Transverse impedance of a periodic array of cavities

A. V. Fedotov; R. L. Gluckstern; M. Venturini

doi:10.1103/PhysRevSTAB.2.064401

Open Access

Transverse impedance of a periodic array of cavities

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

Physics Department, University of Maryland, College Park, Maryland 20742

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Phys. Rev. ST Accel. Beams 2, 064401 – Published 21 June, 1999

DOI: https://doi.org/10.1103/PhysRevSTAB.2.064401

Abstract

We examine the transverse impedance of a periodic array of cavities in a beam pipe at high frequency. The calculation is an extension of a previous one for the longitudinal impedance of a periodic array of azimuthally symmetric pillboxes, for which only TM modes were needed. In the present case, we must include TE modes as well. In addition, we extend the applicability of the previous calculation by including an extra term in the coupling kernel so that the results are valid for all values of the ratio of the cavity length to the period of the structure (all values of the ratio of iris thickness to structure period). In spite of the presence of TE modes, we find that the high frequency limit of the transverse impedance is simply $(2 / {ka}^{2})$ times the corresponding limit of the longitudinal impedance, just as it is for the resistive wall impedances, a relation which occurs frequently for azimuthally symmetric structures. Finally, we present numerical results as well as approximate expressions for the impedance per period, valid for all ratios of cavity length to structure period.

References (20)

R. L. Gluckstern, Phys. Rev. D 39, 2773 (1989).
R. L. Gluckstern, Phys. Rev. D 39, 2780 (1989).
This calculation is similar to an earlier one for the high frequency smoothed longitudinal impedance [[1]].
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 $r = b$ .
At this point we note that, had we used $e_{θ} (z^{'})$ rather than ${de}_{θ} (z^{'}) {/ d z}^{'}$ 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 ${de}_{θ} (z^{'}) {/ d z}^{'}$ .
We note at this point that the behavior of $e_{z} (z^{'})$ near $z^{'} = 0$ is expected to be $({kz}^{'})^{- 1 / 3}$ for ${kz}^{'} ≪ 1$ ( $z^{'}$ small compared with the wavelength $2 π / k$ ). 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 ${kz}^{'} \sim kg ≫ 1$ , 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.
A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley, New York, 1993).
K. Bane and M. Sands, Part. Accel. 25, 73 (1990).
$p_{s}$ and $p_{t}$ are given a small positive imaginary part to ensure the convergence of the sums in Eq. 5.12.
K. Yokoya (private communication); also K. L. F. Bane and K. Yokoya, in Proceedings of the 1999 Particle Accelerator Conference (PAC99), New York (to be published).
G. Stupakov, in Proceedings of the 1995 Particle Accelerator Conference, Dallas, Texas (IEEE, Piscataway, NJ, 1996), p. 3303.
K. L. F. Bane, A. Mosnier, A. Novokhatskii, and K. Yokoya, in Proceedings of the European Particle Accelerator Conference, Stockholm, Sweden, 1998 (Institute of Physics, Bristol, UK, 1998).
L. A. Vainshtein, Theory of Diffraction (Golden Press, Boulder, CO, 1969).
B. Zotter and S. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998).
R. L. Gluckstern, in Proceedings of the 1989 Particle Accelerator Conference, Chicago, IL (IEEE, Piscataway, NJ, 1989), p. 1157.
S. S. Kurennoy and S. V. Purtov, Part. Accel. 34, 155 (1990).
S. S. Kurennoy, Part. Accel. 39, 1 (1992).
V. Balbekov, IHEP Report No. 93-55, 1993.
S. S. Kurennoy (private communication).
Note that we obtain $\mp j$ , 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.

[c1] R. L. Gluckstern, Phys. Rev. D 39, 2773 (1989).

[c2] R. L. Gluckstern, Phys. Rev. D 39, 2780 (1989).

[c3] This calculation is similar to an earlier one for the high frequency smoothed longitudinal impedance [[1]].

[c4] 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 $r = b$ .

[c5] At this point we note that, had we used $e_{θ} (z^{'})$ rather than ${de}_{θ} (z^{'}) {/ d z}^{'}$ 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 ${de}_{θ} (z^{'}) {/ d z}^{'}$ .

[c6] We note at this point that the behavior of $e_{z} (z^{'})$ near $z^{'} = 0$ is expected to be $({kz}^{'})^{- 1 / 3}$ for ${kz}^{'} ≪ 1$ ( $z^{'}$ small compared with the wavelength $2 π / k$ ). 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 ${kz}^{'} \sim kg ≫ 1$ , 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.

[c7] A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley, New York, 1993).

[c8] K. Bane and M. Sands, Part. Accel. 25, 73 (1990).

[c9] $p_{s}$ and $p_{t}$ are given a small positive imaginary part to ensure the convergence of the sums in Eq. 5.12.

[c10] K. Yokoya (private communication); also K. L. F. Bane and K. Yokoya, in Proceedings of the 1999 Particle Accelerator Conference (PAC99), New York (to be published).

[c11] G. Stupakov, in Proceedings of the 1995 Particle Accelerator Conference, Dallas, Texas (IEEE, Piscataway, NJ, 1996), p. 3303.

[c12] K. L. F. Bane, A. Mosnier, A. Novokhatskii, and K. Yokoya, in Proceedings of the European Particle Accelerator Conference, Stockholm, Sweden, 1998 (Institute of Physics, Bristol, UK, 1998).

[c13] L. A. Vainshtein, Theory of Diffraction (Golden Press, Boulder, CO, 1969).

[c14] B. Zotter and S. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998).

[c15] R. L. Gluckstern, in Proceedings of the 1989 Particle Accelerator Conference, Chicago, IL (IEEE, Piscataway, NJ, 1989), p. 1157.

[c16] S. S. Kurennoy and S. V. Purtov, Part. Accel. 34, 155 (1990).

[c17] S. S. Kurennoy, Part. Accel. 39, 1 (1992).

[c18] V. Balbekov, IHEP Report No. 93-55, 1993.

[c19] S. S. Kurennoy (private communication).

[c20] Note that we obtain $\mp j$ , 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.

Physical Review Accelerators and Beams

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