Resistive wall impedance in elliptical multilayer vacuum chambers

Abstract

The resistive wall impedance of a vacuum chamber with elliptic cross section is of particular interest for circular particle accelerators as well as for undulators in free electron lasers. By using the electric field of a point charge and of a small dipole moving at arbitrary speed in an elliptical vacuum chamber, expressed in terms of Mathieu functions, in this paper we take into account the finite conductivity of the beam pipe walls by means of the surface impedance, and evaluate the longitudinal and transverse driving and detuning impedances for any beam velocity. We also extend the definition of the Yokoya form factors, valid in the thick wall regime, at any beam energy, and show that, in the ultra-relativistic limit, they coincide with the ones that are found in literature. The method is also extended to the multilayer vacuum chamber case. Under conditions generally satisfied with particle accelerator beam pipes, the classical transmission line theory can be used to modelling the impedance seen by a bunch in a vacuum chamber with several layers as an equivalent circuit with the same number of load impedances, giving, as result, a surface impedance that can be used in combination with the fields of the elliptic geometry to obtain the resistive wall impedance in an elliptical multilayer vacuum chamber. The results are also compared with a more time consuming 3D electromagnetic code and with solutions for known cases of circular and flat beam pipe.

Physics‐informed neural network method for modelling beam‐wall interactions

“…The domain and boundary sampling points were generated as in Figure 2b, where the coordinates are scaled with s 0 = R . The surface impedance function on the innermost chamber wall can be given by [12] $$\begin{equation}{Z_s}{\rm{\ }}\left( \omega \right) = {\rm{\ }}j{\eta _{ct}}\tan {k_{ct}}d,{k_{ct}} = \omega \sqrt {{{\hat{\mu }}_{ct}}{{\hat{\varepsilon }}_{ct}}} ,{\rm{\ }}{\eta _{ct}} = \sqrt {\frac{{{{\hat{\mu }}_{ct}}}}{{{{\hat{\varepsilon }}_{ct}}}}} \end{equation}$$ $$\begin{equation}{\hat{\mu }_{ct}} = {\mu _0},{\hat{\varepsilon }_{ct}} = {\varepsilon _0} - j\frac{\sigma }{\omega }\end{equation}$$where μ 0 is the permeability of vacuum. Note that the real part is different from the imaginary part at low frequencies.…”

“…When Z s = 0, Equation ( 6) can be reduced to just the PEC-BC e z = 0. We also assume the perturbative treatment of magnetic field, as discussed in [1] and used in [12,13]. This means that the magnetic field on the resistive wall is the same as that of the PEC wall; the longitudinal component of the magnetic field is zero even for non-ultrarealistic beams.…”

“…Results and discussion: To show the feasibility of the proposed method, we apply it to the analysis of a round vacuum chamber with inner radius R = 25 mm and the first layer with a small conductivity σ = 400 S/m and a thickness d = 5 mm followed by a PEC, as shown in Figure 2a. The domain and boundary sampling points were generated as in Figure 2b, where the coordinates are scaled with s 0 = R. The surface impedance function on the innermost chamber wall can be given by [12] Z s (ω) = jη ct tan k ct d, k ct = ω μct εct , η ct = μct εct ( 14)…”

Review of impedance-induced instabilities and their possible mitigation techniques

“…For Eq. ( 2) we have supposed a cylindrically symmetric structure and the speed of light, otherwise also another term, called quadrupolar wake field, would have been necessary [17][18][19][20].…”

Electromagnetic field computation of multilayer vacuum chambers with physics-informed neural networks

“…The resistive wall impedance of multilayer vacuum chamber is of particular interest for modern high energy accelerators and x-ray free electron laser projects. Many efforts have been made to analytically investigate the wakefields and impedance in multilayer vacuum chambers, see e.g., [17][18][19][20][21][22][23]. However, it is still challenging to calculate the impedance of multilayer vacuum chambers with arbitrary cross section.…”