Robinson instability and beam loading

Abstract

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Modes of storage ring coherent instabilities

“…Equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) represents the solution of Maxwell's equation, and it is one of the foundations of all the following discussion in Part I of these lectures.…”

“…(3-17) n -n Let us now separate Z n (to) into the real part and the imaginary part, Z (oi) =m (w) + iX (u)) . (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) n n n The real part ^^(w) is usually called the resistive part and the imaginary part X n ((JJ) ? the reactive part.…”

“…Let us take the current of the form ,t) = I n (u)e in9 -iUt + i^uOe-" 9 ** 1 *' . (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) We include two terms here to ensure that the current is real. The electric field induced by this current is Si (u) > 0 .…”

“…The longitudinal electric field g responsible for the longitudinal coherent instability is induced by the perturbation of the beam current which is in turn related to the beam density modulation [see Eqs. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and ]. We establish here the relationship, to first order in the particle displacement, between the line density modulation and the displacement of the particles from their respective nominal positions in the beam.…”

“…In summary, we have discovered that the linear eigenvalue problem represented by Eqs. (4)(5)(6)(7)(8)(9) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) can be reduced to the problem of solving the dispersion relation (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). We also found that the different eigensolutions are parameterized by the revolution mode number n, and that the eigenfunction is given by with the coherent frequency & satisfying the dispersion relation (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).…”