Basic Beam dynamics. Frédéric ric Chautard. GANIL, Caen, France. F. Chautard - ECPM09

Transcription

1 Basic Beam dynamics Frédéric ric Chautard GANIL, Caen, France 1

2 OUTLINE History Cyclotron review Transverse dynamics Longitudinal dynamics Mass separation

3 CYCLOTRON HISTORY In 1919 Lord Ernest Rutherford ( ) discovered that nitrogen can be brought to emit protons by bombardment with alpha particles, according to the nuclear-reaction equation: N+ He 8O H This discovery meant the initiation of a new era in natural sciences. How then would it be possible, by some method other than the use of radioactive substances, to make available projectiles with sufficient energies to bring about nuclear reactions? 3

4 In this case, use was made of a high electrical voltage, up to about 6 kv, to accelerate protons which, upon bombarding lithium, caused a nuclear reaction: Li+ 1H He IMPROVEMENTS: Idea 1: Large potential difference (Cockroft- Walton, Van de Graaf). But high voltage limit around 1.5 MV (Breakdown) Idea : Linacs (Wideröe). Successive drift tubes with alternative potential (sinusoidal). Large dimensions AC generator Idea 3: Another brilliant idea (E. Lawrence, Berkeley, 199). The device is put into a magnetic field, curving the ion trajectories and only one electrode is used several times. AC generator 4

5 FIRST EXPERIMENTAL DEVICE A circular copper box is cut along the diameter: A half is at the ground the other «Dee» is connected to an AC generator. Insert all under vacuum (the first vessel was in glass) and slip it into the magnet gap 11 inch first cyclotron At the centre, a heated filament ionises an injected gas: This is the ion source. Dee Accelerating gap 5

6 Cyclotrons 1. Uniform field cyclotron. Aimuthally Varying Field (AVF) cyclotron 3. Separated sector cyclotron 4. Spiral cyclotron 5. Superconducting cyclotron 6. Synchrocyclotron 6

7 Conventional cyclotron Let us consider an ion with a charge q and a mass m circulating at a speed v θ in a uniform induction field B. The motion equation can be derived from the Newton s law and the Lorent force F: r d( mv) dt r F and r F r r q( v B) θ r In cylindrical coordinates (r,θ,) d( mr& ) mr & θ q[ r & θb & Bθ ] dt d( mr & θ ) + mr& & θ q[ B & r r& B ] dt d( m& ) q[ rb & θ r & θb r ] dt 7

8 Conventional cyclotron Taking the magnetic field B along the negative -axis: B -B, the equations become: m (& r r & θ ) qr & θb m ( r& θ + r& & θ ) qr& B m & with the beam initial conditions: ( r &, & θ, & ) The trajectory is a circle. It is called closed orbit in the median plane (r,θ). The radius is r and the angular velocity & θ ω rev (f rev Larmor frequency): ω rev πf rev qb m The magnetic rigidity is defined by: B r p q 8

9 Conventional cyclotron II Centrifugal force Magnetic force mv r θ qv θ B (q, m) Angular velocity: qv θ B (Larmor) ω dθ dt v r θ rev qb m θ Closed orbit r 9

10 Conventional cyclotrons means non relativistic cyclotrons low energy γ ~ 1 m / m ~ 1 In this domain ω QB m rev const We can apply between the Dees a RF accelerating voltage: with V V ω RF hω rev cosω RF t h 1,, 3, called the RF harmonic number 1

11 Isochronism condition: The particle takes the same amount of time to travel one turn and with ω rf h ω rev, the particle is synchronous with the RF wave. In other words, the particle arrives always at the same RF phase in the middle of the accelerating gap. h1 11

12 h 1 Harmonic notion 1 beam by turn ω rf ω rev upper gap +V +V lower gap +V +V 1

13 h 3 Harmonic notion 3 beams by turn ω rf 3ω rev The beam goes 3 times slower than the RF frequency. Over 36 the 3 beams are separated by 36 /3 1 (beam phase) ω rf 3ω rev, then while the beam travel 6 the RF shifts of 3x6 18. Therefore, for the 1st beam at the maximum accelerating field (9 RF phase), the second beam is 1 further and the 6 remaining to the gap will be travelled to an equivalent of 3x6 18 RF. The electric field is inverted and accelerates the second beam. Idem for the 3rd beam. ONLY 1 BEAM IS ACCELERATED AT A TIME +V

14 Why several harmonic number are important RF cavities have a fixed and reduced frequency range : f min < f rf < f max And The energy is proportional to f rev! (W particle 1/mv particle1/m(ω rev R) f rev f rf Then /h) Working on various harmonics extend the energy range of the cyclotron Magnetic field limit Frequency/energy limits for h 14

15 Transverse dynamics Steenbeck 1935, Kerst and Serber 1941 Horiontal stability : θ ρ cylindrical coordinates (r,θ,) and define x a small orbit deviation r Median plane x r ρ + x ρ (1+ x ρ ) x<< ρ ( Paraxial or Gauss conditions) Closed orbit 15

16 16 Taylor expansion of the field B around the median plane: ) (1 ) 1 ( ρ ρ ρ x n B x x B B B x x B B B + + index field the x B B n with ρ x B Q v r mv F θ θ ) x n (1 B Q v ) x (1 mv F x ρ ρ ρ θ θ Not on Closed Orbit (Centrifugal force Magnetic force) Horiontal restoring force Centrifugal force - Magnetic force

17 F x mv ρ θ x ( 1 ) Qv B θ ρ (1 n x ) ρ and mv θ x F x (1 n ) ρ ρ Motion equation under the restoring force Fx m& x& v θ & x& + (1 n ) x && x + ω x ρ ω vθ ρ qb m 1 n ω Harmonic oscillator with the frequency ω rev Horiontal stability condition: n < 1 17

18 Vertical stability Vertical restoring force requires B x : (no centrifugal force) Because B B x B x && + ω Motion equation F m && q v θ B x n ω Harmonic oscillator with the frequency ω B x n B ρ o Vertical stability condition : n > 18

19 Betatron oscillation A selected particular solution in the median plane for harmonic oscillator: EXAMPLES: x(t) x cos(ν r ω t) (t) cos(ν ω t) Horiontal oscillation ν r ν 1 Vertical oscillation n n 1 turns in the cyclotron for 9 horiontal oscillations ν r ω > ω (1/ ν r ) 1 vertical oscillation for 9 turns in the cyclotron ν ω < ω (1/9.11 ν ) 19

20 Weak focusing Horiontal stability, ν r 1 n, with n < 1 means: < n < 1 B can slightly decrease n < B can increase as much as wanted Vertical stability, ν n, with n > means: B should decrease with the radius Simultaneous radial and axial focusing : Weak focusing B n 1 slightly decreasing field x

21 First limit Decreasing field ( < n < 1) gives 1 point with a perfect isochronism ω RF ω rev ϕ < ϕ > ω rev ω rf ω rev qb m ω RF ϕ π 1 ωrev Limited acceleration 1

22 Relativistic case Isochronism and Lorent factor m m γ m β 1 β, v c ω rev qb( r) γ ( r) m ω rev constant if B(r)γ(r)B increasing field (n < ) For high energy B(r) should increase. Not compatible with a decreasing field for vertical stability

23 Vertical focusing AVF or Thomas focusing (1938) We need to find a way to increase the vertical focusing : F r (v θ B ): keep ion on the circle F (v θ B r ): vertical focusing (not enough) Remains F with v r, B θ : one has to find an aimuthal component B θ and a radial component v r (meaning a non-circle trajectory) Sectors 3

24 Aimuthally varying Field (AVF) B θ created by: Succession of high field and low field regions B θ appears around the median plane Valley : large gap, weak field Hill : small gap, strong field 4

25 V r created by: Valley: weak field, large trajectory curvature Hill : strong field, small trajectory curvature Trajectory is not a circle Orbit not perpendicular to hill-valley edge Vertical focusing F v r. B θ 5

26 Vertical focusing and isochronism for AVF conditions to fulfill Increase the vertical focusing force strength: F ω ν ω ν AVF Field modulation or Flutter: F l B B B ( B B ) hill 8 B val where <B> is the average field over 1 turn For a cyclotron with N sectors, the betatron number depends on the flutter term such as: ν N n + + K N 1 F l > F l enhances the initial weak focusing term n Keep the isochronism condition true: For high energies, γ(r) Leading to: n 1 γ < then B QB( r) ω rev γ ( r) m B ( r ) r ( r ) γ ( r ) B ( ) > 6

27 Vertical focusing and isochronism for AVF conditions to fulfill Increase the vertical focusing force strength: ν N n + + K N 1 F l Keep the isochronism condition true: > n 1 γ < The focusing limit is: N F l N 1 > n γ 1 7

28 Focusing condition limit: If we aim to high energies: Separated sector cyclotron γ then -n >> Increase the flutter F l, using separated sectors where B val F l N N F 1 > n ( B B ) hill l 8 B val γ 1 High energies PSI PSI 8

29 Spiralled sectors In 1954, Kerst realised that the sectors need not be symmetric. By tilting the edges (ξ angle) : The valley-hill transition became more focusing The hill-valley less focusing. But by the strong focusing principle (larger betatron amplitude in focusing, smaller in defocusing), the net effect is focusing (cf F +D quadripole). N ν n + F (1 + l N 1 tan ξ ) 9

30 Superconducting cyclotron (1985) Most existing cyclotrons utilise room temperature magnets B max T (iron saturation) Beyond that, superconducting coils must be used: B hill ~ 6 T 1. Small magnets for high energy. Low operation cost 3

31 Energy and focusing limits 1. For conventional cyclotron, F l increases for small hill gap (B hill ) and deep valley (B val ) but does not depend on the magnetic field level: F l ( B B ) hill 8 B. For superconducting cyclotron, the iron is saturated, the term (B hill -B val ) is constant, hence F l 1/<B> val consequences on W max 31

32 Max Energy for Conventional Cyclotrons A cyclotron is characterised by its K b bending factor giving its max capabilities Q W max ( MeV / nucleon ) K with K ( ) b b 48,44 B rext A - Wmax r ext ² but iron volume r 3! - for compact cyclotron r extraction ~ m. - For a same ion (or isobar ACst), W max grows with Q : great importance of the ion sources 3

33 Max Energy for Superconducting Cyclotrons Because of the focusing limitation due to the Flutter dependence on the B field, the max energy is given as a function of K f the so-called focusing factor: W max ( MeV / nucleon ) K f Q A 33

34 Synchrocyclotron Machine : n > and uniform magnetic field. (only 4 machines remain around the world) The RF frequency is varied to keep the synchronism between the beam and the RF ω rev QB/γm ω rf Cycled machine (continuous for cyclotrons) F rf Extraction 1 to 5 turns (RF variation speed limitation) low Dee voltage small turn separation W ~ few MeV to GeV Injection t 34

35 Laboratories GANIL(FR) NAC (SA) GANIL (FR) GANIL (FR) RIKEN (JP) PSI (CH) DUBNA (RU) MSU (USA) W Cyclotron name/type C SSC CIME SSC RING Ring U4 K1(cryo) Few K b ( MeV nucleon ) max / K b K (MeV/n) (or proton energy Q/A 1) Q A 1 (K f 4) R extraction (m) more 35

36 1,, TeV 1, TeV 1, TeV A Livingston plot showing the evolution of accelerator laboratory energy from 193 until 5. Energy of colliders is plotted in terms of the laboratory energy of particles colliding with a proton at rest to reach the same center of mass energy. Livingston chart 1, TeV 1 TeV Electron Proton Colliders 1 TeV Proton Storage Rings Colliders 1 TeV 1 GeV 1 GeV Proton Synchrotrons Electron Synchrotrons Electron Linacs Linear Colliders Electron Positron Storage Ring Colliders 1 GeV 1 MeV 1 MeV Betatrons Cyclotrons Synchrocyclotrons Proton Linacs Sector-Focused Cyclotrons Electrostatic Generators 1 MeV Rectifier Generators Year of Commissioning

37 Betatron oscillation and instabilities x(t) x cos(ν r ω t) (t) cos(ν ω t) ν r ν 1 n n If n > 1 or n < ν r or ν imaginary x(t), (t) ~ diverge SRC Riken simulation 37

38 Tunes and resonances K.ν r + L.ν P K, L and P integer K + L is called the resonance order (1,, 3 ) W r² 38

39 Cyclotron as a separator Q For an isochronous ion (Q, m ): ω rev m Constant energy gain per turn: δ T QV cos( ϕ B ) For ions with a Q/m different from the isochronous beam Q /m, ω ω rev ) ( r γ There is a phase shift of this ion compared to the RF field during acceleration ϕ π Nh 1 γ ( m / Q m / Q ) when the phase ϕ reaches 9, the beam is decelerated and lost. 39

40 Cyclotron resolution There is the possibility to have out of the source not only the desired ion beam (m,q ) but also beams with close Q/m ratio. If the mass resolution of the cyclotron is not enough, both beams will be accelerated, extracted and sent to the physics experiments. Mass resolution: R m Q m Q 1 π h CIME example: h3, N 3 R~ 1-4 N We want R small separation of close ions Meaning that ions with a m/q > 1.1 m/q will not be extracted To have R small for a given harmonic h, the number of turn N needs to be increased lowering the accelerating voltage small turn separation poor injection and/ or extraction (great problems for new exotics beam machines : isobar and contamination for new machine ) 4

41 Cyclotron as a separator ϕ π Nh 1 γ ( m / Q m / Q ) Q/m ~ 3/18 ~ 6/36 m 36 and Q6 9 m18 and Q3 41

42 BIBLIOGRAPHY Textbooks S. Humphries, jr., J. Wiley & Sons 1986, Principles of Charged Particle Acceleration. H. Bruck, Bibl. des Scienc. Et Tech. Nucléaires 1966, Accélérateurs Circulaires de Particules. J. J. Livingood, D. van Nostrand Comp. 1961, Principles of Cyclic Particle Accelerators. Conference Proceeding Proceedings of the International Conferences on Sector-focused Cyclotrons and on Cyclotrons and their application : ICC1, Informal Conference on sector-focused Cyclotrons 1959 in Sea Island, NAS-NRC, Publ.656 (1959) ICC, Int. Conference 196 in Los Angeles, Nucl. Inst.& Meth. 18, 19 (196) ICC3, Int. Conference 1963 in Geneva, CERN (1963) ICC4, Int. Conference 1966 in Gatlinburg, IEEE Trans. NS-13(4) (1966) ICC5, 5th Int. Cyclotron Conference 1969 in Oxford, McIllroy, Butterworth (1971) ICC6, 6th Int. Cyclotron Conference 197 in Vancouver, AIP Conf. Proc. N 9 (197) ICC7, 7th Int. Cyclotron Conference 1975 in Zürich, Birkhäuser (1975) ICC8, 8th Int. Cyclotron Conference 1978 in Bloomington, IEEE Trans. NS-6() (1979) ICC9, 9th Int. Cyclotron Conference 1981 in Caen, les Editions de Physique (198) ICC1, 1th Int. Cyclotron Conference 1984 in East Lansing, IEEE, New York (1984) ICC11, 11th Int. Cyclotron Conference 1986 in Tokyo, Ionics Publishing Tokyo (1987) ICC1, 1th Int. Cyclotron Conference 1989 in Berlin, World Scientific Publ. (1991) ICC13, 13th Int. Cyclotron Conference 199 in Vancouver, World Scientific Publ. (1993) ICC14, 14th Int. Cyclotron Conference 1995 in Cape Town, World Scientific Publ Co. (1996) ICC15, 15th Int. Cyclotron Conference 1998 in Caen, Institute of Physics Publ. (1999) ICC16, 16th Int. Cyclotron Conference 1 in East Lansing, American Inst. of Ph., New York (1) Contribution to CERN Accelerator Schools W. Joho, CAS Aarhus 1986, CERN 87-1 (1987) 1 Modern Trends in Cyclotrons H.L. Hagedoorn et al., CAS Jülich 199, CERN 91-4 (1991) 33 Introduction to Cyclotron H.L. Hagedoorn et al., CAS Leewenhorst 1991, CERN 9-1 (199) 1 Hamilton Theroy P. Heikinnen, CAS Jyväskylä 199, CERN 94-1 (1994) Cyclotrons and Injection and Extraction T. Stammbach, CAS La Hulpe, 1994, CERN 96- (1996) Introduction to Cyclotrons F. Chautard, CAS Zeegse, CERN-6-1 (5) Beam dynamics for cyclotrons 4

43 END 43