Longitudinal Beam Dynamics

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2 Longitudinal Beam Dynamics Shahin Sanaye Hajari School of Particles and Accelerators, Institute For Research in Fundamental Science (IPM), Tehran, Iran IPM Linac workshop, Bahman 28-30, 1396

3 Contents 1. Introduction 2. Electrostatic vs RF acceleration 3. RF cavities 4. Longitudinal beam parameters 5. Velocity modulation bunching 6. Adiabatic phase damping 7. Longitudinal dynamics of the IPM linac

4 1. Introduction

5 1. Introduction 1 Charged particle beams Particles traveling in nearly the same direction with nearly the same energy. σ Ek E k p x, p y p z Longitudinal direction & Transverse Plane Independent motion. 3D 2D+1D Acceleration vs Beam Control.

6 1. Introduction 2 The goal of a Beam Dynamics study To study the beam behavior under the influence of electromagnetic fields of accelerator components (magnets and cavities and ) and the beam itself. Maxwell's equations:. E = ρ ε 0. B = 0 E = B t B = μ 0 J + E t Lorentz force: dp dt = q E + v B Determination of the required array of the electromagnetic fields (specification of the accelerator components). Perquisites Beam Dynamics Design Classical mechanics (including Hamiltonian dynamics) Electromagnetic theory. Special relativity. Statistical mechanics (only for Space charge dynamics)

7 2. Electrostatic vs RF acceleration

8 2. Electrostatic vs RF acceleration 3 Electrostatic acceleration limitations Breakdown effect Highest voltage ever: ~12 MV Large and expensive machines. Holifield Heavy Ion Accelerator, ORNL

9 2. Electrostatic vs RF acceleration 4 Electrostatic acceleration limitations Non-repeatable Electrostatic forces are conservative! Work done by a conservative force on a particle in a closed loop is zero! Solution: Electromagnetic waves confined in cavities. The frequency range determines the cavity dimension. Radio Frequency (RF) acceleration

10 2. Electrostatic vs RF acceleration 5 RF acceleration and the concept of bunching Sinusoidal field: Alternative acceleration and deceleration depending on the arrival time. Bunched beam crest zero crossing

11 3. RF cavities

12 3. RF cavities 6 RF cavities An RF cavity is simply an empty space surrounded by metallic walls in which the electromagnetic waves resonate. In accelerators, one usually uses cavities with axisymmetric geometries. Cavity modes Different solutions to Maxwell s equations for any specific boundary condition. The excited mode depends on the cavity geometry and the frequency. Transverse electric (TE), transverse magnetic (TM), transverse electric and magnetic (TEM) modes. transverse magnetic (TM) or E mode is characterized by a longitudinal electric field (E z ).

13 3. RF cavities 7 RF cavities Example: TM 01 mode in a cylindrical waveguide [1] (the lowest frequency mode). E z = E 0 J 0 k c r e ik zz e iωt k = ω c /c k c = ω c /c = p 01 /b k z 2 = k 2 k c 2 first root of J 0 x E z = E 0 cos θ Longitudinal field on axis θ = k z z ωt RF phase seen by the particle Synchronization condition: dθ dt = 0 v = ω k z v ph = c ω ω 2 ω c 2 > c definition of phase velocity,v ph [1] D.M. Pozar, Microwave Engineering, third edition, John Wiley & Sons, 2005 (chapter 3).

14 3. RF cavities 8 Disc loaded structure; Slowing down the waves! Introducing some obstacles, e.g. irises and providing a periodic structure. A perturbed version of the cylindrical waveguide Periodic function of z E z = E d r, z e ik 0z e iωt r < a Fourier expansion of E d r, z and applying Maxwell s equations E z r, z, t = E 0 C n J 0 K n r e i ωt k nz n= k n = k 0 + 2πn/d K n 2 = ω/c 2 k n 2 v ph,n = ω k n = ω k 0 + 2πn/d Infinite number of waves with different phase velocities (called Space Harmonics).

15 3. RF cavities 9 Structure modes Slater theorem dispersion relation [2]: ω = p 01c b 1 + κ 1 cos k n d e αh κ = α = 4a 3 3πJ 1 2 p 01 b 2 d 1 p 01 a 2 ω c 2 Phase advance per cell ψ = k 0 d 0, π π 2, 2π 3, π, v ph,0 = ω = ω d adjustable phase velocity k 0 ψ Principle wave (n = 0) Structure modes [2] T.P. Wangler et al., RF Linear Accelerator, 2nd edition, John Wile & Sons, 2008 (chapter 3).

16 3. RF cavities 10 Traveling wave (TW) vs Standing wave (SW) cavities TW structure E TW z r, z, t = ℇ TW i kz ωt z r, z e Short filling time (< 1μs). SW structure E z SW r, z, t = ℇ z SW r, z e iωt (Fixed nodes) Longer filling time (~10s μs).

17 4. Longitudinal beam parameters

18 4. Longitudinal Beam parameters 11 Longitudinal phase space Kinetic energy vs phase(ωt) At a longitudinal position (z) z vs t Beam parameters: rms bunch length: σ φ = φ φ 2 rms energy spread: σ Ek = E E 2 rms longitudinal emittance: a measure of the area of phase space ellipse. σ Ek = E E 2 φ φ 2 E E φ φ

19 4. Longitudinal Beam parameters 12 Longitudinal phase space Convergent phase space: φ E k > 0 Early particles Late particles After 30 cm Divergent phase space: φ E k < 0 After 30 cm

20 5. Velocity modulation bunching

21 5. Velocity modulation bunching 13 Thin lens cavity and velocity modulation Energy gain in short cavity L/2 E k = q Re E SW z 0, z, t dz L/2 L/2 = q ℇ SW z 0, z cos ωt dz L/2 = qv cos ωt Definition of the cavity voltage

22 5. Velocity modulation bunching 14 Velocity modulation bunching

23 5. Velocity modulation bunching 14 Velocity modulation bunching

24 5. Velocity modulation bunching 14 Velocity modulation bunching

25 5. Velocity modulation bunching 14 Velocity modulation bunching

26 5. Velocity modulation bunching 14 Velocity modulation bunching

27 5. Velocity modulation bunching 14 Velocity modulation bunching

28 5. Velocity modulation bunching 14 Velocity modulation bunching

29 5. Velocity modulation bunching 14 Velocity modulation bunching

30 5. Velocity modulation bunching 14 Velocity modulation bunching

31 5. Velocity modulation bunching 14 Velocity modulation bunching

32 6. Adiabatic phase damping

33 6. Adiabatic phase damping 15 Longitudinal Beam dynamics in a TW structure Longitudinal electric field E z r, z, t = E 0 C n J 0 K n r e i ωt k nz n= 1. In most of structures of interest C n C 0, n The structure is design so that v = v ph,0, therefore the interaction of the particles with other space harmonics (v v ph,n, n 0) can be neglected. Re E z 0, z, t = E 0 cos θ, θ = ωt k 0 z Longitudinal equation of motions E k = ee 0 cos θ z = mc 2 γ θ = ωt k 0 z = ω t 1 v ph z dγ = e E dz mc 2 0 cos θ dθ = ω 1 1 dz v v ph Definition of synchronous particle A theoretical particle whose velocity is kept (approximately) equal to v ph derfore dθ s /dz 0.

34 6. Adiabatic phase damping 16 Motion of non-synchronous particle γ = γ s + Δγ θ = θ s + Δθ d 2 Equation of motion dz 2 Δθ + 2α d dz Δθ + Ω2 Δθ = 0 dδγ = ee 0 dz sin θ mc 2 s Δθ dδθ = ω γ dz c s 2 1 3/2 Δγ with 3/2 e dγ dz = e mc 2 E 0 cos θ dθ dz = ω 1 v 1 v ph Ω 2 = ω γ c s 2 1 E mc 0sin θ 2 s α = 3γ s e E 2 γ 2 s 1 mc 0cos θ 2 s Beam acceleration Damped oscillatory motion The acceleration procedure At the entrance of the structure: v ph = v s and cos θ s = 0 or θ s = π/2 Then v ph is slowly increased. v ph > v s dθ s dz > 0

35 6. Adiabatic phase damping 16 Motion of non-synchronous particle γ = γ s + Δγ θ = θ s + Δθ d 2 Equation of motion dz 2 Δθ + 2α d dz Δθ + Ω2 Δθ = 0 Beam acceleration dδγ = ee 0 dz sin θ mc 2 s Δθ dδθ = ω γ dz c s 2 1 3/2 Δγ with Damped oscillatory motion dγ dz = e mc 2 E 0 cos θ dθ dz = ω 1 v 1 v ph Ω 2 = ω γ c s 2 1 3/2 ee 0 sin θ mc 2 s With a Hamiltonian dynamics approach one finds [3]: The acceleration procedure At the entrance of the structure: v ph = v s and cos θ s = 0 or θ s = π/2 Then v ph is slowly increased. v ph > v s dθ s dz > 0 α = 3γ s 2 γ s 2 1 Δθ max = C emc3 ω sin θ s E 0 γ s 2 1 3/2 Δγ max = C emc3 ω sin θ s E 0 γ s 2 1 3/2 ee 0 mc 2 cos θ s 1/4 [3] J. Le Duff, Dynamics and Acceleration in Linear Structures, CERN Accelerator School (CAS), CERN /4

36 6. Adiabatic phase damping 17 Motion of non-synchronous particle Example: particle trajectories in three sample TW structure dγ dz = e mc 2 E 0 cos θ dθ dz = ω 1 v 1 v ph v ph constant v ph linearly increased v ph linearly increased E 0 constant E 0 constant E 0 linearly increased

37 7. Longitudinal dynamics of the IPM linac

38 8. Longitudinal dynamics of the IPM linac 18

39 Thanks for your attention!