S Neglecting the term in X and X 2 with respect to the right-hand side, and substituting , yields X ' 2 3 h � / 8 S 2 2 2 � 9 Sh 9 Sh Q 0 / � / � 3 S whose solutions are � 16 Q S � � � 1 1 # � � 2 2 � 9 3 S h � . � � / 1 , 2 2 The solution of interest is the one close to the stable point P 2 , � � 8 Q 2 Q � S S H 1 1 1 (5.15) / F � � � 2 2 2 2 9 3 S h 3 3 3 H H Note that 1- / F >> O . This justifies the omission of O in equation (5.10) in the previous section. The spill will thus end at 1 n 2 3 A 3 � / � � F T ln / | | s , d F n 3 A 1 3 � � / 2 3 A H � � � F �� H � � / F 1 ln � � � � � � 2 � A 1 3 � � � / H F � � � � 1 � � �� � �� � �� � � � 2 2 � 2 1 n 2 3 A 3 3 3 3 2 3 A 3 3 � � � H H H � � � � � | � ln 1 ln � � � � � � � � � 3 n A � � A � 3 � � � � � � H H H H � � where the term in O in the last fraction has been neglected as in (5.6). Expressing this time in units of T c,d , yields 1 � 2 2 2 1 n 2 3 A 3 3 3 3 � 2 3 A 3 3 � �� � �� � �� � H H H � � ln � 1 ln � � � � � � � � n 3 A � � A � 3 � � � � � � � � H H H � � H � T � � � � � � / � � s , d F | T 2 1 n 3 3 c , d � � H ln � � � n 3 � � � 3 H H � � � 84

  81. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ 2 2 � � n 2 3 A 3 3 2 3 A 3 3 � � � � � � � � � H � H ln ln 2 ln ln � 1 ln � � � � � � � � � � � � � � � n 3 � A � � � A � � � � � � � H H � � � � � � � � � � 2 . | 2 n 3 3 �� � H � � ln ln � � � � � n 3 � � � � H � � � � 2 Where all the addenda have been considered negligible with respect to ln . Hence this � H H complicated derivation converges to the simple result, � � T 2 T . (5.16) / s , d F c , d Thus the first particle reaches the electrostatic septum at T c,d , the last one at 2 T c,d , and the spill length is T c,d . 5.5 Width of the initial spike It is now possible to evaluate the time needed for the last particle to be overtaken by the separatrix. Consider the velocity of the particle along the side of the stable triangle and, similar to Chapter 4, add the velocity of the separatrix: dX ' 8 � � S 2 2 � 3 Sh 3 3 Q . � / � / � O � dt 3 S This yields 2 � � H S � 2 � . 3 3 Q / / � / � � O 3 H Finally, the time for a particle starting at / i to be overtaken by separatrix B is 2 � H / 1 1 1 i 3 � H T d ln / | 2 2 � 3 H S H 2 � � � H 3 3 Q 2 3 1 / H / � H / � O � H / � � i i 3 3 H H For / i -> / F , T goes to infinity, as expected since this is the time needed for the separatrix to reach a particle which moves with the same velocity. So it is necessary to consider / i = / F + , that is one step of the separatrix away from / F . Then 2 O 3 1 1 T ln (5.17) 6 H 3 H which may be a large fraction of T c,d for small H . However, for the sake of simplicity, and considering that most of the particles are near the stable points and thus will be overcome in a few turns, this delay will be neglected and all the particles starting near P 2 will be considered as if they were starting near P 3 . 85

  82. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ 5.6 Population of spike and tail It is interesting to evaluate the fraction of the beam contained in P spike and P tail . This can be done in a straightforward way by integrating the density U ( / ). It is easier to integrate P tail between / = 0.1 and / = / F , as in this range O can be neglected in the density. It results, O ln / F 27 3 1 H � � � d U (5.18) / / | O 2 2 ln 0 . 1 3 neglecting ln(27 � 3 H ) and ln(3) with respect to ln( O ) yields the result that 1/2 of the beam is in P tail and 1/2 in the initial spike. Thus the two components of the strip profile are equally important. This result is of consequence for the response to ripple and the efficiency of feedback systems. For perturbations up to frequencies corresponding to the width of the initial spike (of the order of 100 kHz for revolution periods of the order of 1 P s), half of the beam behaves coherently with a definite delay while the other half is extended over a period that will cause overlap with ripple frequencies as low as 1 kHz. 5.7 Time profile without ripple Up to now, only the microscopic behaviour has been considered. When performing an extraction, the whole beam is present and the total spill is the important feature. In the following the aim is to derive the global behaviour from the elementary strip profile. In the Steinbach diagram, which shows the stable and unstable regions in the amplitude vs tune (or momentum) space, the strip corresponds to a fixed amplitude and tune (momentum), as shown in Figure 5.3. Amplitude Strip Resonance region BEAM Stable region ' p / p Figure 5.3. A strip in the Steinbach diagram corresponds to a certain amplitude and momentum. In general the beam may have any shape in the amplitude-momentum space, and the beam distribution shown in the figure is meant to represent a generic distribution. The particular cases of wide and narrow beam, with respect to the resonance, will be analysed later. 86

  83. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ The total time profile of the beam is obtained by summing the contributions of the different strips. In a schematic way this is shown in Figure 5.4: dN/dt 0.02 Pstrip,1 Pstrip,2 Pstrip,3 0.0175 0.015 0.0125 0.01 0.0075 0.005 0.0025 t 100 200 300 400 500 600 Figure 5.4. Strip profiles for different strips. The total spill is the sum of all the strips. where P strip,1 , P strip,2 , P strip,3 represent the profiles in time for strips that become unstable at different times or with different amplitudes and so on. In principle they are all different and the extracted current is given by: I ( t ) = 6 i P strip,i ( t ). The evaluation of I ( t ), can be greatly simplified when the total extraction time is much longer than the duration of a strip. In this case, indeed, the relative variation of H , and of the other parameters, during the strip extraction is small and thus all the terms P strip,i ( t ), relative to the same H , effectively contributing to the total extracted current are equal. Then, the sum of all these terms just gives the integral over one of them. Since the same argument applies to all the amplitudes, or modified tune distances H , involved then the extracted current equals the number of particles that become unstable as a function of time N T ( t )! This means that: x if the extraction is performed in amplitude the time profile is the amplitude distribution, x if the extraction is performed in momentum the time profile is the momentum distribution, x combinations of amplitude and momentum yield the combined distribution. 5.8 Ripple As it has just been shown, under ideal extraction conditions, apart from an initial and a final transient, the intuitive result is found, and the effort of this complicated analysis seems worthless. Unfortunately, ideal conditions are never met in reality and the fields in the magnets, and thus all the parameters that depend on them, are not perfectly stable, but oscillate around their value. This effect is called ripple and in general many ripple 87

  84. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ frequencies with different amplitudes are present at once. In the following only one ripple frequency will be considered. The effect of a ripple in tune is to move the “V” of the resonance back and forth while a ripple in the resonant sextupole strength causes the slope of the “V” to increase and decrease, as shown in Figure 5.5. In the very simple model in which a particle is extracted as soon as it enters the resonance, the profile in time d N/ d t of the extracted beam is given by the distribution of particles that become unstable per unit tune variation F = d N /d Q and by the relative motion between the resonance and the stack: dN dN dQ � � � � � � � � � Q Q Q ZG Q cos Z t (5.19) � � � � F F r R 0 0 dt dQ dt where a sinusoidal ripple has been assumed. If the product of the ripple frequency Z and � , the extracted beam is ripple amplitude G Q R equals or exceeds the stack velocity Q 0 completely or over modulated, respectively, and there are instants during which no beam is extracted because the resonance is “escaping” faster then the beam. Amplitude Resonance excitation ripple Tune ripple ' p / p Figure 5.5. Beam and resonance with ripples. This approximation holds until the transit time of the particles is short compared to the period of the ripple. As soon as the transit time is no longer negligible, a more accurate analysis is needed. As it has already been shown, the transit time is not the same for all the particles, thus a mixed concept considering both the transit time and the number of particles having a certain fixed delay has to be developed. In practice, this means evaluating the time profile for the set of particles that become unstable in one step of the ejection process, taking into account all the possible amplitudes. The “width” of this profile is the quantity that has to be compared to the ripple period to have an indication of the overlapping of these elementary profiles. If these overlap, the continuity of the extracted beam will not be broken even if the product ZG Q R exceeds � Q 0 . 88

  85. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ 5.9 Comparison of extraction methods To extract the beam, either the particles have to be progressively moved into the resonance stopband or the stopband must be moved onto the beam. This means that some physical quantities have to be changed. The varying parameter, the way in which it is changed and the initial geometry in the amplitude momentum space distinguish the extraction method. Of the many possibilities illustrated in Section 3.11, the three cases in which the average extracted momentum does not vary during extraction are considered: x Amplitude selection by moving the resonance : the beam is narrow in momentum and the tune of the particles is changed, by varying the focusing of the machine, see Figure 5.6 ( a ). x Amplitude-momentum selection by moving the beam [13]: the beam is wide in momentum and the tune changes, due to chromaticity, when accelerating the particles, see Figure 5.6 ( b ). x RF knock-out [20]: the chromaticity is zero (or quasi-zero), so that all particles have the same tune. The particles are made unstable by blowing up the beam with transverse RF excitation. Thus the resonance is reached in amplitude, see Figure 5.6 ( c ). Amplitude Amplitude Amplitude ( a ) ( b ) ( c ) ' p / p ' p / p ' p / p Figure 5.6. Comparison of the main extraction methods: ( a ) moving the resonance, ( b ) moving the beam, ( c ) increasing the particle amplitude. In the amplitude selection case, the movement of the resonance maroons the large amplitude particles first in the unstable region. Since the momentum spread is small, the basic element of spill marooned in the elementary extraction process (one elementary movement of the resonance) is nothing more than the strip profile just described. So the “building block” of this particular spill starts with a very narrow peak containing half of the particles involved. Moreover in the initial phase the transit time is very short and sensitivity to the ripples is high. At the end of the spill the transit times are longer and the particles in the tail are distributed over a longer period. In the RF-knockout method the blow-up velocity is fixed by the spill length. When looked at in tune, the Steinbach diagram is similar to the amplitude selection case one. The beam 89

  86. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ in this case has null width and the resonance is reached in amplitude. A ripple in tune would move the beam right and left, which causes the resonance line to move up and down in the “ ' p/p diagram”. As the particles enter the resonance at high amplitudes, and thus large tune distances, the transit time is short and the sensitivity to ripples is high. In the amplitude-momentum selection case, particles of all amplitudes become unstable at the same moment. This leads to an enlarged leading peak, which can fill the time interval during which no beam enters the resonance in a much more efficient way. A more detailed study is presented to estimate the smoothing of the ripples with this method in the following sections. 5.10 Elementary "band" for a wide momentum spread Consider the amplitude-momentum selection case. The beam is wide in tune and a stationary situation develops in which all betatron amplitudes are extracted simultaneously. In Figure 5.7 an amplitude-momentum selection extraction is shown and the elementary band * of beam that becomes unstable in one step is highlighted. Amplitude H 0 , T 0 No particles with small amplitude (hollow beam) H F , T F Q res G Q min G Q max Figure 5.7. Amplitude-momentum selection extraction. A complete band of particles becomes unstable in one step. A hollow beam is considered to set a minimum H which can be considered large with respect to its variation in the transit time. If the number of particles that would normally be in the hole is small, then a good representation for the beam with no hole is found. This approximation is not so restricting since firstly the radial particle density across a beam rises from zero at the centre and secondly particles with sufficiently small amplitude will cross the resonance without being extracted. Let the band that is extracted contain particles with tune shifts between G Q min and G Q max . The time profile of this band, is given by considering the time profile (5.4) for each H value between H F = 6 SG Q min , corresponding to T F = T c,d ( H F ), and H 0 = 6 SG Q max , corresponding to T 0 = T c,d ( H 0 ), and summing at each instant all these contributions, as shown in Figure 5.8. * Here the word “band” is used to indicate the series of strips corresponding to different momentum and emittance that become unstable simultaneously. 90

  87. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ The resultant spill can be written as, H H F F � � � � N P ( t ) N ( ) P ( t , ) d N ( ) P ( t , ) P ( t , ) d (5.20) � H H H H H H H B band strip strip strip spike tail H H 0 0 where N B is the total number of particles in the band and N strip ( H ) d H is the number of particles marooned with modified tune distance between H and H + d H . dN/dt 0.02 Pstrip,1 Pstrip,2 Pstrip,3 H 1 H 2 0.0175 H 3 0.015 0.0125 0.01 0.0075 0.005 0.0025 t 100 200 300 400 500 600 Figure 5.8. The total flux of particles is obtained summing the “strip profiles” for all the amplitudes present. 5.11 Simplified model The exact integral is far too complicated to be evaluated analytically. So, to facilitate the task, each P strip ( t ) is approximated by one delta function plus a rectangle representing P spike and P tail respectively. The width of the rectangles, as mentioned in section 5.4, is equal to the time needed for the first particle to reach the ES, that is T c,d , and hence the height is 1/ T c,d . Not all the P strip ( t , H ) give their contribution at any given instant. Most of them are zero, either because they have not yet “started” or because they have already “finished”. At time t , only the H for which t /2 < T c,d ( H ) < t give a non zero contribution. Thus the part of the integral arising from P tail , becomes ( t / 2 ) H N ( ) 1 H � T ( ) . N P t d H B band , tail 2 T ( ) H c , d ( t ) H Only half of the particles are in the rectangle To be correct the integration limits should be max( H ( t ), H F ), which corresponds to min( t , T F ), and min( H ( t /2), H 0 ) which corresponds to max( t /2, T 0 ). 91

  88. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ After a change of the integration variable, min( ( t / 2 ), ) max( t / 2 , T ) H H 0 N ( ) 1 0 N ( ( T )) 1 d H H � � H T T d dT . (5.21) H 2 T ( ) 2 T dT H c , d max( ( t ), ) min( t , T ) H H F F It is necessary to find H ( T ), that is to invert 1 n 3 . T d ln c , � � � n 3 3 1 � H H � � � � � O � � 3 H H To simplify the calculation, it is assumed that the ES is far away so that n >> 1, thus n /( n +3) | 1. Neglecting O as usual, gives 2 1 3 1 3 3 H ln ln T � c , d 1 � 3 3 � H H H H � 3 H H which can be written 1 1 3 3 � H T ln H � � c , d 2 � 3 3 3 H H � � H It is necessary to invert the equation Y = ln( X )/ X . A simple fit to the inversion, valid to within 10% in the range 4 < X < 30 000, is X = -1.42 ln( Y )/ Y. 0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 Figure 5.9. Fit to the inversion (Logarithmic scale). 92

  89. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ � H 2 Note that X = 10 corresponds to 5 10 and values of X smaller than this should be � | � 2 H 2 avoided to stay within the limits of the transit time formulae ( measures the relative � H H variation of the stable region during the transit time). Thus 1 1 � � � H � � H � ln T ln T � � � � � � 2 2 � 3 � � 3 � � 2 H � � � � 1 . 42 1 . 42 (5.22) H � � � � T 3 3 1 � 3 H T � 2 3 from which d 2 1 1 � � � � � H H 1 . 42 1 ln T . (5.23) � � � � � � � � 2 dT 2 3 T 3 � � � � � � � � It is now necessary to estimate N T ( H ) d H . This is the number of particles that are in the 2 � � 2 H 2 � � border of thickness dH of a triangle of surface 3 3 3 3 . This border H � 3 S � corresponds to the annulus of width d R around a circle of the same surface S R 2 in the initial beam. Thus, 3 3 2 8 3 � � H 2 N ( ) d N ( R ( )) d 3 3 H N ( ) d (5.24) H H U H U H H T B B 2 3 S 3 S S where N B is the total number of particles in the band. 5.12 Uniform distribution in phase space Assuming a uniform phase space distribution in the initial beam, of value U 0 = 1/Triangle 2 3 S Area = , yields for the integration of the tails from all the differential strips in the 2 4 3 H 0 band, max( t / 2 , T ) 1 0 ( ( )) 1 N T d H � H T ( ) P t dT band , tail N 2 T dT B min( t , T ) F 2 1 1 � � � � � � � � H H ln ln T T � � � � � � � � � max( t / 2 , T ) 2 2 � 3 � � 3 � � � 8 3 4 1 0 � 2 � � � � � � � � 1 . 42 dT U � 0 2 4 3 2 3 S T min( t , T ) F 2 � � � � � � 1 1 � � � � � H � � H � � � � ln ln T T � � � � � � max( t / 2 , T ) 2 2 2 3 3 � � � � � � � 4 1 . 42 0 � � � . dT 2 4 3 T H 0 min( t , T ) F 93

  90. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ Using the standard formulae ln( ax ) ln( ax ) 1 � dx � � p � � p 1 2 � � p 1 x p 1 x 1 p x � � � � and � � 2 � � 2 ln( ) ln( ) 2 ln( ) 2 x x x � , dx � � � p � � p 1 � � 2 p 1 � � 3 p 1 1 x p x p 1 x p 1 x � � � � � � the integral can be evaluated. It results that 2 � � 4 1 . 42 � ( ) P t band , tail u 2 3 H 0 max( t / 2 , T ) 2 0 � � 1 1 � 1 � � � � � H � � H � � H � ln T ln T 2 ln T � � � � � � � � � � � � � 2 2 3 2 3 � � 3 � � 1 � � � � 2 � � � � � � � � � � � � � � � � � 3 3 3 3 3 3 T 9 T 3 T 9 T 27 T � � � � � � min( t , T ) � � F max( t / 2 , T ) 2 0 � � 1 1 � � � � � H � � H � 9 ln 3 ln 1 T T � � � � � � � � � � � � 2 2 � � 2 3 3 � � � � 4 1 . 42 � � � � � � � � � � � � � 2 3 3 27 T H 0 � � � � � � � � min( t , T ) F and finally, for times 2 T 0 < t < T F , that is apart from the initial and final parts � � 2 � 4 1 . 42 u P ( t ) band , tail 2 3 H 0 � � 2 � � � � � � 1 1 � � � � � � � � � H � � H � (5.25) � � � � � � � 2 � � 63 ln t 144 ln( 2 ) 21 ln t 72 ln( 2 ) 24 ln( 2 ) 7 � � � � � � � � 2 2 3 3 � � � � � � � � � 3 � 27 t � � � � � � To correctly evaluate the initial and the final part of the beam, the correct integration limits have to be considered. In fact for times t < T 0 , no particle has reached the septum and P band,tail = 0. For times T 0 < T < 2 T 0 , the integral has to be performed between T and T 0 (there are no particles with H = H ( T /2)). For the same reason for times T F < T < 2 T F , the integral has to be performed between T F and T /2. For times greater than 2 T F , P band,tail = 0 again. 94

  91. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ The contribution from the initial spikes of the elementary strips in the band, gives T 0 N ( ( T )) d � H H T � � � P ( t ) t T dT G band spike , 2 dT T F 2 � � � � � � 1 � 1 � � � � H � � H � � � � ln t ln t � � � � � � 2 2 2 � � � � 3 � � 3 � � � 4 1 . 42 � . 2 3 3 t H 0 The graph of the spill shape for a particular case is drawn in Figure 5.10. P band (t) Sum of the leading peaks 0. 01 0. 00 8 Sum of the tails 0. 00 6 Total flux 0. 00 4 0. 00 2 t 20 0 40 0 60 0 80 0 Figure 5.10. Band profile for G Q max = 3.782 10 -3 , G Q min = 3.782 10 -4 , � Q =-5.673 10 -8 , S = 27.7897 m -1/2 . The two contributions and the sum are shown. 5.13 Gaussian distribution in phase space Beams are often assumed to have a gaussian distribution in phase space 2 R 1 � � � 2 E R e x (5.26) U S E 2 x where E x is the horizontal one- V emittance * . Assuming that the area of the biggest stable triangle corresponds to an emittance of n - V , the one- V emittance is 2 Area 4 3 H 0 E x (5.27) 2 2 2 n 3 S n S S Thus 2 2 n H 2 2 3 S n 2 � 2 � � e H (5.28) 0 U H 2 8 3 H 0 * The definition of emittance used is phase space area divided by S 95

  92. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ 2 2 n H 2 n � 2 � � 2 N d N e H d (5.29) 0 H H H H T T 2 H 0 and max( t / 2 , T ) 2 2 � � � 0 ( ( )) N T 1 d 2 n 1 . 42 � H H T u P ( t ) dT band , tail 2 2 T dT 3 H 0 min( t , T ) F � � 2 � � � � � � 2 � � 1 1 � � 1 � � � H � � � H � � � 2 � H � � � � 2 � � � � 1 . 42 2 n ln T ln T ln T � � (5.30) � � � � 2 3 � � � � � � max( t / 2 , T ) 2 2 � 3 3 0 � � � � � � � 2 2 � � 3 T e H dT . 0 � 4 � T min( t , T ) � � F � � � � The integral can be evaluated numerically and plotted, as shown in Figure 5.11. P band Sum of the leading peaks 0. 00 5 Sum of the tails 0. 00 4 Total current 0. 00 3 0. 00 2 0. 00 1 t 20 0 40 0 60 0 80 0 Figure 5.11. Band profile for a gaussian beam for G Q max = 3.782 10 -3 , G Q min = 3.782 10 -4 , � Q =- 5.673 10 -8 , S = 27.7897 m -1/2 , E tot = 9 E rms . The two contributions and the sum are shown. If the gaussian and the uniform cases are plotted on the same graph, the comparison in Figure 5.12 is obtained P band 0. 01 5 0. 01 25 0. 01 0. 00 75 0. 00 5 0. 00 25 t 20 0 40 0 60 0 80 0 Figure 5.12. Band profile for gaussian and uniform beam. A 3 V emittance of the gaussian beam has been considered as total emittance. 96

  93. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ From the plots above, it appears that the width of the “band profile” is of the order of T 0 . Thus it can be expected that when the ripple period is of the order of T 0 , the width of the profile partially fills the time interval during which no beam enters the resonance and smoothes the modulation of the extracted beam. The band profiles have been worked out in this Chapter considering a constant � Q . Thus considering them as “filling the gap” in the presence of ripple is an approximation which neglects the instantaneous variation and uses an average � Q ( � Q = � Q 0 + Z G Q R cos( Z t ) | � Q 0 ). This can be accepted by considering that � Q appears in the expression for the transit time inside a logarithm and visually by comparing the profiles in time for a variation in � Q of 2 orders of magnitude, as shown in figure 5.13: P band P band . . 0. 01 2 Q = -5.673 10 -7 Q = -5.673 10 -9 0. 02 0. 01 0. 01 5 0. 00 8 0. 00 6 0. 01 0. 00 4 0. 00 5 0. 00 2 t t 20 0 40 0 60 0 80 0 20 0 40 0 60 0 80 0 Figure 5.13. Time profiles for gaussian and uniform beams for different values of � Q . Even a large variation in � Q does not vary the orders of magnitude and, to a first approximation, the band profile can be considered when analysing the spill with ripple. The enlarged leading peak quoted in Section 5.9 can be thus estimated in hundreds of turns to be compared with the tens of turns of the strip profile, demonstrating the superior ripple tolerance of the amplitude-momentum selection scheme. 5.14 Simulations of extraction with ripple All the theory and approximations made so far, call for a check to verify the correctness of the conclusions drawn. With this in mind, three numerical simulations have been made: 1) amplitude selection extraction with a mono-energetic beam; 2) amplitude-momentum selection with a uniform distribution in phase space; 3) amplitude-momentum selection with a gaussian distribution in phase space. In order to be consistent with the numerical examples shown so far, the following parameters have been used for the simulations: T rev = 0.734 P s Q = -5.673 10 -8 => � Q = -0.0257 s -1 . � It should be noted that these simulation parameters correspond to the extraction of protons at 60 MeV from the PIMMS ring and are therefore representative of a practical situation. 97

  94. Chapter 5 Time profile of the extracted beam ________________________________________________________________________________________________________________________________________________________________________________________________________ From inspection of Figure 5.12, it results that the width of the “band profile” should be able to fill gaps of the order of ' T = 100. To demonstrate this effect a round number for the tune ripple frequency has been chosen as f ripple = 4000 Hz => ' T = 113. The corresponding amplitude for 100% modulation of the extracted beam in the “instantaneous transfer” model is A ripple = 10 -6 . The tune ripple is excited by a quadrupole gradient ripple on a single quadrupole family. The corresponding normalised quadrupole gradient ripple amplitude is easily obtained with a lattice program. It should be noticed that the relative ripple amplitude, in this case, is just 1.3 10 -6 , which is a very tough constraint on the quality of the power supplies! Since the binning in the analysis of the results cannot be infinitely large, a slightly larger (20% larger) amplitude has been used in the simulations. The program used to simulate the extraction is MAD [21]. Note that the particle tracking does not follow the approximated hamiltonian used in the theoretical derivation, but is an exact tracking through the PIMMS synchrotron. In order to limit the computing time and to have a sufficiently large number of extracted particles, only a small fraction of a beam is generated which occupies the region of interest. This is schematically shown in Figure 5.14. monochromatic “band-like” beam beam Figure 5.14. The initial distributions are generated only in the region of interest. The resonant sextupole strength is raised linearly from zero to its final value in 4000 turns. The beams are generated such that a small part of them is unstable once the sextupole is fully raised. In order to extract those particles that are already unstable, after the sextupole has reached its full strength, the program lets the particles circulate for 6000 turns before extraction starts. As an example the main steps in the preparation of the beam are shown in Figures 5.15 to 5.17 where the initial distribution, the distribution after 4000 turns (sextupole reaches its final value) and 10000 turns (just before extraction starts) are shown for the uniform, band-like distribution. The time profile is analysed using bins 30 turns wide, which correspond to an integration time of 22 P s for an hypothetical extracted current measurement. 98

  95. Time profile of the extracted beam Chapter 5 ________________________________________________________________________________________________________________________________________________________________________________________________________ Figure 5.15. Initial beam distribution for the uniform band case. Figure 5.16. Beam distribution after raising the sextupole. Figure 5.17. Beam distribution just before extraction starts. 99

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