Modeling of second order space charge driven coherent sum and difference instabilities

Abstract

Second order coherent oscillation modes in intense particle beams play an important role for beam stability in linear or circular accelerators. In addition to the well-known second order even envelope modes and their instability, coupled even envelope modes and odd (skew) modes have recently been shown in [Phys. Plasmas 23, 090705 (2016)] to lead to parametric instabilities in periodic focusing lattices with sufficiently different tunes. While this work was partly using the usual envelope equations, partly also particle-in-cell (PIC) simulation, we revisit these modes here and show that the complete set of second order even and odd mode phenomena can be obtained in a unifying approach by using a single set of linearized rms moment equations based on "Chernin's equations." This has the advantage that accurate information on growth rates can be obtained and gathered in a "tune diagram." In periodic focusing we retrieve the parametric sum instabilities of coupled even and of odd modes. The stop bands obtained from these equations are compared with results from PIC simulations for waterbag beams and found to show very good agreement. The "tilting instability" obtained in constant focusing confirms the equivalence of this method with the linearized Vlasov-Poisson system evaluated in second order.

Structure resonance crossing in space charge dominated beams

“…Recently, the sum and difference instabilities are studied in Ref. [17] with Chernin's model [18], which gives exactly the same result as that from the 2nd order odd structure resonance. From the experiment point of view, great efforts have been put into the spectrum measurement of the I 0;2,0 and I 2;0,2 in GSI [31] and CERN [32].…”

“…As to the study of collective instability due to space charge effect with ion beams, one branch starts from the rms envelope equations [12], with which the 2nd order (envelope instability) structure resonance is well studied [13][14][15][16][17][18], while another branch is to solve the Vlasov-Poisson equations self-consistently [19][20][21][22].…”

Three-dimensional envelope instability in periodic focusing channels

“…It has been studied theoretically [1][2][3][4][5][6][7][8] and experimentally [9][10][11] since the 1980s. In recent years, there was growing interest in further understanding this instability and other structural resonances [12][13][14][15][16][17][18][19][20][21][22]. Some of those studies were summarized in a recently published monograph [23].…”