Transverse resistive-wall impedance from very low to very high frequencies

The transverse resistive-wall impedance of a circular beam pipe derived from field matching in Ref. [1] is extended here, following exactly the same approach, but(i) without making the “low-frequency” approximation k b / γ << 1 (where k is the wave number, b the beam pipe radius and γ the beam relativistic mass factor), and(ii) without assuming (necessarily) a good conductor for the first layer. The formulae obtained for a vacuum chamber consisting of one or two layers, which were used in Ref. [2] to compute the impedance of the LHC graphite collimators, are given explicitly. It is found that Zotter’s formalism unifies the approaches of Burov-Lebedev [3] for “low frequencies” and Bane [4] for high frequencies, and that it is also valid for any beam velocity.