Statistical and Collective Effects in Accelerator Physics

Accelerator Physics Statistical and Collective Effects II A. S. Bogacz, G. A. Krafft, S. DeSilva, B. Dhital Jefferson Lab Old Dominion University Lecture 16 USPAS Accelerator Physics January 2020

Beam Temperature K-V has single value for the transverse Hamiltonian ( ) 2 ( ) 2 + + 2 y y y + + 2 x x x 1 1 y y + = x x C x x y y ( ) ( ) , , , x x y y 1 C No Temperature! Temperature in beam introduces thermal spreads Transverse Temperature Debye length Longitudinal Temperature Landau Damping USPAS Accelerator Physics January 2020

Warm Beam Models Model beam as fluid with pressure scalar Bennet model for self-pinched beam in plasma Newton s law for fluid mass in volume dV (no electric field in plasma) dv mn dV F pdV dt = = + env BdV In equilibrium = p env BdV Equation for magnetic field r 1 r r e ( ) ( ) ( ) n r r dr = = 0 rB ev n r B v 0 z z r 0 USPAS Accelerator Physics January 2020

Bennet Profile Using the ideal gas law the density solves r e dnkT dr ( ) n r r dr = 2 z 0 n v r 0 Bennet profile n ( ) n r = 0 ( ) + 2 2 2 1 / r r b Beam size 8 n e kT = 2 r b 2 z 2 / 0 0 USPAS Accelerator Physics January 2020

Beam Profile Long Solenoid Radial focusing in rotating frame = 2 L r nm r Using the ideal gas law the density solves dnkT dr = 2 L nm r Gaussian profile ( ) ( ) n r = 2 2 r 0exp n / 2 r Beam size kT m = 2 r 2 L USPAS Accelerator Physics January 2020

Waterbag Distribution Lemons and Thode were first to point out SC field is solved as Bessel Functions for the 2D adiabatic equation of state (P/n2constant). Later, others, including my advisor and I showed the equation of state was exact for the waterbag transverse distribution. ( 0 2 2 2 2 T A H H = = + = ) + 2 2 2 m x y 2 z p ( ) = + + + H x y e T SC m ( ) = 0 ( 2 ) + 2 0 2 2 m x y ( ) n r = n 1 / SC dx dy H e 0 0 b H 0 0 ( ) ( 2 ) 2 z 2 2 + 2 0 2 2 p x y dx dy m x y H m = 2 v 1 0 SC dx dy H 2 m 0 0 USPAS Accelerator Physics January 2020

Self-consistent potential solves ( 2 ) + 2 0 2 2 m x y en = 2 1 SC b SC 2 D H 0 0 m H e n m H = = = Debye Length 0 2 0 0 0 v D 2 e n p b b Analytic solutions in terms of Modified Bessel Functions = + ( ) + 2 0 2 2 m x y A I ( ) ( ) r ( ) ( ) + / 1 / e r BK r 0 0 SC D D 2 = 0 by boundary condition chosen so that solution without solution to inhomogeneous eqn. I B A 0 USPAS Accelerator Physics January 2020

Equation for Beam Radius Now 1 r r 2 r = 2 r 2 r ( ) = 2 D 2 0 2 p 2 A m = At the density vanishes r r b ( ) ( ) ( ) = 2 D 2 0 2 p 2 1 / H m I r 0 0 b D 2 p ( ) + = 1 / I r 0 b D 2 0 2 p 2 ( ) ( ) / / I r I r ( ) n r = 0 0 b D r D n ( ) b / 1 I 0 b D USPAS Accelerator Physics January 2020

Debye Length Picture* *Davidson and Qin USPAS Accelerator Physics January 2020

Collisionless (Landau) Damping Other important effect of thermal spreads in accelerator physics Longitudinal Plasma Oscillations (1 D) n t + = 0 v n z d eE m en = z z dt E = z z 0 USPAS Accelerator Physics January 2020

Linearized v z n t + = 0 n z 0 e = E z z t E z m e n = z 0 2 2 en m e n E z n = = 0 0 n z 2 t m 0 2 e n i t = 0 n e p p m 0 In fluid limit plasma oscillations are undamped USPAS Accelerator Physics January 2020

Vlasov Analysis of Problem ( ) , , e z F z p t = + + = ( ) , z p dp dp x y = 0 v e F z e t z z p z ( ) 2 e F dp n e z i 2 z 0 0th order solution F F p = ( ) = , 0 0 0 e z linearized + F p = 0 v F e z e t z z z 2 e = F dp e z 2 z 0 USPAS Accelerator Physics January 2020

Initial Value Problem Laplace in t and Fourier in z = ( ) ( ) i t Im large enough to converge F dte F t 0 1 ( ) ( ) C i t = d e F t F 2 d dt ( ) ( ) ( ) 0 i t = i F = 0 dte F t F t = ( ) ( ) ilz L = 2 / , , z t l t e l ( v ) i F l p t = ( ( z F l p t v , , 0 2 / e l F v p ( ) ( ) F l p = + e z , , , 0 2 z l ( ) ( ) ( ) e z l L 2 / / l L L z z ) 2 / l F p p 2 e ( ) 0 v z z , dp l ( ) = ) z 2 / L l L 2 L ( ) 0 = , l ( ) ) 2 l , , 0 ei + e z dp ( z 2 / l L 0 z USPAS Accelerator Physics January 2020

Dielectric function Landau (self-consistent) dielectric function ( ) ( ) ( ) = , , , D l l N l ( ) / F p p 2 e L ( ) = + 0 z z , 1 D l dp ( ) z 2 2 / l v l L 0 z Solution for normal modes are ( ) = , 0 D l ( 2 ) F p 2 e ( ) = 0 z l L , 1 D l dp ( ) ) z ( ) 2 m / v 0 z ( / F p n = 0 v 0 z 2 p 1 dp ( ) z ( ) 2 l L 2 / z USPAS Accelerator Physics January 2020

Collisionless Damping For Lorentzian distribution F n = 0 + 2 z 2 p 0 1 = 2 p 1 dp ( ) z ( ) 2 + 2 z 2 p 2 / p l Lm z 2 p = ( ) ( ) 2 + l Lm 2 / i Landau damping rate 2 l = iLm p USPAS Accelerator Physics January 2020

Negative Mass Instability Simplified argument: assume longitudinal clump on otherwise uniform beam Particles pushed away from clump centroid If above transition, come back LATER if ahead of clump center and EARLIER if behind it The clump is therefore enhanced! INSTABILITY; particles act as if they have negative mass (they accelerate backward compared to force!) USPAS Accelerator Physics January 2020

Longitudinal Impedance longitudinal wake function distance between exciting charge and test charge 1 , z q arrival ring q W q ( ) ( ) + / units V/C W E z t c dz trailing particle (singly charged) picks up voltage per turn of " " e ( ) ( ) z W ( ) z V z = z z dz total energy loss " " ( ) z dz ( ) z W ( ) = U e e z z dz z USPAS Accelerator Physics January 2020

Frequency Domain ( ) ( 1 ) = , , note the coordinate moves with beam I z t c z t z " " z z ( ) ( ) V z t = + , , I z t W z z dz c c z Fourier Transform " " 1 ( ) ( ) ( ) ( ) ( ) I = / i c V I e W dz Z c z 1 ( ) ( ) = / i c Z e W d c 1 ( ) z ( ) i z = / c W e Z d 2 Loss factor = 2 q U q ( ) ( ) 0 2 = Re k Z I d 2 2 USPAS Accelerator Physics January 2020

NMI Simple Analysis revolution frequency of particle d dt d dt dE dt = + t t d dE dt dE dE dt = = 0 c 2 E 0 ( ) i n t = = 0 0 qV qZ I e zn n 2 2 ( ) i n t = + oscillation frequency of disturbance Z q E e 0 n n I 2 0 2 ( ) = c n i 0 n 2 0 USPAS Accelerator Physics January 2020

Linearized Continuity Equation = = 2 I v r v z b z ( ) + = 0 v z t z 1 R ( ) + = v z t + + = 0 0 0 t ( ) = n I nI 0 0 n n USPAS Accelerator Physics January 2020

Oscillation Frequency 2 0 E nq I ( ) 2 = = 2 0 c n i Z 0 2 2 0 1 mode has positive imaginary part instability Resistive impedance has positive real part "Resistive wall instability" If Re 0 (e.g. space charge impedance at long Z = stability/instability depends on sign of RHS Im 0 (inductive, stable if Im 0 (capacitive, space charge is this way, stable if Later case is negative mass instability Re 0 Z wavelengths) 0,unstable if 0) Z Z c c 0,unstable if 0) c c USPAS Accelerator Physics January 2020

NMI Growth time Impedance? = e r e r r e r r r 0 c r r rc b 2 b 2 r 2 2 0 b = b E B r rb e r r 0 c r r b 2 b r 2 0 e z ( ) ( ) ( ) = 1 2ln + 2 1 / E r r z c b 4 z 0 ( ) i n t e n in in ( ) ( ) ( ) ( ) = 1 2ln + = 1 2ln + / / V r r I r r SC n c b n c b 2 2 2 2 c 0 0 2 0 E 2 2 0 2 nq I n q I ( ) ( ) ( ) 2 = = 1 2ln + / 0 0 c c c E n i Z r r 0 SC c b 2 2 2 4 0 0 0 USPAS Accelerator Physics January 2020

Stabilization by Beam Temperature? Canonical variables , + + = + / p p 0 = 0 t ( ) i n t e n 0 n ( ) = 0 i n ( ) n n i n t e n = = 0 0 0 0 c current perturbation is = I q d 0 n n USPAS Accelerator Physics January 2020

Dispersion Relation ( ) ( ) dE dt = 0 0 0 c 1 ( ) = = 2 / E 0 0 c 2 3 0 q Z / = c E 1 0 i d 2 2 n 0 n recover before N = ( ) / 2 0 0 b / N n = 0 b d ( ) 2 n 2 n n 0 n USPAS Accelerator Physics January 2020

Landau Damping Use our favorite analytic distribution 1 = 1 ( ) ( ) 0 ( ) 0 0 + 2 2 0 2 + 2 0 = 0 0 c 2 0 q Z I n E 0 c 1 i d ( ) ( ) ( ) 2 2 2 2 + 2 n 0 0 n q Z I n E 2 0 = 0 c 1 i ( + ) 2 2 2 + n ni 0 0 n = + n ni V iU 0 n USPAS Accelerator Physics January 2020

LD from another view Single Oscillator u u Fe u t = i t + = 2 Fe i t 1 1 ( ) + 2 Many oscillators distributed in frequency simultaneously excited 1 dN N d ( ) = N u i = = 1 N i U i t 1 1 Fe = U d = + 2 ( ) Fe ( ( ) ) for i t = U d USPAS Accelerator Physics January 2020

Resonance Effect ( ) i t Fe ( ) = + + . . U i PV d ( ) ( ) i t = . . U Fe iPV d For our analytic Lorentzian ( ) = ( ) + 2 2 i t i t Fe i Fe + ( ) = = U i 2 2 Energy goes in! Where does it go? USPAS Accelerator Physics January 2020

Inhomogeneous Solution F ( ) = + sin sin u t a t t 2 2 Solution with zero initial excitation F a = 2 2 F = sin sin u t t 2 2 No energy flow sin F t = cos u t t = 2 2 Resonant particles capture energy and oscillation genera ted out of phase USPAS Accelerator Physics January 2020

Multipass BBU Instability USPAS Accelerator Physics January 2020

BBU Theory following Krafft, Laubach, and Bisognano = k R Q ( ) HOM /2 HOM Q sin HOM HOM transverse W e HOM 2 HOM Single cavity/Single HOM case t ( ) ( ) ( ) ( ) t I t d t dt = V t transverse W t On the second pass T eV t d t = ( c ) t ( ) 12 r With no initial displacement t T e ( ) ( ) ( ) ( t I t V t ) = V t 12 c transverse W t t dt r Delay differential (integral) equation USPAS Accelerator Physics January 2020

Beam current = ( ) I t ( ) = 0 0 I t t mt 0 m Normal mode V nt ( ) i nt = V e 0 0 0 Sum the geometric series for eigenvalue equation e Ke e e + i t HOM /2 t HOM Q sin e t 0 0 i t = 1 0 HOM r ( ) 2 i t HOM i t HOM /2 /2 t HOM Q t HOM Q 1 2 cos e e t 0 0 0 0 0 HOM ( T ) = 2 HOM / / 2 1 at threshold K For R Q sin k t 12 0 0 eT I t HOM 0 K 12 HOM r USPAS Accelerator Physics January 2020

Perturbation Theory Works iK i t HOM HOM i t /2 t HOM Q i t 1 e e e e 0 0 0 r 2 Growth rate ( 2 ) HOM r sin K t ( ) Im HOM Q 2 t 0 HOM Threshold current 1 th I e R Q 2 = HOM ( ) ( ) HOM r 2 HOM / sin Q k T t 12 HOM HOM USPAS Accelerator Physics January 2020

CEBAF (Design) Simulations Stable Bunch Number USPAS Accelerator Physics January 2020

Close to threshold Bunch Number USPAS Accelerator Physics January 2020

Unstable Bunch Number USPAS Accelerator Physics January 2020

Chromatic (Landau) Damping When threshold current is modified to 1 / HOM e R Q depends on energy offset , T 12 2 k = HOM I ( ) ( ) th HOM r 2 HOM sin Q T t 12, HOM eff ( ) ( ) ( ) f T T f d 12 = T 12, eff d If 1, 0 12, eff USPAS Accelerator Physics January 2020