Dietrich, J.

Paper	Title	Page
TUA1C03	Necessary Condition for Beam Ordering	87
	A. V. Smirnov, I. N. Meshkov, A. O. Sidorin JINR, Dubna, Moscow Region J. Dietrich FZJ, Jülich A. Noda, T. Shirai, H. Souda, H. Tongu Kyoto ICR, Uji, Kyoto K. Noda NIRS, Chiba-shi
	The very low momentum spread for small number of particle was reached on different storage rings. When the sudden reduction of the momentum spread ("phase transition") was observed during decreasing of the particle number it was interpreted as ordered state of ion beams. The most extensive study of ordered ion beams was done on storage rings ESR (GSI, Darmstadt) and CRYRING (MSL, Stockholm). Recently, for the first time, the ordered proton beam has been observed on S-LSR (Kyoto University). From analysis of the ESR experimental results we assumed that the ordered state can be observed if the dependence of momentum spread on the particle number can be approximated as ∆P/P ~ Nk for k < 0.3. In pioneering experiments at NAP-M (INP, Novosibirsk) and, in recent years, at COSY (FZJ, Juelich) the phase transition was not observed and the coefficient was found equal k > 0.5. This report presents the experimental investigations of low intensity proton beams on COSY and S-LSR which have the aim to formulate the necessary conditions for the achievement of the ordered state. The experimental studies on S-LSR and numerical simulations with the BETACOOL code were done for the dependence of the momentum spread and transverse emittances on particle number with different misalignments of the magnetic field at the cooler section. As result of both experimental and numerical studies one can conclude that the necessary condition for the phase transition appearance is k < 0.3.
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