Circulating Beams Søren Pape Møller ISA / DANFYSIK A/S Chapter 4 i Wilson - 1 hour

Transkript

1 Circulating Beams Søren Pape Møller ISA / DANFYSIK A/S Chapter 4 i Wilson - 1 hour

2 Particles in space En partikel har to transversale koordinater og en longitudinal og tilsvarende hastigheder. Ofte er bevægelsen i de tre planer uafhængig og man kan beskrive partiklerne i de tre planer uafhængigt af hinanden. I hver af de transversale planer er der altså en koordinat x (y og en tilsvarende hastighed v x (v y. I hver af de to transversale planer kan hastigheden til en meget god approksimation repræsenteres af en vinkel x (y - v x = x v x.

3 Phase space emittance A beam of N particles correspond to N points in 6-dim phase space, when internal forces << external forces 6-dim 3 2-dim Harmonic forces ellipses in phase space y Unit for phase space [position] [momentum] y Unit for emittance π mm mrad (upright ellipse at symmetry point

4 Phase space and Liouville Liouville: For hamiltonian systems, the phase space density is constant (when measured along a trajectory The phase space volume (emittance is conserved Often the two transverse and the longitudinal degrees of freedom are decoupled p x pdq = constant q x Quadrupole focusing

5 Kort simpel udgave af Liouvilles teorem Liouvilles teorem: For et system af partikler der kun påvirkes af positionsafhængige kræfter er faserumstætheden bevaret. Dette gælder f.eks. for elektriske og magnetiske kræfter, men ikke for f.eks. gnidningskræfter der afhænger af hastighed. Det kommer vi til at beskæftige os mere med i forelæsningen om køling.

6 Coffee, cream, Liouville and Stochastic cooling

7 Example of emittances A typical ionsource in this house, the Nielsen source has an emittance of around 5-10 π mm mrad at 100 kev The electron beam in ASTRID at 580 MeV has a horisontal emitance of 0.14 mm mrad=140 nm and a vertical emittance of around 10 nm The normalised emittance of the electron beam in the XFEL in Hamburg is around 1 mmmrad Since the energy is 25 GeV, γ=25gev/511kev=50000 and the emittance will be around 20 pm

8 Phase space ellipse in beamline

9 Phase space ellipse 3 kapitel twissparametre fra,, ( ( 2 ( ( cos( ( γ β α ε β α γ φ φ β ε = = y s yy s y s s s x

10 Phase space emittance y y m c vy m ds dy dt ds m dt dy m v m c m p x y q y y y = = = = = ( id for ; βγ γ γ γ γ γ β γ y y

11 Adiabatic damping Hamiltonian mechanics canocical coordinates ε pdq m c( βγ q dq Emittance = πε = m y p = & 0, q = y v / c 0 = 0 = constant ( βγ ε invariant or normalised emittance y dy = πε βγ 1 p

12 Beam distributions in real space Emittance defined to ~90% of particles Gaussian distribution with truncated tails Parabolic distribution ε protons = σ 2 /β Emittance defined as rms value ε electrons = σ 2 /β

13 Acceptance How large an emittance can fit inside a given vacuum chamber? A( s = r( s A = 2 min[ r( s / β ( s 2 / β ( s]

14 Electron beam size emittance

15 Slit-grid emittance measurement

16 Målt med slit-grid arrangement Emittance

17 Pepperpot measurement

18 Measurements of emittance Ionisation monitor

19 Adiabatic damping using rest gas ionisation detection

20 Mountain range display Observation of adiabatic damping

21 Betatron oscillations from kicked beam Δ Pick-up kicker

22 Tune measurement by kicking

23 Tune measurement by kicking functions at each turn the rest is Q part of fractional ( slowest wave ] ( sin(2 ( [sin(2 ½ ( ( cos(2 ( betatron oscillations sin(2 ( ( bunch δ π π ρ π π δ ρ Q n t f Q n t f Q n y a t y t Qt f y t y t nf a t t n n + + = = = =

24 Knock-out tune measurements Δ Pick-up kicker ~

25 Knockout tune measurement

26 ELISA signal [μv] 50 Q H =1.39 V H Q V = frequency [khz]

27 SLUT

28

29 Tune measurement by Schottky

30 Phase space, Liouville and cooling Liouvilles theorem means that cooling is not possible for Hamiltonian systems, that is systems with forces that can be derived from potentials. In addition particles cannot be injected into already filled areas of Phase space. All you can do is to change the form of phase space. However, with velocity-dependent forces drag, friction (dissipative forces electron, radiation, Laser, ionisation cooling cooling is indeed possible!!

31 Phase space and Liouville Liouville sætning gælder for et Hamiltonsk system for såkaldt konjugerede koordinater som x og p x Et hamiltonsk system er et konservativt system, dvs. kræfterne er alene afhængige af position og ikke hastighed og dermed kan udledes fra et potentiale En partikel beskrives ved q hvor L( q, q&, t T U Hamiltonfunktion H = H ( q, p, t Hamiltons ligninger i og p = L / q p& = H / q + Q i er Lagrangefunktionen p q& i i i L q& = H / p