Beam Dynamics with Space- Charge

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1 Beam Dynamics with Space- Charge Chris Prior, ASTeC Intense Beams Group, RAL and Trinity College, Oxford 1

2 1. Linear Transverse Review of particle equations of motion in 2D without space-charge - Courant-Snyder parameters, envelope equation Examples of space-charge Particle and envelope equations with linear space-charge Space-charge (Laslett) Tune shift Image effects Envelope oscillations, resonances 2

3 2. Non-linear Transverse General particle equations under space-charge Non-linear beams - rms beam sizes, rms emittance - rms envelope equations - evolution of rms emittance Examples of 2D distributions - KV, waterbag, Gaussian - concept of stationary distributions Beam halo causes, measurement (kurtosis) 3

4 3. Longitudinal Effects Longitudinal space-charge Self-consistent distributions - Hofmann-Pederson model Microwave instability Acceleration cycle in a synchrotron Bunch compression Long and short bunches 4

5 4. Injection Techniques to build up beam intensity Liouville s Theorem and implications optimum injection parameters 2-plane injection Non-Liovillean injection charge exchange injection transverse phase-space painting treatment of partially stripped ions foil heating Longitudinal painting bunching factor control 5

6 5. Modelling Techniques Envelope codes with linear space-charge Modelling beams with space-charge Principles of tracking codes - Particle-in-cell, finite differences, finite elements - Boundaries and images - Space-charge calculations; multiprocessors - Generation of model distributions; number of particles - Leap-frog, Runge-Kutta and other integration techniques - Charge allocation Available codes, capabilities and limitations 6

7 6. High Intensity Proton/Ion Facilities Spallation sources - ISIS/SNS/ESS/C-SNS Neutrino Factory/Muon Collider Accelerator Driven Systems (ADSR) - Chinese ADSR - Transmutation Heavy Ion Fusion - HIDIF Radioactive Ion Beam Facilities - FAIR, FRIB, SPIRAL2, EURISOL 7

8 7. Other Possible Topics Theoretical Envelope oscillations Instabilities Vlasov Equation - Practical Techniques to mitigate against spacecharge effects Beam loss control 8

9 Reading E.J.N. Wilson: Introduction to Accelerators S.Y. Lee: Accelerator Physics M. Reiser: Theory and Design of Charged Particle Beams D. Edwards & M. Syphers: An Introduction to the Physics of High Energy Accelerators M. Conte & W. MacKay: An Introduction to the Physics of Particle Accelerators R. Dilao & R. Alves-Pires: Nonlinear Dynamics in Particle Accelerators R. Davidson & Hong Qin: Physics of Intense Charged Particle Beams in High Energy Accelerators 9

10 Notation and Basic Formulae Velocity of light c = m/sec Relative velocity = v c, v = c 1 Relativistic gamma = r = v 2 1 c 2 Rest mass m 0 1 p 1 2 Relativistic mass m = m 0 Momentum p = mv = m 0 v = m 0 c Energy E = mc 2 = m 0 c 2 Kinetic energy T = E m 0 c 2 = m 0 ( 1)c 2 Note: E 2 c 2 = p2 + m 2 0c 2 10

11 Maxwell s Equations E B j Electric field Magnetic flux density Charge density Current density E = 0 B =0 E = B t B = µ 0 j + 1 E c 2 t µ 0 Permeability of free space, Permittivity of free space, µ 0 = 1 c 2 James Clerk Maxwell 11

12 Gauss Flux Theorem E = 0 Equivalent to Gauss Flux Theorem:! r E = 0!!! ZZZ V r E dv = ZZ SE ds = 1 0 ZZZ The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface. dv = Q 0 12

13 Ampère s Circuital Law r ^ B = µ 0 j + 1 c 2 E t r B = µ 0 j ZZ r B ds = S I B dl = µ 0 ZZ j ds = µ 0 I S For a straight line current: B = µ 0I 2 r 13

14 Review of Simple Particle Dynamics Equation of motion in electromagnetic fields: dp dt = q E + v ^ B Lorentz Force produces acceleration produces bending Total energy E = mc 2 and E 2 = p 2 c 2 + m 2 0c 4 = E de = de dt = c2 p dp dt = qc2 p E dt = qv E Energy change only from Electric fields 14

15 When E = 0, E =constant = =constant = dv dt = q m v Suppose B = (0, 0,B), B constant, then, in cartesian coordinates (x, y, z), ẍ = qb m ẏ (1) Put Z = x + iy; (1)+i(2) = Z = iqb m If Z = 0 when Z = 0, solution is ÿ = qb m ẋ (2) z = 0 = ż = constant Z B Z = Z 0 e iqbt/m = Z = constant, circular motion about B-field lines qb at a rate m = C, with constant motion in direction of B 15

16 Axisymmetric Motion When E = 0, equation of motion is d dt (m 0 v) =qv B Assuming B = (0, 0,B), motion is axisymmetric, use cylindrical polar coordinates (r,,z) Magnetic field only = v = const = = const, so m 0 r r 2 = qbr (1) m 0 1 r d dt (r2 ) = qṙb (2) 16

17 Busch s Theorem -equation m 0 1 r d dt (r2 ) = qṙb = m 0 r 2 + q Brṙ dt = K, constant = m 0 r 2 + q 2 = K where = B ds is the magnetic flux through the particle s path For constant B = r 2 B so m 0 r qbr2 = K True%generally,%equivalent%to%constant%canonical%momentum% p θ %in%hamiltonian%formalism. 17

18 1. Particle launched at point 1 with 1 =0 At any other point (r,,z) m 0 r 2 = q For uniform field B z (r, z) =B, = qb 2m 0 1 r 2 1 r = ± 1 2 c 1 r 2 1 r 2 where c = qb m 0 is the cyclotron frequency (Sign depends on polarity of the charge) 18

19 2. Particle launched where B = 0, 1 =0 Then 1 = 0, so m 0 r 2 = q 2 Near axis where r 2 B, = = qb 2m 0 = 1 2 c = L where L = qb 2m 0 is the Larmor frequency 19

20 r-equation m 0 r r 2 = qbr (1) From Busch: = K r 2 L = r r K r 2 = r + 2 L r 2 = 2 L r L K 2 r 3 = 0 1. Circular motion, r =0 = r = 2. If K = 0, r cos / sin L t = can be focused to a point. 3. If K = 0, particles cannot be focused to a point. K L K r 2 L 20

21 External Forces Maxwell s time-independent equations, no sources: r^ B =0 =) 9 such that B = r Scalar magnetic potential B =0 = 2 =0, Laplace s equation Simplest solutions, with z = x + iy (i = p 1) are n =1 / x + iy dipole n =2 / (x 2 y 2 )+2ixy quadrupole n =3 / x(x 2 3y 2 )+iy(3x 2 y 2 ) sextupole Then 21

22 6 y design orbit Q QQQs x + s Write x 0 = ẋ ṡ,y0 = ẏ ṡ Equations of Motion m 0 d dt ( v) =q v ^ B = q =) m 0 d dt No bends 2 4 ẋ ẏ ṡ ẋ ẏ ṡ 3 5 = Knq 5 ^ Kn i 0 iṡ ṡ iẋ ẏ, and note in an accelerator ẋ, ẏ ṡ z n 1 5 z n 1 so that v =(x 0,y 0, 1)ṡ, x 0,y 0 1 and d dt =ṡ d ds 22

23 From above, = 1 d dt ( ṡ) 0 ẋ 2 +ẏ 2 +ṡ 2 c 2 =( (x 02 + y ) 2 ) second order terms Paraxial Equations Ignoring second and higher apple velocity terms, apple equations of motion are m 0 2 c 2 x 00 i y 00 = Knq c z n k = q Field strength m 0 c = Field strength/b 23

24 Equations of Motion in Dipoles apple x x y 00 = 2 4 x x y = apple 0 0 apple x 00 +2ky 0 + k 0 y y 00 2kx 0 k 0 x = apple

25 Mathieu-Hill Equations Paraxial equation of motion in periodic systems: x (s)+k x (s)x = 0 y (s)+k y (s)y = 0 where s is distance along beam axis k x (s), k y (s) periodic focusing functions, k(s + L) =k(s) George Hill Floquet s Theorem confirms two independent solutions: u = w(s)e i (s), v = w(s)e i (s) The Wronskian is W (u, v) =uv vu = 2iw 2 = C, a constant Choose C = 2i = d ds = = 1 w 2. Then, substitute u or v into Mathieu-Hill equation: Wronskian 25

26 u =(w + iw )e i =(w + i/w)e i, u =(w iw /w 2 + iw /w)e i =(w 1/w 3 )e i u + ku =0 = w + kw 1 w 3 =0 Any solution of Matthieu-Hill is a linear combination of u, v, so set x = Aw(s) cos( (s)+ ). = A w 2 sin( + ) x w = A cos( + ), d ds = x2 w 2 + wx w x 2 = A 2. Or: ˆ x 2 +2ˆxx + ˆ x 2 = A 2 where ˆ = w 2, ˆ = ww = 1 2 ˆ, ˆ = 1 w 2 + w 2 = 1+ˆ2 x w ˆ 26

27 Phase-Space Ellipse; Emittance ˆ (s)x 2 +2ˆ(s)xx 0 + ˆ (s)x 02 = A 2 Area of ellipse is A 2 ( ˆ ˆ ˆ2) = A 2 ˆ x 2 +2ˆxx 0 + ˆ x 02 apple is beam ellipse in x-x 0 phase space and is called the beam emittance ˆ, ˆ, ˆare Courant-Snyder parameters q Beam size (half-width) is a(s) = ˆ(s) Z Z Z ds Phase advance = 0 ds = w 2 = ds ˆ 27

28 is a constant of the motion, independent of s ˆ, ˆ, ˆ determine the shape of the ellipse in x-x phase space For a ring ˆ, ˆ, ˆ are periodic functions, so that the ellipse rotates and shears with position s, while its area is conserved (no acceleration) Phase advance around a ring gives the tune (number of oscillations per revolution) Q = 1 I ds 2 ˆ(s) Beam envelope given by maximum value of x: a = A max w(s) = w(s) Equation for w then gives the envelope equation: w 00 + kw 1 w 3 =0 =) a00 + ka 2 a 3 =0 28

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