Understanding Longitudinal Particle Motion in Accelerators

Longitudinal Motion 1 Eric Prebys, FNAL

Acceleration in Periodic Structures We consider motion of particles either through a linear structure or in a circular ring Always negative 1 p p = 2 p p In both cases, we can adjust the RF phases such that a particle of nominal energy arrives at the the same point in the cycle s E Goes from negative to positive at transition 1 1 p p = 2 2 p p t Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 2

Slip Factors and Phase Stability The sign of the slip factor determines the stable region on the RF curve. >0 (above transition) <0 (linacs and below transition) (t ) (t ) V V bunch Particles with lower E arrive earlier and see greater V. Particles with lower E arrive later and see greater V. t t Nominal Energy Nominal Energy Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 3

Longitudinal Acceleration Consider a particle circulating around a ring, which passes through a resonant accelerating structure each turn Harmonic number (integer) h ( ) = + = 2 sin ; V V t E 0 0 Period of nominal energy particle The energy gain that a particle of the nominal energy experiences each turn is given by eV E E sin 0 1 + = + Synchronous phase n n s Where the this phase will be the same for a particle on each turn A particle with a different energy will have a different phase, which will evolve each turn as p rf n n = + = + 1 1 p E rf + E = use n 2 p E 2 p E S Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 4

Phase Stability Thus the change in energy for this particle for this particle will evolve as ( n n eV E E sin 0 1 + = + d ) sin n s So we can write rf = E 2 dn E S d E ( ) = sin sin eV 0 n s dn 0 eV 2 2 d ( ) rf = sin sin n s 2 dn E S d Multiply both sides by and integrate over dn dn d2j dn2 dj dndn =eV0wrfth = -eV0wrfth ESb2 )dj dndn ( sinj(n)-sinjs ESb2 2 dj dn 1 2 ( )+constant cosj +jsinjs 2+2eV0ESb2 wrfth ( ) = constant ( ) DE cosj +jsinjs Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 5

Synchrotron motion and Synchrotron Tune Going back to our original equation + dn eV 2 d ( ) 0 rf = sin sin 0 n s 2 2 E S For small oscillations, sin ( ) = sin cos cos n s s n s s And we have E rf eV 2 d 0 + = cos 0 s 2 2 dn S This is the equation of a harmonic oscillator with 0 0 eV eV 1 rf rf = = cos cos n s s s 2 2 2 E E S S Angular frequency wrt turn (not time) synchrotron tune = number of oscillations per turn (usually <<1) Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 6

Longitudinal Emittance We want to write things in terms of time and energy. We have can write the longitudinal equations of motion as 1 = ( ) ( ) t n n rf ( ) 1 ( ) d t n d n = = ( ) E n 2 dn dn E rf S We can write our general equation of motion for out of time particles as ( dDt dn 1 )+ ( ) Dt(n)= Dt0cos 2pnsn sin 2pnsn 2pns 0 th ( )+ ( ) = Dt0cos 2pnsn DE0sin 2pnsn 2pESb2ns DE(n)=ESb2 dDt(n) dn th )-2pESb2ns ( ( ) = DE0cos 2pnsn Dt0sin 2pnsn th Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 7

So we can write th ( ) ( ) cos 2pnsn sin 2pnsn = Dt0 DE 2pESb2ns Dt(n) DE(n) -2pESb2ns th ( ) ( ) sin 2pnsn cos 2pnsn 0 We see that this is the same form as our equation for longitudinal motion with =0, so we immediately write ( ) ( ) t cos 2 sin 2 ( ) n n t n 0 s L s = ( ) ( ) E sin 2 cos 2 ( ) n n E n L s s 0 Where 0 eV 1 rf = cos s s 2 2 E S 1 = = ; = L L 2 0 2 2 cos E eV E S s rf S s L Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 8

E We can define an invariant of the motion as t Area= L units generally eV-s What about the behavior of t and E separately? Note that for linacs or well-below transition ( ) ( )4 1 1 1 = 3 2 3 2 ; E t 4 2 Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 9

Large Amplitude Oscillations We can express period of off-energy particles as dt = p = n E n 2 dn p E s ( ) d ( ) = sin sin E eV t 0 n rf n s dn d d dn dt So ( ) ( ) = E E d d dt d d d ( ) ( ) E E Use: d d = = dt d dt = 2 L 0 2 rf cos eV E dn dt dn rf S s 2 eV E = 0 sin sin s s E rf 1 = sin sin s 2 2 cos E rf s L Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 10

Continuing sin sin ( ) = s Ed E 2cos rf 2 L s Integrate ( 2 cos + + cos sin cos sin 1 ) 2 = 0 0 s s E 2 rf 2 L s The curve will cross the axis when E=0, which happens at two points defined by sin cos 1 1 + = + cos sin 0 0 s s Phase trajectories are possible up to a maximum value of 0. Consider . ); sin (cos 0 0 = + s s 1 . Limit is at maximum of 1.5 1 or = sin sin 0 0.5 , 0 max s 0 = 0 -6 -1 4 , 0 max s s -0.5 unbound , 0 -1 bound max -1.5 -2 Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 11

Longitudinal Separatrix The other bound of motion can be found by sin cos max , 1 max , 1 + = + + ( sin cos( ) ) sin s s s s = ( cos ) s s s The limiting boundary (separatrix) is defined by 2 sin cos 2 E cos ( ) + + cos sin ( ) = s s s s 2 rf 2 L s The maximum energy of the bucket can be found by setting = s + cos 2 cos 2 2 sin sin ( ) 2 = 2 s s s s E b 2 rf s L 2 1 tan s s = 2 E b rf L Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 12

Bucket Area The bucket area can be found by integrating over the area inside the separatrix (which I won t do) 16 A ( ) ( )= s = ; b f f s rf L Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 13

Transition Crossing T We learned that for a simple FODO lattice so electron machines are always above transition. Proton machines are often designed to accelerate through transition. ( ) ( ) ( ) 0 = 0 0 As we go through transition Recall At transition: eV 1 Dtmax constant DEmax 0 rf = cos s s 2 2 E S t = = max E L 2 cos eV E 0 max rf S s so these both go to zero at transition. To keep motion stable 2 cos below 0 transitio n; 0 s s 2 cos above 0 transitio n; s s Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 14

Effects at Transition As the beam goes through transition, the stable phase must change Problems at transition (pretty thorough treatment in S&E 2.2.3) Beam loss at high dispersion points Emittance growth due to non-linear effects Increased sensitivity to instablities Complicated RF manipulations near transition Much harder before digital electronics Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 15

Accelerating Structures The basic resonant structure is the pillbox = ) t ( , E E r z ) = ( , B B r t Maxwell s Equations Become: 1 1 1 E E ( ) = = z B rB 2 2 c t r r c t Boundary Conditions: B B E = = 0 E B = = z E || t r t Differentiating the first by t and the second by r: 1 rB r r t 2 1 r 1 c B B E ( ) = + = z 2 2 r t t t 2 B E = z 2 r r t 2 2 1 r 1 c E E E + = z z z 2 2 2 r r t Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 16

General solution of the form + = i t ( ) E E r e z Which gives us the equation 0thorder Bessel s Equation 2 1 z z + + = = 0 ( ) E E E E r E J r 0 0 z 2 r c c 0th order Bessel function First zero at J(2.405), so lowest mode c 2 0= . 2 405 f R 0 B E = = = i t i t ( ) z E J r e i B r e 0 r c c t 1 0 1 = = ( ) B r i E J r i E J r 0 0 1 c c c c Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 17

Transit Factor In the lowest pillbox mode, the field is uniform along the length (vp= ), so it will be changing with time as the particle is transiting, thus a very long pillbox would have no net acceleration at all. We calculate a transit factor Assume peak in middle / 2 L z v fL fL L cos 2 eE f dz sin eE sin 0 0 v energy gain sin u f v v = = = / 2 T fL eE L eE L eE L u 0 0 0 v Example: 5 MeV Protons (v~.1c) f=200MHz T=85% u~1 c = 57 = . 2 405 cm R 2 f v = = 9 . 4 cm L u f Sounds kind of short, but is that an issue? Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 18

Power dissipation in RF Cavities Volume=L R2 Energy stored in cavity R 1 1 1 1 2 1 f ( . 2 ) = + = = = 2 2 2 0 2 0 2 0 2 dr 2 405 U E B dV E dV LE rJ r VE J 0 0 max 0 0 1 2 2 2 2 c 0 0 =(.52)2~25% Power loss: Magnetic field at boundary Surface current density J [A/m] l d = = = B LB I LJ . 0 0 enclosed 1 = J B B 0 2 ends Cylinder surface Average power loss per unit area is 1 1 2 2 R = = 2 1 E p J B = RLJ + 2 2 0 c 2 2 2 P R J r rdr s s 2 0 2 1 1 s 2 c c 0 0 2 Average over cycle + 1 E R ( . 2 ) = 2 0 2 1 405 RL J 1 s 2 Z L 0 1 = = = (impedance 0 where 376 73 . of free space) Z c 0 0 c 0 0 Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 19

The figure of merit for cavities is the Q, where ( ( E 2 ) Stored Energy U = 2 Q ) Energy Lost per Cycle P 2 0 2 2 . 2 ( 405 ) R LJ = 0 1 + 1 E R 2 2 . 2 ( 405 ) Z RL J 0 1 Z 1 s = Z L c 0 0 0 2 0 2 0 Z R c = = . 2 405 0 0 + 1 + 1 R R 2 2 Z s s L L = 0 . 2 405 + 1 R 2 s L So Q not very good for short, fat cavities! Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 20

Drift Tube (Alvarez) Cavity Put conducting tubes in a larger pillbox, such that inside the tubes E=0 Bunch of pillboxes Drift tubes contain quadrupoles to keep beam focused Gap spacing changes as velocity increases v d = f Fermilab low energy linac Inside Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 21

Shunt Impedance If we think of a cavity as resistor in an electric circuit, then V V P = 2 2 V = R R P By analogy, we define the shunt impedance for a cavity as ( ) gain voltage P ( ) 2 2 E LT = 0 R s P 2 0 2 Z L T = We want Rs to be as large as possible + R R 2 1 . 2 ( 405 ) J s 1 L Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 22

Other Types of Accelerating Structures cavities E v d = Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 23

Sources of RF Power For frequencies above ~300 MHz, the most common power source is the klystron , which is actually a little accelerator itself Electrons are bunched and accelerated, then their kinetic energy is extracted as microwave power. Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 24

Sources of RF Power (contd) For lower frequencies (<300 MHz), the only sources significant power are triode tubes, which haven t changed much in decades. 53 MHz Power Amplifier for Booster RF cavity FNAL linac 200 MHz Power Amplifier Lecture 8 - Longitudinal Motion 1 USPAS, Knoxville, TN, Jan. 20-31, 2014 25