Coupling impedances of a resistive insert in a vacuum chamber

We have developed a theory to calculate both longitudinal and transverse impedances of a resistive short (typically shorter than the chamber radius) insert with cylindrical symmetry, sandwiched by perfectly conductive chambers on both sides. It is found that unless the insert becomes extremely thin (typically a few nm for a metallic insert) the entire image current runs on the thin insert, even in the frequency range where the skin depth exceeds the insert thickness, and therefore the impedance increases drastically from the conventional resistive-wall impedance. In other words, the wakeﬁelds do not leak out of the insert unless it is extremely thin. Formulas of the impedance valid for various cases of the insert are categorized in summary.


I. INTRODUCTION
The impedance of a finite-length, resistive insert in a beam pipe has recently been studied in several reports [1][2][3][4]. To simplify this complicated problem, it is often assumed that the skin depth is smaller than the radius of the chamber and its thickness, and therefore investigations have been limited to the behavior of the impedance in a high frequency region [2,3]. In this case, the wakefields inside the chamber are completely shielded and do not leak out of the insert. However, in the frequency region where the skin depth exceeds the chamber thickness, these theories are no longer valid.
In proton synchrotrons, the inner surface of a short ceramic break is normally coated by a thin (typically about 10 nm) titanium nitride (TiN) to suppress the secondary emission of electrons. The skin depth can be larger than the thickness of the TiN coating in low frequency, and the wakefields may interact with the outside world through the coating. It is thus important to construct a theory of resistive insert taking into account its thickness effects.
The resistive-wall impedance has been studied for many years and many formulas are obtained [5][6][7][8]. But, those formulas are sometimes applicable only to a relativistic beam with a limited frequency range. Recently, Burov and Lebedev, and Metral et al. have calculated the resistivewall impedance of chambers composed of more than one layer [9][10][11]. Their theories, however, assume a beam pipe with translational symmetry. For a finite-length insert in a beam pipe, a new theory is needed.
Recently, we have developed the theory to describe the impedance of a gap that is sandwiched by perfectly conductive chambers [12]. This theory gives us basic understanding of the interaction between a beam and a gap. Replacing this gap by an insert, we can construct a theory of the impedance of the insert as a three-dimensional problem. The main difference between a gap and an insert is that the insert has a finite skin depth, and this skin depth effect will modulate how wakefields propagate in the chamber and the impedance of the insert.
We consider only cylindrical symmetric problems in this paper. The main objective is to study how the impedance will change from that of the conventional resistive-wall theory to that of a gap when the thickness of the insert is changed compared to the skin depth. We develop a theory to describe the impedance of a short insert with cylindrical symmetry by generalizing the theory of impedance of a gap. In Secs. II and III, we develop the theory of both the longitudinal and transverse impedances, respectively. In these sections, we compare the impedance of the resistive wall with finite thickness and that of the short insert. The paper is summarized in Sec. IV.
In numerical examples shown in the figures, unless specified otherwise, we consider a beam pipe radius a ¼ 5 cm with an insert of length g ¼ 8 mm, and conductivity c ¼ 6 Â 10 6 = m. This can be a model for a short ceramic break with TiN coating in a copper beam pipe.

II. LONGITUDINAL IMPEDANCE OF A RESISTIVE INSERT IN A BEAM PIPE
Let us start with deriving electromagnetic fields generated by the interaction between a beam and an insert in the cylindrically symmetric system. We use the cylindrical coordinates ð; ; zÞ as shown in Fig. 1. We assume that both the sandwiching chambers and the insert have thickness t and the inner radius of a. The insert is located in the region where Àw < z < w (namely, the length of the insert g is equal to 2w).
In order to obtain formal solutions of the fields, we apply the field matching technique to this system. We assume that the beam has the cylindrically uniform density with the PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 12, 094401 (2009) 1098-4402=09=12(9)=094401(19) 094401-1 Ó 2009 The American Physical Society radius of and its total charge is 1C. Namely, its current density is given by where ÂðxÞ is the step function, k ¼ !=c, ¼ v=c, v is the velocity of the beam, c is the velocity of light, ! ¼ 2f, and f is the frequency. The formal solutions of the fields at the frequency ! inside the chamber are given by (see Appendix A) for < , and for > , where " k ¼ k=, is the Lorentz factor, AðhÞ is the expansion coefficient, Z 0 ð¼ 120Þ is the impedance of free space, J m ðzÞ is the Bessel function, and Ã ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi . The time dependence of the fields is assumed to be harmonic and it is expressed as the complex exponential e j!t . Since E z on the inner surface of the chamber should be zero except in the insert, the expansion coefficient AðhÞ should satisfy the following relation: for À w < z < w 0 otherwise; (6) whereṼ 1 is the voltage inside the insert at ¼ a. Here the assumption was made that the longitudinal length of the insert g is short (typically shorter than the radius of the chamber) and thus, the z dependence of E z on the insert can be negligible. Then, the expansion coefficient AðhÞ is rewritten by the insert voltageṼ 1 as Substituting Eq. (7) into Eqs. (2)-(5), we obtain for < , and for > . For the fields outside the chamber, we follow Silver and Saunders's theory [13] and describe those as (see Appendix B) for > a þ t, where H ð2Þ m ðzÞ is the Hankel functions of the second kind andṼ 2 is the voltage inside the insert at ¼ a þ t. Here we should notice that Eq. (12) satisfies the condition that it should be zero on the outer surface of the chamber except in the insert. These insert voltagesṼ 1 and V 2 will be determined by boundary conditions.
In order to match the solution for < a and that for > a þ t, we have to find the relation of fields at ¼ a and ¼ a þ t, especially inside the insert. Hence, as shown in Fig. 2, we consider the one-dimensional problem of a wall with thickness t in free space. We call region (x < 0) region I, (0 < x < t) region II, and (t < x < 1) region III. The beam running in region I creates fields on the inner surface of the wall (x ¼ 0), which are written as E z ð0Þ and H ð0Þ.
If we assume that the insert obeys Ohm's law,j ¼ cẼ , Maxwell equations in region II are written as follows: @H ðxÞ @x where 0 is the relative dielectric constant of the insert, usually negligible compared to the first term in Eq. (15) [14]. The solutions are Since the fields inside the chamber [given by Eqs. (4) and (5)] and outside the chamber [given by Eqs. (12) and (13)] must be connected through the relations (16) and (17), one of the insert voltagesṼ 1 can be solved as follows: where J and Y are defined as is satisfied in most of the cases, it can be approximately expressed as where L is the length of the beam pipe. The first term in Eq. (21) represents the nonrelativistic space charge impedance (see Appendix A 1) [12]. The second term Z insert;L in Eq. (21) is the coupling impedance of the insert. The integration of the Bessel functions can be done simply by picking up residues in the complex plane h. However, the integration of the Hankel functions is a quite complicated and difficult task since there are branch points at h ¼ AEk in the complex plane h. To proceed further, we follow the manipulation explained in Ref. [12]. We first rewrite the integration of the Hankel functions as shown in Eq. (C3). This manipulation enables the integration of the Hankel functions over h with the usual residue theorem. We choose the path of integration to be below the poles for h < 0 and above the poles for h > 0. Finally, we obtain the expression of the longitudinal coupling impedance as að2 À e Àjðb s =aÞðzþwÞ À e jðb s =aÞðzÀwÞ Þ wb 2 Here, b 2 s ¼ k 2 2 a 2 À j 2 0;s ¼ À 2 s ; j 0;s are sth zeros of J 0 ðzÞ and H ð1Þ m ðzÞ is the Hankel function of the first kind. We should notice that b s approaches Àj s for j 0;s > ka.
In the above derivation, we used Eqs. (C1) and (C3) of Appendix C. The integration in Eq. (25) over is much more straightforward than the integration in Eq. (22) over h, since there is no singular point except ¼ 0 along the integration path.

A. Frequency dependence and length dependence of the impedance
In this subsection, we assume that the chamber thickness t satisfies the condition We exclude an extremely thin insert case here. This assumption allows us to neglect the effect from radiation terms such as Y pole and Y cut in Eq. (23) in the low frequency region where the skin depth exceeds the insert thickness t.
The thickness t min is typically a few 10 nm for a metallic insert. Krinsky et al. and Stupakov [2,3] studied the impedance of a short insert. Their results indicate that when g ( ðZ 0 a 4 =4Þ 1=3 and (the frequency f D is typically of the order of THz in our short insert), then and is proportional to ffiffiffi g p .
Let us consider the case that the thickness of the insert is larger than 2 1=2 3=4 t min and see if our theory can reproduce Eq. (28) in the extremely high frequency region f ) f D . In this frequency region, we may take a limit of t to infinity in Eq. (23), and the following inequality can be applied to Eq. (23): Then, Eq. (23) becomes by choosing the matching point z as zero for an extremely thin beam (the beam radius ¼ 0). Specifically for a relativistic beam, Eq. (30) reproduces Eq. (28). These results show that the impedance decreases in proportion to k À1=2 in the extremely high frequency, as predicted by the diffraction theory [15].
In the intermediate region of f ( f D where the skin depth is still smaller than the insert thickness t, we can apply the following inequality to Eq. (23): We then obtain the conventional formula of the resistivewall impedance for a relativistic beam [7]: which is proportional to the length of the insert g.
In the low frequency region where the skin depth exceeds the insert thickness t, but the effect from radiation terms such as Y pole and Y cut are still negligible in Eq. (23), we obtain When the thickness of the insert is smaller than 2 1=2 3=4 t min but larger than t min , Eq. (34) becomes valid all the way up to f D .

B. Dependence of the insert impedance on its thickness
Before studying the thickness dependence of the insert impedance, let us study the thickness dependence of the resistive-wall impedance in order to compare them with our results afterwards. We numerically calculate the resistive-wall impedance for different thicknesses of the chamber by borrowing the general formulas of the resistive-wall impedance with finite thickness from Metral et al.'s recent work [10] (see Appendix A 2): where The results for a relativistic beam are shown in Fig. 3. The red, the blue, and the black lines show the cases that the insert thickness t is equal to infinity, 10 m and 1 m, respectively. The impedance starts to deviate from that for the infinitely thick chamber when the skin depth exceeds the chamber thickness. Apparently, the wakefields leak out at low frequency. The dependence of Z L on the conductivity c , the frequency f, and the chamber thickness t, for the case that the skin depth exceeds t, can be approximately written as Contrary to our intuition, the impedance becomes larger as the conductivity of the material c increases. Now, let us discuss the properties of the impedance of the insert by changing the thickness of it. The thickness dependence of the real part of the insert impedance obtained by Eq. (23) is shown in Fig. 4. The red, the blue, the black, the black dashed, the black dot, the green and the blue dashed, and the red dashed lines represent the cases that the thickness t is equal to 100 m, 10 m, 1 m, 100 nm, 10 nm, 1 nm, 100 pm, and 10 pm, respectively. When the skin depth is smaller than the thickness of the insert but the frequency f is lower than f D , the impedance of the insert is identical to the resistive-wall impedance given by Eq. (32) (see the results for the case that t is equal to 100 m). As we find from the result of t ¼ 10 m in Fig. 4, if the skin depth exceeds the insert thickness [f ' 0:42 GHz in this case; see Eq. (33)], the real part of the impedance becomes independent of the frequency. The imaginary part is still inductive for the insert with this thickness. This indicates that the whole wall current runs in the thin insert, despite the fact that the skin depth exceeds the insert thickness in most of the frequencies.
In other words, the beam current is completely shielded by the wall current in the insert, and the wakefields do not propagate out of the chamber. If this picture is correct, the real part of impedance should be equal to the resistance of the wall current Z wall . Actually, the results of t ¼ 1 m to t ¼ 100 nm (even including the result of 10 nm that is smaller than t min ) described in Fig. 4 are equal to the resistance of the wall current Z wall : This behavior of the insert impedance is quite different from that of the resistive-wall impedance of the chamber with finite thickness for a relativistic beam, which was discussed in the first paragraph of this subsection. When the thickness of the insert is extremely thin like t ( t min , the situation is quite different from the above plays a more important role in the impedance. In the frequency region f ( f c , the contribution from the wall current dominates in the impedance. In the rest of the frequency, the radiation effects become dominant contributions. The dips for these cases in Fig. 4 correspond to the cutoff frequencies of the chamber. The imaginary part of impedance becomes capacitive, which is opposite to the result of t > t min . Figure 5 represents the dependence of the impedance on the length of the insert g when the thickness of the insert t is equal to 1 nm and Lorentz factor is equal to 1000. The red, the blue, and the black lines show the longitudinal impedances per a unit length for the cases that the length of the insert g is equal to 2 mm, 8 mm, and 18 mm, respectively. The impedances themselves (not normalized by the gap length g) are proportional to ffiffiffi g p in f > f c , while they are proportional to g in f ( f c . As the insert becomes longer, the frequency f c becomes lower [see Eq. (40)], and the frequency range where the radiation effects dominate in the impedance becomes wider.
The physical reason of why the whole wall current tends to run on the thin insert except for the extremely thin insert case is that the nature tries to minimize the energy loss of a beam, which is smaller when the wall current runs on the thin insert with large resistance than when it converts to the radiation out to free space ð¼ gap impedanceÞ. When t ( t min , the real part of the correct impedances using the present theory is smaller than the hypothetical impedances calculated by extending the simple formula (39) to these extreme thicknesses. The real part of the impedance, i.e. the energy loss of a beam, becomes smaller by the wall current converting to outer radiation than it staying in the extremely thin insert. Finally, we briefly mention the nonrelativistic beam case for the insert that satisfies t > t min as a typical case. The same discussion as in the relativistic beam case is valid in this case as well. Simple formula which is based on the formulas and Eq. (39) in the frequency region where the skin depth is smaller than the thickness t, and is larger than the thickness t, respectively, provides good approximation to the correct impedance of the insert.

III. TRANSVERSE IMPEDANCE OF A RESISTIVE INSERT IN A BEAM PIPE
Following the manipulation for the longitudinal case, we consider the situation where a beam is traveling in the chamber with the charge distribution of the azimuthal dependence as j z ¼ qcð À r b Þ cose Àjkz =r b . At first, we describe fields inside the chamber as in Ref. [12] (see Appendix A): for < r b and dhAðhÞe Àjhz J 1 ðÃÞ J 1 ðÃaÞ cos; for > r b , where i 1 ¼ qr b , and AðhÞ and BðhÞ are expan-sion coefficients. Since E z and E on the inner surface of the chamber should be zero except in the insert, the expansion coefficients AðhÞ and BðhÞ should satisfy the following relations: whereṼ 1 andṼ 2 are the voltages inside the insert at ¼ a.
Then, the expansion coefficients AðhÞ and BðhÞ are rewritten by the insert voltagesṼ 1 andṼ 2 as Substituting Eqs. (49) and (50) into Eqs. (43)-(46), we obtain for > r b . Second, we have to know fields outside the chamber to apply the field matching technique [6]. They are (see Appendix B) where H ð2Þ m ðzÞ is the Hankel function of the second kind, the prime means the differential by its argument z, V 1 , and V 2 are the voltages inside the insert at ¼ a þ t. In the process of obtaining the above equation, we used the approximation that V 1 and V 2 are almost constant.
We have to find the relation between fields at ¼ a and those at ¼ a þ t. Similarly to the longitudinal impedance, let us assume that the relation is approximately obtained by solving the one-dimensional problem as shown in Fig. 2. Maxwell equations in region II depicted in Fig. 2 are given by in addition to Eqs. (14) and (15). In addition to Eqs. (16) and (17), the solutions are Since the fields given by Eqs. (51)-(54) and those given by Eqs. (55)-(58) must be connected through these relations [Eqs. (16), (17), (61), and (62)], one of the insert voltagesṼ 2 can be solved as where z is the matching point of magnetic fields. If we substitute Eq. (63) into Eq. (42) and use the Panofsky-Wenzel theorem [7,16], we can finally obtain the expression for the transverse impedance as The first term of Eq. (66) represents the transverse space charge impedance for a nonrelativistic beam (see Appendix A) [12], while the second term gives the transverse insert impedance Z T;insert . Using Eqs. (C1), (C2), (C4), and (C5), we can rewrite Z T;insert as  1;s À k 2 2 a 2 q for j 02 1;s > k 2 2 a 2 . Again, we reach the equation where the integration over is just straightforward for numerical calculations, since there is no singular point along the integration path.

A. Parameter dependence of the transverse impedance
Now let us examine the frequency dependence of the transverse impedance. We start to study from the extremely high frequency and then will gradually lower the frequency. When the thickness of the insert is larger than 2 1=2 3=4 t min [see Eq.  27)]. In this frequency range, the transverse impedance is approximately given by which becomes for a relativistic beam In the frequency region where f ( f D but still the skin depth is smaller than the insert thickness, the impedance is approximately written as which reproduces the conventional resistive-wall impedance for a relativistic beam [7]: In a lower frequency where the skin depth exceeds the thickness of the insert, the impedance becomes When the thickness of the insert is smaller than 2 1=2 3=4 t min but larger than t min , Eq. (74) gives correct impedance all the way up to f D . Finally, in the region of the wakefields leak out of the insert again. It is almost identical to the gap impedance Z gap;? described in Ref. [12], and goes down toward zero as the frequency approaches to zero. Now, we consider the thickness dependence of the insert. The dependence of the real part of the insert impedance on the insert thickness is shown in Fig. 6. The red, the blue, the black, the black dashed, the black dot, the green, the blue dashed, and the red dashed lines represent the cases for the thickness t equal to 100 m, 10 m, 1 m, 100 nm, 10 nm, 1 nm, 100 pm, and 10 pm, respectively. Similar to the longitudinal case, we at first consider the case that the thickness of the insert t is larger than t min . The result of t ¼ 100 m in Fig. 6 corresponds to the case that the skin depth is smaller than the thickness of the insert t, which reproduces Eq. (73). The results of t ¼ 10 m to t ¼ 100 nm in Fig. 6 represent the case that the skin depth exceeds the thickness of the chamber t except at the low frequency extreme f ( f L [see Eq. (75)]. These impedances (even including the result for 10 nm which is smaller than t min ) agree very well with those obtained from the simple formula, where Z wall is identical to Eq. (39). The case of t ¼ 10 m especially helps us to understand the behavior of the real part of the impedance, which starts to deviate from Eq. (73) and becomes proportional to f À1 when the skin depth exceeds the insert thickness [f < f ; see Eq. (33)]. In the frequency region specified by f L < f < f , the whole wall current runs on the thin insert, and wakefields are still confined inside the chamber. Contrary to the longitudinal impedance, this picture of the insert impedance is applicable to that of the resistive-wall impedance for the transverse impedance. We numerically calculate the resistivewall impedance for different thicknesses of the chamber by borrowing the general formula (A49) of the resistive-wall impedance with finite thickness from the recent work of Metral et al. [10] (see Appendix A 2). The results for a relativistic beam are shown in Fig. 7. The red, the blue, the black, the black dashed, and the black dotted lines show the cases that the insert thickness t is equal to infinity, 10 m, 1 m, 100 nm, and 10 nm, respectively. The entire wall current runs on the chamber for the resistive-wall impedance as well, after the skin depth exceeds the chamber thickness. But at the region f < f L (but not quite lower as in the short insert) where the skin depth is much larger than the chamber thickness, the resistive-wall impedance starts to fall off. In the case that the thickness of the insert t is extremely thin like t ( t min , the situation becomes significantly different. The results of t ¼ 1 nm to 10 pm in Fig. 6 corre-FIG. 6. (Color) The thickness dependence of the transverse impedance of the insert for the relativistic beam case. The red, the blue, the black, the black dashed, the black dot, the green, the blue dashed, and the red dashed lines represent the cases that thickness t is equal to 100 m, 10 m, 1 m, 100 nm, 10 nm, 1 nm, 100 pm, and 10 pm, respectively. The impedance becomes proportional to f À1 , when the skin depth exceeds the insert thickness. The matching point is z ¼ 0.
spond to this case. The parameter f L should be replaced by a new parameter: : Contrary to the longitudinal case, f r as well as f c [see Eq. (40)] are used to classify the property of the impedance along the frequency axis. In the frequency region f r < f ( f c , the contribution from the wall current dominates in the impedance, while the radiation effects dominate in the rest of the frequency. Figure 8 shows the dependence of the impedance on the length of the insert g where the thickness of the insert t is equal to 10 nm and Lorentz factor is equal to 1000. The red, the blue, and the black lines show the transverse impedance per a unit length for the cases that the length of the insert g is equal to 2 mm, 8 mm, and 18 mm, respectively. Since the wall current effect dominates in the impedance in the frequency region f r < f ( f c , the impedance is proportional to the length of the insert g. The impedance is proportional to ffiffiffi g p in the higher frequency region, as the contributions from the radiation dominate in the impedance. The wakefield makes dips in the impedance curve for the frequency that is larger than the cutoff frequency of the chamber. Especially, in the case of the infinitesimally thin insert, the impedance is identical to the gap impedance, in the entire frequency. Like the longitudinal case, the transition thickness of the insert at which the wall current starts converting to the outer radiation from running on the thin insert is deter-mined by which case minimizes the impedance and thus the energy loss of a beam. Figure 9 demonstrates this fact by comparing the correct impedances with the hypothetical ones obtained by extending the simple formula (76) to these extreme thicknesses.
Finally we briefly mention the nonrelativistic beam case. The same discussion as in the relativistic beam case is valid in this case as well. The transverse resistive-wall impedance [10,11] well describes that of a short insert except the extremely thin case like t < t min and in the low frequency extreme: f ( f L . FIG. 7. (Color) The dependence of the real part of the transverse resistive-wall impedance of a uniform beam pipe (no insert) on the thickness of the chamber. The length of the beam pipe g ¼ 8 mm. Lorentz factor ¼ 1000. The red, the blue, the black, the black dashed, and the black dotted lines show the cases that the chamber thickness t is equal to infinity, 10 m, 1 m, 100 nm, and 10 nm, respectively. The impedance for finite thickness starts to deviate from that for infinite thickness at the frequency where the skin depth exceeds the chamber thickness.

IV. SUMMARY
We have developed the theory to describe the longitudinal and the transverse impedances of a short insert with cylindrical symmetry sandwiched by the perfectly conductive chambers by generalizing the theory of a gap impedance. When the thickness of the insert t is larger than 2 1=2 3=4 t min ½¼ ð16g=Z 3 0 3 c Þ 1=4 , the impedance by our theory in the frequency region of f ð¼ c=Z 0 c t Þ reproduces the results of the resistive-wall impedance. Moreover, the result by our theory in the higher frequency region is consistent with the prediction of the diffraction theory. Formulas valid for various cases of inserts (including the resistive-wall impedance) are categorized in Table I.
An interesting finding for the short insert is that when the thickness of the insert t is larger than t min ½¼ ð4g= 2 Z 3 0 3 c Þ 1=4 (typically a few 10 nm for a metallic insert and a few times larger than the thickness of TiN coating inside a short ceramic break in a proton synchrotron), the entire wall current runs in the thin insert even when the skin depth exceeds the thickness of the insert [strictly speaking, except the low frequency extreme given by f ( f L ð¼ 3c= 4Z 0 c taÞ for the transverse impedance] and therefore the impedances increase drastically from the conventional resistive-wall impedance. This feature of the short insert does not depend on whether a beam is relativistic or not. In other words, this behavior is quite different from that of the resistive-wall impedances, especially from the longitudinal impedance for a relativistic beam. In this resistive-wall case the wakefields immediately propagate out of the chamber once the skin depth exceeds the thickness of the chamber.
The physical reason of why the whole wall current tends to run on the thin insert except for the extremely thin insert case is that the nature tries to minimize the energy loss of a beam, which is smaller when the wall current runs on the thin insert with large resistance than when it converts to the radiation out to free spaceð¼ gap impedanceÞ.
Only when the thickness of the insert t is smaller than t min , the contribution from the wall current in the impedance of the short insert starts to diminish. For the longitudinal impedance, the parameter f c ð¼ 2 c Z 2 0 t 2 c=4gÞ specifies the upper limit of the frequency where the wall current effects are dominant in the impedance. As the insert becomes thinner, this upper limit moves to a lower frequency. For the transverse impedance, the another parameter f r ½¼ ðgc 3 =4 3 Z 2 0 2 c a 4 t 2 Þ 1=3 specifies the lower limit of the frequency region where the wall current effects are dominant. As the insert becomes thinner, the lower limit moves to a higher frequency, and as a result, the frequency region where the wall current effects dominate in the impedance becomes narrower from the both sides. Finally, for the extremely thin insert, both the longitudinal and the transverse impedances by our theory converge to those of the gap impedance.
Since these parameters f c and f r , that specify the wall current dominant region, are proportional to t 2 =g and TABLE I. Valid formulas categorized according to various cases of inserts, where a is the radius of the chamber, g is the length of the insert, t is the thickness of the insert, c is the velocity of light, Z 0 ð¼ 120Þ is the impedance of free space, c is the conductivity of the insert, f is the frequency, ð¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi c=Z 0 f c p Þ is the skin depth, f ¼ c=Z 0 c t 2 , f L ¼ 3c=4Z 0 c ta, f D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 Z 0 c =2 2 g p , f c ¼ 2 c Z 2 0 t 2 c=4g, f r ¼ ðgc 3 =4 3 Z 2 0 2 c a 4 t 2 Þ 1=3 , and t min ¼ ð4g= 2 Z 3 0 3 c Þ 1=4 .  [12] ðg=t 2 Þ 1=3 , respectively, the increase of the length of the insert g has a similar effect on the impedance as the reduction of the insert thickness, especially when the thickness of the insert is smaller than t min . It is remarkable that the parameters t min and f D are proportional to g 1=4 and g À1=2 , respectively. Then, the longer the length of the insert g is, the more easily the radiation effect appears in the impedances. In general, as the insert becomes longer (but still shorter than the radius of the chamber), the wakefields tend to propagate out of the insert.