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crossing space charge resonances

Delve into the complexities of space charge resonances in beam dynamics, exploring octupolar and decapolar forces in LINAC and RING accelerators. Learn about scaling laws, trapping, scattering, and resonance crossings, supported by simulations and analytic models. Discover insights on emittance growth and adiabatic behavior in beam structures.

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crossing space charge resonances

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  1. crossing space charge resonances I. Hofmann

  2. space charge and resonances in high intensity beams space charge nonlinearity can itself be a source of resonance in LINAC and RING forces considered: “octupolar” and “decapolar” 4th order resonance (Montague) 6th order resonance transverse/longitudinal emittance exchange 2kz - 2kx ~ 0 4th order x/y emittance exchange 2Qx – 2Qy ~ 0 when crossing the resonance beam emittances exchange totally or partially according to the speed of crossing.

  3. Does exists a scaling law ? Scaling laws relate one measurable quantity to another one Scaling law are simple if proper variables are used Scaling variable for crossing Montague resonance S= (Qx2) / d(Qx0) / n which yields  /  S Validity tested in linacs with PARMILA and in rings with MICROMAP

  4. example: 4Q=12 4th order nonlinear resonances Crossing this resonance produces trapping into resonances if the resonance is crossed from above or scattering if crossed from below. Above Below trapping scattering Scaling law for crossing the 6th order resonance (relevant for non-scaling FFAG ) trapping scattering

  5. The emittance growth scaling laws in resonance crossing X. Pang

  6. Scaling law relevant for FFAG Space charge modeled with a new analytic formulation expanded in series for non round beams Advantage: space charge analytic model allows to compute the resonance strength Gnm Simulations show the formation of 6 islands when crossing the 6th order structure resonance Scaling law for crossing resonance (ansatz): EGF = (-d/dn)a Simulations show that there exists a critical tune ramp rate (d/dn)c such that for ramp rate smaller EGF ~ 1. By imposing that (resonance strength) x (turns inside stop-band) is constant it is found (d/dn)c  ()2. Authors claim that this law is valid for 4th and 6th order resonance

  7. Ingo’s comment in the discussion

  8. (Ingo’s comment is a little dogmatic) • S is adiabaticity parameter • S large: adiabatic behavior with slow tune change in 2D coasting beam limit • 3D: S < Ssynch ~ Q2/(Q/nsynch) = Q nsynch • Example: Q = -0.2 nsynch = 20  Ssynch ~ 4  loss of adiabatic condition for S > 4! • no full exchange for Montague (experimental evidence at CERN PS!) • no trapping with 4th or 6th order structure resonance • probably no trappping in linac (S^2 or S^3 power law suppressed)

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