Resistive-wall impedances of a thin non-evaporable getter coating on a conductive chamber

...................................................................................................................Theresistive-wallimpedancesofathinnon-evaporablegetter(NEG)withconstantconductivity coating on a copper chamber are studied in both longitudinal and transverse directions. The copper chamber has a ﬁnite thickness and is surrounded by air.As the frequency increases, wake ﬁelds see mostly the air ﬁrst and then the copper chamber next, and then ﬁnally the NEG coating. Both longitudinal and transverse impedances slowly undergo transitions from the resistive-wall impedances of the copper-only chamber to those of the NEG-only chamber over a 0.1–100 GHz frequency range. They start to deviate from the conventional impedance lines for the NEG-only chamber at ∼ 100 GHz. They increase ﬁrst to ∼ 1THz and then decrease rapidly as a function of the frequency. Numerical examples of the resistive-wall impedances are presented up to ∼ 100THz using similar parameters to those of the SLS-II (Swiss Light Source) upgrade. We brieﬂy discuss the characteristic of a loss factor for the chamber with NEG coating.

In this paper we study the longitudinal and transverse impedances of a copper chamber coated with a non-evaporable getter (NEG) and the effects of the NEG thickness over a wide range of frequencies from very low frequency (∼Hz) to very high frequency (∼100 THz). Here, we assume for simplicity that all physical parameters such as the conductivity have no frequency dependence. For example, the relaxation time of the conductivity is ignored. For more accurate calculations, the frequency dependencies of the physical parameters would need to be measured by experiment. However, this is outside the scope of this paper.
In most parts of this paper we deal with the resistive wall impedance of a thin coating on a conductive chamber with finite conductivity and thickness. A thin NEG coating on vacuum chambers has been successfully used to achieve ultra-high vacuum in many accelerators, such as CERN LHC [19], ESRF [20], etc. Some numerical examples are given in this paper using beam and machine parameters similar to those of the SLS-II (Swiss Light Source) upgrade [17,18]. We hope that the analysis will provide some clues on how to optimize the thickness of the NEG coating for similar electron machines.
In Sect. 2, we first try to simulate the resistive-wall impedance of a copper-only chamber, and discuss the limits and reliability of the method. In Sect. 3, theoretical approaches are applied to the case of a thin coating on the copper chamber up to THz frequencies, where the impedance has a peak in the THz range. In Sect. 4, we present simple formulae for the longitudinal and transverse impedances for the case of a thin coating on a conductive chamber, where the contribution from the induction term of the Maxwell-Ampère equation is taken into account. In Sect. 5, we briefly discuss the feature of a loss factor for the resistive-wall impedance with a NEG coating. The paper concludes in Sect. 6.
The formulae for the longitudinal and transverse impedances of a thin coating on a chamber with a finite conductivity and thickness are derived in Appendix A using the field-matching technique. The longitudinal impedance is approximated in Appendix B for an extremely thin NEG case. General formulae for longitudinal and transverse impedances for a perfectly conductive chamber (space charge impedances [21]) are derived in Appendix C.

Simulation approach to obtain resistive-wall impedances
When the conductivity of a chamber is much higher than 0 ω, where 0 is the dielectric constant of vacuum and ω is the angular frequency, the surface impedance technique [22], where the induction term of the Maxwell-Ampère equation is neglected, is applied to most simulations to estimate resistive-wall impedances. The three-dimensional simulation code CST STUDIO [23] has implemented a material with high conductivity by introducing "Lossy Metal." The Wake Solver of the code may estimate the impedances by letting a beam pass through the chamber. Figure 1 shows the simulation results for a chamber with inner radius (d = 10 mm and outer radius a = 11 mm whose conductivity is σ 3c = 5.9 × 10 7 S m −1 . The red and the blue lines show the real and the imaginary parts of the impedance. The results demonstrate that the method is not appropriate to estimate the resistive wall impedances due to the oscillation of the real part of the impedance,  because it must not be negative from a physical point of view (a beam must lose its energy passing through the chamber).
From an experimental point of view, one way to measure the resistive-wall impedance is to stretch wires inside the chamber. Thus, it may be possible to obtain the resistive impedances by simulating this method [24]. Off-line measurements of the longitudinal and transverse impedances of a device under test (DUT) are typically done by stretching single or twin wires inside the DUT (see Fig. 2). By connecting both ends of the DUT to a network analyzer, the measured S-parameters are converted into impedances by using the standard log formulae [24]: where is the interval between the twin wires, Z cc and Z dd are the characteristic impedances for the common and the differential modes, respectively, S DUT 21 , S DUT dd21 , and S (ref ) dd21 are the transmission coefficients for the resistive chamber (DUT) and those for the perfectly conductive chamber, respectively. The subscripts cc and dd denote the common and differential modes, respectively.
Here, let us calculate the impedances by simulating the measurement setup. The Micro-Wave Solver in CST STUDIO [23] calculates S-parameters for the coaxial structure composed of the wires and the resistive chamber by adopting "Lossy Metal" as the ingredient of the chamber. The Micro-Wave Solver utilizes the technique of the surface impedance [22] to obtain the S-parameters, even if a resistive material with high conductivity is present in the simulation setup. Figures 3 and 4 show the simulation results for a chamber with inner radius d = 10 mm and outer radius a = 11 mm. In these simulations, we use a copper (whose conductivity is σ 3c = 5.9 × 10 7 S m −1 ) chamber as the restive material. The black lines show the longitudinal and transverse impedances obtained by using Eqs. (1) and (2), respectively. The brown lines denote the theoretical impedances using the conventional formulae [1] given by where j is the imaginary unit, Z 0 = 120π is the impedance of free space, C is the chamber length, c is the velocity of light, and σ 3c = 5.9 × 10 7 S m −1 , for reference. The results demonstrate that the simulation technique by using the Micro-Wave Solver reproduces reasonably well the conventional formula of the resistive-wall impedance, except below around 1 GHz. The upper limit of the frequency of the impedance evaluated by the wire method is determined by the cut-off frequency of the waveguide structure, which is about 10 GHz in this setup. The lower limit of the frequency is given by δ (a − d), which is roughly about f 4 kHz, at which the skin depth (δ) exceeds the thickness of the chamber (a − d). Theoretically, this is a limit of the application of the surface impedance theory. But, the simulation results shown in Figs. 3 and 4 seem to deviate from the conventional formulae (3) and (4) at less than around 1 GHz, which is much higher than the frequency (∼kHz).

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Downloaded from https://academic.oup.com/ptep/article-abstract/2017/12/123G01/4746618 by guest on 30 July 2018 3. Overall behavior of the resistive wall impedance of a thin NEG coating on a conductive chamber The formulae for the longitudinal and transverse impedances of a thin coating with a finite conductivity (σ 2c ) and thickness (d − b) on a chamber with a finite conductivity (σ 3c ) and thickness (a − d) are derived in Appendix A using the field-matching technique. In this section, we present some numerical examples using parameters similar to those of the vacuum chambers of the SLS-II upgrade plan [17,18]: d = 10 mm, σ 3c = 5.952×10 7 S m −1 for the copper chamber, and σ 2c = 1.098×10 6 S m −1 for the NEG coating [14]. From now on, we deal only with a relativistic beam, for simplicity. It is noticeable that all the physical parameters such as the conductivity are assumed to have no frequency dependence.
The black lines in Fig. 5 show the exact solutions of the real parts of the longitudinal (left) and transverse (right) impedances per unit length for a copper thickness of a − d = 1 mm and NEG thickness of d − b = 2 μm, respectively. The brown lines in both figures show the real part of the resistive-wall impedance of the copper-only chamber using the conventional formulae given by Eqs. (3) and (4). On the other hand, the purple lines show the real part of the resistive-wall impedance of the infinitely thick NEG coating by replacing σ 3c with σ 2c in the formulae.
We can see clearly that both the longitudinal and transverse impedances slowly undergo transitions from the resistive-wall impedances of the copper-only chamber to those of the NEG-only chamber over a 1-100 GHz frequency range for a NEG thickness of d − b = 2 μm. Both longitudinal and transverse impedances start to deviate from the conventional impedance lines for the NEG-only chamber at ∼100 GHz. They increase first to ∼1 THz and then damp rapidly as a function of the frequency. Now let us investigate the effects of the thickness of both the copper chamber and the NEG coating. We use the outer radius of the copper chamber a (that gives the thickness of the copper chamber) and the inner radius of the NEG coating b (that gives the thickness of the NEG coating) as free parameters. Figure 6 shows the real parts of the longitudinal impedances for a copper thickness of a−b = 1 mm (left) and a − b = 2 mm (right). The green, black, blue, and red lines show the real parts of the  longitudinal impedances when the thickness of the NEG coating is 1 μm, 2 μm, 5 μm, and 10 μm, respectively. The slow transitions from the resistive-wall impedances of the copper-only chamber to those of the NEG-only chamber can be found over a 0.1-100 GHz range in both left and right figures. The only difference between the two figures appears at very low frequencies (less than a few kHz) where the wake fields leak out of the copper chamber to the outside air. This difference may not be too important in ordinal light source rings where the revolution frequency is of the order of MHz.
In the high-frequency region where the skin depth is much smaller than the NEG thickness, all the lines converge to the same result. The frequencies where the skin depth becomes comparable to the thickness of the NEG coating are 230 GHz for 1 μm, 57.6 GHz for 2 μm, 9.2 GHz for 5 μm, and 2.3 GHz for 10 μm, respectively. We can see in the figures that the transitions of impedances from the copper-only case to the NEG-only case are almost completed at around those frequencies. Figure 7 shows the real parts of the transverse impedances for a copper thickness of a − b = 1 mm (left) and a − b = 2 mm (right), respectively. The green, black, blue, and red lines show the real parts of the transverse impedances when the thickness of the NEG coating is 1 μm, 2 μm, 5 μm, and 10 μm, respectively. The brown and purple lines show the impedances of the copper-only chamber and the NEG-only chamber using the conventional formula (4), respectively. The slow transitions from the resistive-wall impedances of the copper-only chamber to those of the NEG-only chamber can be observed over a 0.1-100 GHz range in these transverse impedance cases as well.
Now, let us see more closely the impedances in the frequency range up to 100 GHz on a linear scale. Figures 8 and 9 show the real parts of the longitudinal and transverse impedances when the thickness of the copper chamber is 1 mm and the thickness of the NEG coating is 1 μm (green), 2 μm (black), 5 μm (blue), and 10 μm (red), respectively. The brown and purple lines show the real part of the impedances when the formulae (3) and (4) are used for the copper-only case and the NEG-only case, respectively. It can be seen that the real part of the longitudinal impedance increases almost linearly to the frequency when the thickness of the NEG coating is small (a few μm). Also, the impedance shape almost saturates at around 5 μm as a function of the NEG thickness. In other words, the real part of the longitudinal impedance is almost unchanged even if the thickness of the NEG coating is increased above 5 μm. 6   The transverse impedance behaves very differently from the longitudinal one. As the left panel of Figs. 7 and 9 show, it varies widely as a function of the thickness of the NEG coating. Above around 1 THz, the transverse impedance looks saturated as a function of the NEG thickness, but below that frequency, the transverse impedance increases significantly for a thicker NEG coating. When the NEG coating is very thin (such as 1 μm), the effect of the NEG coating may be unnoticeable and the impedance due to the NEG coating becomes important beyond 20 GHz. From a few kHz to 10 MHz, the wake fields see mostly the copper chamber so that the transverse impedance has a little dependence on the thickness of the NEG coating.
To see the reliability of the result, we compare our results with those of the ImpedanceWake2D code [15] for the same set of parameters. The ImpedanceWake2D code generally calculates the  Fig. 9. Dependence of the real part of the transverse impedance per unit length on the thickness of the NEG coating for the 1 mm-thick copper chamber, in a different frequency range from Fig. 7 and depicted on a linear scale. The green, black, blue, and red lines show the real part of the transverse impedances for 1 μm, 2 μm, 5 μm, and 10 μm, respectively. The brown and purple lines show the impedances of the copper-only chamber and the NEG-only chamber using the conventional formula (4), respectively. longitudinal and transverse impedances by using sophisticated matrix formalism for multi-layered chambers [15]. Since the ImpedanceWake2D code includes the beam energy (Lorentz γ ) dependence of impedances, we restore it in our formulae for the comparisons (γ = 7460.52). Figures 10 and 11 show comparisons of the longitudinal and transverse impedances, respectively. The red lines show the present theoretical results. The blue lines show the results by ImpedanceWake2D [15]. Both results show excellent agreement.
Finally, let us see some simulation results for the same set of parameters. We applied the simulation technique explained in the previous section to the calculations. The "Lossy Metal" is chosen to be the copper, while the "Normal Metal" is the NEG. Otherwise, the wake fields cannot see the effect of the copper chamber. The results of the longitudinal and transverse impedances are shown in Figs Fig. 11. Comparisons of the transverse impedances per unit length between the present theory (red) and the ImpedanceWake2D code [15] (blue). The left and right figures show the real and imaginary parts of the impedance, respectively. The purple line shows the impedance of the NEG-only chamber using the conventional formula (4). lines show the impedances of the copper-only chamber by using the conventional formulae (3) or (4), for reference. As in the previous result for a single-layered resistive chamber shown in Figs. 3 and 4, the discrepancy between the simulation (black) and the theoretical (blue) results is also clearly identified below around 1 GHz in Figs. 12 and 13. However, from 1 GHz to 10 GHz, the agreement between them is rather good. Moreover, we can clearly see that the impedances rise apart from the impedance of the copper-only chamber (brown) in the frequency region. Notwithstanding these interesting findings about the simulation results, the theoretical approaches turn out to be more accurate from a practical point of view.  expressed as where I n (z) and K n (z) are the modified Bessel functions [25], and k = ω/c. Here, it is noticeable that we do not assume that the conductivities (σ 2c and σ 3c ) are much larger than ω 0 in order to retain the effect due to the induction term, which is typically neglected in the surface impedance theory [22]. For the transverse impedance, the formula is divided by the frequency region for simplification.
where 10 for a thin coating (d − b a − b), and the conductivity σ 3c is reasonably assumed to be much larger than ω 0 at the frequency, where the prime in the modified Bessel functions means the differential by their argument.
At high frequency ( jkZ 0 σ 3c b 1 and jkZ 0 σ 2c b 1 ), we obtain , (12) where Equation (8) is derived by assuming that the chamber is made only of copper, whose thickness is finite, and approximated for low frequency. Equation (11) is obtained by assuming the copper thickness is infinite and a thin coating (b d), where the Bessel functions are approximated by the hyperbolic functions by using the conditions ( jkZ 0 σ 3c b 1 and jkZ 0 σ 2c b 1). Equation (12) is obtained by considering that the thickness of the NEG is infinite compared to the skin depth at high frequency, where the effect of the induction term is restored. Figures 14 and 15 show the longitudinal and transverse impedances for a copper chamber thickness of 2 mm and a NEG coating thickness of 10 μm. The solid and dotted lines show the exact and approximated solutions given by Eqs.   both the real and imaginary parts of the impedance. The formulae (5), (8), (11), and (12) have no convergence problem that some of the other methods may suffer [13].

Loss factor
Once the longitudinal impedance is calculated, a loss factor [16] is calculated by using where we assume a Gaussian beam profile, in the longitudinal direction. When the RMS bunch length σ s is 3 mm, the loss factor per unit length is 0.0122 V (pCm) −1 and 0.0741 V (pCm) −1 for a NEG coating thickness of 1 μm and 10 μm, respectively. If we assume an SLS-II-like ring (circumference = 288 m) and use a total beam current 12 of 400 mA and single bunch current of 1 mA, the power deposition on the chamber per unit length is about 30 W m −1 for a NEG thickness of 10 μm. It may not be significant power deposition that requires a special cooling system for the copper chamber. The estimation significantly depends on the bunch length σ s . However, when we assume an infinitesimal beam, σ s = 0, we find that the loss factor does not depend on the conductivity of the material in most cases.
This can be proved as follows. First, let us modify Eq. (16) as where W L (t) is the longitudinal wake function. For an infinitely thick chamber with thin TiN coating, the impedance can be approximated as The wake function is calculated as where ω is scaled to be a dimensionless variable and The order of magnitude of the functions F 1 (ω) and F 2 (ω) is less than or equal to one. If we neglect their frequency dependencies and express them as F 3 , the ω-integration is performed as Equation (24) reproduces Eq. (A.87) in Ref. [7] when F 3 is identical to one. The conductivity appears in the parameter ζ 3 , which means that the wake function at the origin t = 0 (the loss factor) does not depend on the conductivity.

Conclusions
We have found that both the longitudinal and transverse impedances of a thin NEG coating on a copper chamber are in the middle of the transit states between a copper-only chamber and NEGonly chamber in the frequency range of 0.1-100 GHz in terms of the conventional resistive-wall impedance formulae. In this frequency range, the longitudinal impedance seems to be saturated at around 5 μm as a function of the thickness of the NEG coating, but the transverse impedance still has a large dependence on the thickness of the NEG coating.
At the SuperKEKB [26], they have considered using a NEG coating on chambers in the interaction region to achieve a high vacuum there. They think that the NEG thickness needs to be at least several μm for effective pumping and for a long lifetime. 1 In this regard, the choice of the NEG thickness may have a great impact on the impedance budget of a machine.
At very high frequencies over ∼100 GHz, the impedances deviate from the conventional impedance lines for the NEG-only chamber due to the effect of the induction term. We need precise calculations using the elaborate formulae such as the present theory, or the more sophisticated ImpedanceWake2D code [15]. Only for an impedance between a few kHz and 10 MHz, where the resistive-wall impedance may excite a Robinson instability [1], one may use the conventional formula for the copper-only chamber.
The present theory and findings can provide guidance for the design of vacuum chambers with a thin coating and a tool to estimate their longitudinal and transverse resistive-wall impedances, though constant conductivities are assumed over a wide frequency range to 100 THz.
Once the longitudinal impedance is found, the loss factor can be calculated. In most cases, the loss factor for an infinitesimal beam does not depend on the conductivity of the chamber material as in the case of a single-layered resistive chamber. The characteristic can be a benchmark to evaluate the reliability for impedance calculations. whereρ and j are the charge density and the current density of the beam, respectively, k = ω/βc, and Z 0 = 120π is the impedance of free space. In the cylindrical coordinates (ρ, θ , z) for an axially symmetric structure, the wave equation for the longitudinal components of the electric and magnetic fields contain no transverse field component. They are decoupled. For the longitudinal field, there is a source term cZ 0 ∂ρ/∂z + jkβZ 0 j z , while the z-component of ∇ × j vanishes for particles with a longitudinal velocity only.
When the beam with its charge q travels along the pipe at a constant radial offset position ρ = r b , θ = θ b , the charge density is expressed as where δ(x) is the δ-function, δ p (θ) is the periodic δ-function, and δ m,n is the Kronecker δ. Since the general solutions of Maxwell equations are obtained by the superposition of those for i m ρ m , we choose i m ρ m as the source term. Let us define the source field specified with superscript S as the solution which satisfies the Maxwell equations with ρ m , j m and vanishes at ρ → ∞. It is given by for m = 0, and for m > 0, where j is the imaginary unit, γ is the Lorentz γ , k = 2π f /βc,k = k/γ , and K m (z) and I m (z) are the modified Bessel functions, respectively [21,25]. 15 Next, let us solve the Maxwell equation so that it satisfies the boundary conditions due to the conductive chamber with the thin coating, and obtain the longitudinal impedance of the beam.

A.2. Longitudinal impedance
General solutions (in particular E z , H θ ) for m = 0 are expressed as in the vacuum chamber (ρ < b); 16) in the coating film with conductivity σ 2c (b < ρ < d); in the conductive material with conductivity σ 3c (d < ρ < a); and outside the chamber (ρ > a), where ν 2 = k 2 /γ 2 + jkβZ 0 σ 2c , ν 3 = k 2 /γ 2 + jkβZ 0 σ 3c , and A(k), C 1 (k), C 2 (k), D 1 (k), D 2 (k), and E(k) are arbitrary coefficients. The matching conditions on each surface are specified by ρ = b, ρ = d, and ρ = a: These equations are composed of six linear equations with six unknown coefficients (A, C 1 , C 2 , D 1 , D 2 , and E). Consequently, they give a unique solution for the coefficient A(k) straightforwardly by using Mathematica [27]. 16 The longitudinal impedance Z L is defined as the average of the longitudinal electric field (normalized by the beam current) over the beam cross-section. When we assume a pencil beam, we obtain the final expression of the resistive-wall impedance by extracting the space charge impedance Eq. (C.5): where C is the chamber length. Next, let us calculate the transverse impedance.