APS » Journals » Phys. Rev. ST Accel. Beams » Volume 2 » Issue 6

Phys. Rev. ST Accel. Beams 2, 064401 (1999)

Transverse impedance of a periodic array of cavities

A. V. Fedotov, R. L. Gluckstern, and M. Venturini

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  3. This calculation is similar to an earlier one for the high frequency smoothed longitudinal impedance [[1]].
  4. We shall avoid an ambiguity which later occurs by allowing λ and μ to depart slightly from Eq. 3.4 to take into account the effect of a high but finite conductivity in the wall at r=b.
  5. At this point we note that, had we used eθ(z′) rather than deθ(z′)/dz′ in Eq. 4.9, the corresponding kernel would have been (z-z′)-3/2, thus making that integral divergent. This has been avoided by using deθ(z′)/dz′.
  6. We note at this point that the behavior of ez(z′) near z′=0 is expected to be (kz′)-1/3 for kz′≪1 (z′ small compared with the wavelength 2π/k). Thus (kz′)1/2 in the denominator of Eq. 4.11 should be replaced by (kz′)1/2+const  (kz′)1/3, where const is O(1). Nevertheless, the dominant region for the integration over z′ in Eq. 4.15 is for kz′∼kg≫1, and thus (kz′)1/2≫(kz′)1/3. It is for this reason that Eq. 4.11 does not accurately reflect the behavior for values of kz′≪1, but this has no effect on the final result in Eq. 4.15.
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  9. ps and pt are given a small positive imaginary part to ensure the convergence of the sums in Eq. 5.12.
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  19. S. S. Kurennoy (private communication).
  20. Note that we obtain ∓j, depending on whether 1-ε lies inside or outside the unit circle. It is for this reason that we used the losses at r=b to determine the appropriate sign in Eq. C16.