The Short-Range Resistive Wall Wakefields

Abstract

Abstract. In an accelerator when the bunch length becomes comparable to a characteristic distance S O , one which depends on the radius and the conductivity of the beam tube and in typical structures is on the order of tens of microns, the usual formulas for the resistive wall wakefield do not apply. In this report we derive the short-range resistive wall wakefields of an ultra-relativistic point particle in a metallic, cylindrical tube, both for a model in which the wall conductivity is taken to be independent of frequency and for one in which a frequency dependence is included. On this scale the wakefield is found to be dominated by a damped, high frequency resonator component. For the caSe of constant conductivity the resonant frequency is given by w = &c/so and the &-factor equals &/2. We provide a physical model to explain these results. For the cme of a frequency dependent conductivity the resonator parameters depend also on the relaxation time of the metal T . For CT/SO 2 0.5 the frequency w FZI d w , with w, the plasma frequency of the free electrons in the metal and b the tube we are interested in the wakefield of a short bunch, by which we mean that the bunch length is comparable to the characteristic distance (see, e.g. Ref. 2) with c the speed of light, b the lube radius, and u the conductivity of the metallic walls. For copper at room temperature* u = 5.8 x 1017 s-', For a copper tube with b = 1 cm, SO = 20pm. Although in accelerators the bunch length is normally much larger than this value the above LCLS example falls within the short bunch category if the tube radius is comparable to 1 cm. Note also that, €or a given bunch length, as we increase b we will also increase so. For the above example, if we increase b to 10 cm then so becomes 93pm.The conductivity of a metal is not independent of frequency. In the presence of oscillating fields Ohm's law becomes J = C E with (3) with w the frequency of oscillation and r the relaxation time of the metal. We note that it is only at higher frequencies that 6 (which we will call the radius, and the l / e damping time becomes 4r. Finally, we calculate the wakefield and loss factor of a short Gaussian bunch. ac conductivity) differs significantly from u (the dc conductivity). For copper at room ternperatixe the re!axatinE time r = 2.7 x s ~r c7 = e..~pm. [Note that for Ag, T = 4.0 x s. Note also that 'UT = e, with 6 the average velocity of the free electrons in the metal and the mean free path between collisions.] Let us introduce the dimensionless relaxation factor I' = cr/so. For a 1 cm copper tube I' = 0.4. Therefore, for this example, for frequencies w -c/so, the second term in the denominator of Eq. (2), in amplitude, becomes comparable to the first term; we might therefore expect the short range iic wakefield to be noticeably different from the short range dc wake.Down to what length scale do we expect that our calculations are valid? An important assumption in all our calculations is that Ohm's law applies in the metallic walls. One effect, the anom...

Resistive wall wakefields in the extreme anomalous skin effect regime

“…It was made so that s 0 is exactly equal to the efolding distance for the resonator term. This definition is consistent with that of Bane and Sands [32], who introduced the characteristic distance for the NSE regime,…”

“…The integration is performed using standard techniques of complex calculus, see e.g. [32] for a similar derivation for the case of NSE resistive wall wake. Note that Z k ð " kÞ has no singularities in the upper half plane, guaranteeing the causality of the wake function, W k ðz > 0Þ ¼ 0.…”

“…The first term is a damped broadband resonator due to pole residues. The second contribution, which, by analogy with [32], we will call the diffusion term, comes from the contributions along the branch cut. The final expression is…”

Wakefield Calculations for Radiation Stopper 1 (RST1)

“…If, in addition, the cavity radius is sufficiently large (b > ∼ a + √ 2gσ z ; b radius and g length of cavity), then the diffraction model of cavity wakes applies. For a Gaussian bunch the longitudinal wakefield is given by [3] what was obtained before without the absorber (the lower-half contribution) and twice the effect for a cavity with gap g p = 22 mm (the upper-half contribution). The wake becomes…”

“…1, with the stopper in its nominal position. In Appendix A we develop a modification of the diffraction model of cavity wakes [3] that allows us to obtain an analytical estimate that we compare with the MAFIA result. …”