By Dieter Möhl
This lecture notice describes the most analytical methods to stochastic cooling. the 1st is the time area photo, during which the beam is swiftly sampled and a statistical research is used to explain the cooling behaviour. the second one is the frequency area photo, that is rather priceless because the observations made at the beam are more often than not during this area. This moment photo is built intimately to evaluate key parts of contemporary cooling idea like blending and sign protecting and to demonstrate a few of the diagnostic equipment. ultimately using a distribution functionality and the Fokker-Plank equation, which supply the main whole description of the beam through the cooling, are discussed.
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Extra resources for Stochastic Cooling of Particle Beams
Example text
Above we tacitly implied that at the kicker the dispersion function DK as well as its derivative, DK are zero. Otherwise the momentum correction leads to an excitation of betatron oscillations. The reason is that the momentum kick introduces an 26 2 Simplified Theory, Time-Domain Picture abrupt change of the equilibrium orbit and the particle starts to oscillate around this new displaced orbit. This is an example of ‘mutual heating’, a problem common to most cooling systems. The more general case where both xp and xβ are present at the pickup and where D is non-zero at the kicker was analysed by Hereward [12].
4). To work out the Schottky noise we assume, once more, a coasting beam with random longitudinal particle positions 0 ≤ ωrev tp ≤ 2π , random ‘betatron phases’ 0 ≤ μ0 ≤ 2π , ˜ of besingle out a group of N = dN/dω particles and assume a distribution n (x) tatron amplitudes. We may interpret N (ω) as the distribution of particles versus the sideband frequencies ωm = (m ± Q)ωrev . 1 Schottky Noise Signals from a Coasting Beam 53 moment D 2 = (I · x)2 in the bands. s. amplitude comes from averaging x˜ 2 and a factor 1/2 from averaging the cos2 terms, Eq.
E. = 12 C/β), such that signals travel to the end of the line in half of the nominal revolution time. The line is shorted at the far end (“stub line”); its open end is connected to the cooling path so that—for the synchronous particle (δp = 0)—the reflected signal from the previous turn annihilates the direct signal. For particles with finite p the cancellation is only partial and acceleration or deceleration occurs. Modern cooling filters (proposed by G. Carron) use a correlator as sketched in Fig.