LONGITUDINAL DYNAMICS �������� ������������������ �������� ���������������� Boris Vodopivec Joël Le DuFF
summary • Radio-Frequency Acceleration and Synchronism • Properties of Radio-Frequency cavities • Principle of Phase Stability and Consequences • Synchronous linear accelerator • The Synchrotron • RF cavities for Synchrotron • Energy-Phase Equations in a Synchrotron • Phase space motions
Bibliography : Old Books M. Stanley Livingston High Energy Accelerators (Interscience Publishers, 1954) J.J. Livingood Principles of cyclic Particle Accelerators (D. Van Nostrand Co Ltd , 1961) M. Stanley Livingston and J. B. Blewett Particle Accelerators (Mc Graw Hill Book Company, Inc 1962) K.G. Steffen High Energy optics (Interscience Publisher, J. Wiley & sons, 1965) H. Bruck Accelerateurs circulaires de particules (PUF, Paris 1966) M. Stanley Livingston (editor) The development of High Energy Accelerators (Dover Publications, Inc, N. Y. 1966) A.A. Kolomensky & A.W. Lebedev Theory of cyclic Accelerators (North Holland Publihers Company, Amst. 1966) E. Persico, E. Ferrari, S.E. Segre Principles of Particles Accelerators (W.A. Benjamin, Inc. 1968) P.M. Lapostolle & A.L. Septier Linear Accelerators (North Holland Publihers Company, Amst. 1970) A.D. Vlasov Theory of Linear Accelerators (Programm for scientific translations, Jerusalem 1968)
Bibliography : New Books M. Conte, W.W. Mac Kay An Introduction to the Physics of particle Accelerators (World Scientific, 1991) P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings (Cambridge University Press, 1993) D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators (J. Wiley & sons, Inc, 1993) H. Wiedemann Particle Accelerator Physics (Springer-Verlag, Berlin, 1993) M. Reiser Theory and Design of Charged Particles Beams (J. Wiley & sons, 1994) A. Chao, M . Tigner Handbook of Accelerator Physics and Engineering (World Scientific 1998) K. Wille The Physics of Particle Accelerators: An Introduction (Oxford University Press, 2000) E.J.N. Wilson An introduction to Particle Accelerators (Oxford University Press, 2001)
Methods of Acceleration 1_ Electrostatic Field Energy gain : W=n.e(V 2 -V 1 ) limitation : V generator = Σ V i Electrostatic accelerator 2_ Radio-frequency Field L=vT/2 Synchronism : v=particle velocity T= RF period Wideroe structure
The advantage of Resonant Cavities - Considering RF acceleration, it is obvious that when particles get high velocities the drift spaces get longer and one loses on the efficiency. The solution consists of using a higher operating frequency. - The power lost by radiation, due to circulating currents on the electrodes, is proportional to the RF frequency. The solution consists of enclosing the system in a cavity which resonant frequency matches the RF generator frequency. r r -Each such cavity can be independently H ou J powered from the RF generator. r - The electromagnetic power is now E z constrained in the resonant volume. - Note however that joule losses will occur in the cavity walls (unless made ω RF of superconducting materials)
The Pill Box Cavity From Maxwell’s equations one can derive the wave equations : ∂ 2 A − μ = ∇ ε A = 2 A 0 ( E ou H ) 0 ∂ 2 0 t Solutions for E and H are oscillating modes, at discrete frequencies, of types TM o r TE. For l<2a the most simple mode, TM 010 , has E z H θ the lowest frequency ,and has only two field components: ( ) } = kr E z J j ω 0 t e ( ) j = − kr J H θ 1 Z 0 π ω 2 = = λ = = Ω k 2 , 62 a 377 Z λ 0 c
The Pill Box Cavity (2) The design of a pill-box cavity can be sophisticated in order to improve its performances: -A nose cone can be introduced in order to concentrate the electric field around the axis, -Round shaping of the corners allows a better distribution of the magnetic field on the surface and a reduction of the Joule losses. A good cavity is a cavity which efficiently transforms the RF power into accelerating voltage.
Transit Time Factor Oscillating field at frequency ω and which amplitude is assumed to be constant all along the gap: V = ω = ω cos t cos t E z E 0 g Consider a particle passing through the middle of the gap at time t=0 : z = vt g / 2 eV z Δ = cos ω The total energy gain is: ∫ W dz g v − g / 2 ω g θ = θ transit angle sin / 2 Δ = = v W eV eVT θ / 2 T transit time factor ( 0 < T < 1 )
Transit Time Factor (2) Consider the most general case and make use of complex notations: ( ) ω Δ = ℜ z g j t ω = ω − ψ ∫ e E z e dz W t e z p 0 v Ψ p is the phase of the particle entering the gap with respect to the RF. ⎡ ⎤ z ( ) ω − ψ j Δ = ℜ j g ∫ v p W e ⎢ e E z e dz ⎥ e z 0 ⎣ ⎦ ⎡ ⎤ z ( ) ω − ψ ψ j Δ = ℜ j j g ∫ v ⎢ p ⎥ e e e E z e dz W i e z 0 ⎣ ⎦ φ = ψ p ψ − Introducing: i z ( ) ω j Δ = φ g ∫ v e E z e dz cos W z 0 ( ) ω g j t ∫ E z e dz and considering the phase which yields the = z 0 T ( ) maximum energy gain: g ∫ E z dz z 0
Principle of Phase Stability Let’s consider a succession of accelerating gaps, operating in the 2 π mode, for which the synchronism condition is fulfilled for a phase Φ s . For a 2 π mode, the electric field is the same in all gaps at any given time. is the energy gain in one gap for the particle to reach the next = ˆ Φ eV e V sin s s gap with the same RF phase: P 1 ,P 2 , …… are fixed points. If an increase in energy is transferred into an increase in velocity, M 1 & N 1 will move towards P 1 (stable), while M 2 & N 2 will go away from P 2 (unstable).
A Consequence of Phase Stability Transverse Instability ∂ ∂ V E z > ⇒ < Longitudinal phase stability means : 0 0 ∂ ∂ t z defocusing RF force ∂ ∂ ∂ The divergence of the field is E E E ∇ = ⇒ + = ⇒ > x z x . E 0 0 0 zero according to Maxwell : ∂ ∂ ∂ x z x External focusing (solenoid, quadrupole) is then necessary
The Traveling Wave Case ( ) = ω − E E 0 cos t kz z RF ω = RF k v ϕ ( ) = − z v t t 0 = v phase velocity The particle travels along with the wave, and ϕ k represents the wave propagation factor. = v particle velocity ⎛ ⎞ v = ω − ω − φ ⎜ ⎟ E E 0 cos t t z RF RF 0 v ⎝ ⎠ ϕ = z = φ If synchronism satisfied: v v and E E 0 cos ϕ 0 where φ 0 is the RF phase seen by the particle.
Multi-gaps Accelerating Structures: A- Low Kinetic Energy Linac (protons,ions) Mode 2 π L= vT = βλ Mode π L= vT/2 In « WIDEROE » structure radiated power ∝ ω CV In order to reduce the radiated power the gap is enclosed in a resonant volume at the operating frequency. A common wall can be suppressed if no circulating current in it ALVAREZ structure for the chosen mode.
CERN Proton Linac
The Synchrotron The synchrotron is a synchronous accelerator since there is a synchronous RF phase for which the energy gain fits the increase of the magnetic field at each turn. That implies the following operating conditions: ^ Energy gain per turn Φ e V sin Φ = Φ Synchronous particle s ω = ω h RF synchronism RF r ρ , R Constant orbit Variable magnetic field ρ = ⇒ P B B e If v = c, ω r hence ω RF remain constant (ultra-relativistic e - )
The Synchrotron (2) Energy ramping is simply obtained by varying the B field: ′ π ρ 2 e R B dp ′ ′ = ρ ⇒ = ρ ⇒ Δ = ρ = p eB e B ( p ) e B T turn r dt v 2 2 2 2 Since: = + ⇒ Δ = Δ E E p c E v p 0 ( ) ( ) = = π ρ = Δ Δ φ ˆ 2 e RB ' e V sin E W turn s s • The number of stable synchronous particles is equal to the harmonic number h. They are equally spaced along the circumference. • Each synchronous particle satifies the relation p=eB ρ . They have the nominal energy and follow the nominal trajectory.
The Synchrotron (3) During the energy ramping, the RF frequency ω ω = = ω RF increases to follow the increase of the ( B , R ) r s revolution frequency : h 2 hence : f ( t ) v ( t ) 1 e f ( t ) 1 ec r RF RF = = < > ⇒ = B ( t ) B ( t ) π π π h 2 R 2 m h 2 E ( t ) R s s s 2 2 2 2 Since , the RF frequency must follow the variation of the = + E m c p c 0 1 ⎧ ⎫ 2 2 f ( t ) c B ( t ) B field with the law : which asymptotically tends RF ⎨ ⎬ = 2 2 2 ⎩ ⎭ π + h 2 R ( m c / ecr ) B ( t ) s 0 c towards when B becomes large compare to (m 0 c 2 / 2 π r) which corresponds to → f r π 2 R v c (pc >> m 0 c 2 ). In practice the B field can follow the law: ω B 2 = − ω = B ( t ) ( 1 cos t ) B sin t 2 2
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