The resistive-wall impedance of cylindrical vacuum chambers was first calculated more than forty years ago under some approximations. Since then many papers have been published to extend its range of validity. In the last few years, the interest in this subject has again been revived for the LHC graphite collimators, for which a new physical regime is predicted. The first unstable betatron line in the LHC is at 8 kHz, where the skin depth for graphite is 1.8 cm, which is smaller than the collimator thickness of 2.5 cm. Hence one could think that the resistive thick-wall formula would be about right. It is found that it is not, and that the resistive impedance is about two orders of magnitude lower at this frequency, which is explained by the fact that the skin depth is much larger than the beam pipe radius. Starting from the Maxwell equations and using field matching, a consistent derivation of both longitudinal and transverse resistivewall impedances of an infinitely long cylindrical beam pipe is presented in this paper. The results, which should be valid for any number of layers, beam velocity, frequency, conductivity, permittivity and permeability, have been compared to previous ones

Métral, E

Salvant, B

Zotter, B

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E. Métral, B. Salvant1),  B. ZotterEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsRESISTIVE-WALL IMPEDANCE OF AN INFINITELY LONGMULTI-LAYER CYLINDRICAL BEAM PIPEThe resistive-wall impedance of cylindrical vacuum chambers was first calculated more than forty years agounder some approximations. Since then many papers have been published to extend its range of validity. Inthe last few years, the interest in this subject has again been revived for the LHC graphite collimators, forwhich a new physical regime is predicted. The first unstable betatron line in the LHC is at 8 kHz, where theskin depth for graphite is 1.8 cm, which is smaller than the collimator thickness of 2.5 cm. Hence one couldthink that the resistive thick-wall formula would be about right. It is found that it is not, and that the resistiveimpedance is about two orders of magnitude lower at this frequency, which is explained by the fact that theskin depth is much larger than the beam pipe radius. Starting from the Maxwell equations and using fieldmatching, a consistent derivation of both longitudinal and transverse resistivewall impedances of aninfinitely long cylindrical beam pipe is presented in this paper. The results, which should be valid for anynumber of layers, beam velocity, frequency, conductivity, permittivity and permeability, have beencompared to previous ones.Large Hadron Collider ProjectAbstractLHC Project Report 1014CERN,CH-1211 Geneva 23Switzerland1) EPFL, Lausanne, SwitzerlandCERN, Geneva, SwitzerlandPresented atPAC07, 22nd PAC Conference, June 25-29, 2007,Albuquerque, USARESISTIVE-WALL IMPEDANCE OF AN INFINITELY LONG MULTI-LAYER CYLINDRICAL BEAM PIPE E. Métral, B. Zotter (CERN, Geneva) and B. Salvant (EPFL, Lausanne)  Abstract The resistive-wall impedance of cylindrical vacuum chambers was first calculated more than forty years ago under some approximations. Since then many papers have been published to extend its range of validity. In the last few years, the interest in this subject has again been revived for the LHC graphite collimators, for which a new physical regime is predicted. The first unstable betatron line in the LHC is at 8 kHz, where the skin depth for graphite is 1.8 cm, which is smaller than the collimator thickness of 2.5 cm. Hence one could think that the resistive thick-wall formula would be about right. It is found that it is not, and that the resistive impedance is about two orders of magnitude lower at this frequency, which is explained by the fact that the skin depth is much larger than the beam pipe radius. Starting from the Maxwell equations and using field matching, a consistent derivation of both longitudinal and transverse resistive-wall impedances of an infinitely long cylindrical beam pipe is presented in this paper. The results, which should be valid for any number of layers, beam velocity, frequency, conductivity, permittivity and permeability, have been compared to previous ones. INTRODUCTION Renewed interest in this old subject [1] is due to the important role of resistive-wall effects for collective beam stability in large circular proton accelerators or colliders such as the LHC. In particular, the numerous collimators in the machine are made of highly resistive graphite to withstand the high temperatures generated by the impact of high-energy protons, and will be moved into positions very close to the beam to protect the surrounding super-conducting magnets from stray protons. These devices could create such high impedances as to severely limit the beam current and hence the performance of the collider. A number of papers have been published on this subject in the last few years [2,3,4], which were already discussed in Ref. [5]. A more recent one was published in Ref. [6]. In this report, whose transverse part is a summary of Refs. [7,8] and whose results were already briefly discussed in Ref. [9], the exact calculation of the electro-magnetic fields excited by an infinitesimally thin, annular particle beam of finite radius in a surrounding coaxial vacuum chamber of infinite length is described. The method used to compute the fields is reviewed in the first section, while the longitudinal and transverse impedances are discussed in the following sections. It is worth mentioning that the “low-frequency” regime, which is of primary importance for the LHC collimators, was already obtained in Ref. [10] under some approximations. FIELD CALCULATIONS Starting from Maxwell equations, assuming that in the frequency domain all the field quantities are proportional to tje ω  (which will be omitted below for simplicity), and combining the conduction and displacement current terms, yields the following scalar Helmholtz equations for the longitudinal field components in the circular cylindrical coordinates (r, q, z)  ( ),111 222222ρυθυεμωθθ ⎥⎦⎤⎢⎣⎡∂∂−∂∂=⎥⎥⎦⎤⎢⎢⎣⎡+∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂rrrHzrrrrrrzc (1)  .111 222222zzcjzEzrrrrrυρμωρεεμωθ+∂∂=⎥⎥⎦⎤⎢⎢⎣⎡+∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂ (2) Here, H  and E  are the magnetic and electric fields respectively, fπω 2=  is the angular frequency, j  is the imaginary unit, ρ  is the (source) charge density, υ  is the source velocity, and  ,00ωσεεεεεjrc+=′=      ,0 rεεε =  (3)  ( ) ,tan1 M00 ϑμμμμμ jr +=′=  (4) where ε ′  ( μ′ ) is the relative complex permittivity (permeability) of the pipe with 0ε  ( 0μ ) the permittivity (permeability) of vacuum and rε  ( rμ ) the relative permittivity (permeability) of the pipe, σ  is the electric conductivity, and Mϑtan  is the loss tangent. The homogeneous equations can be solved by separation of variables, writing the longitudinal component of the magnetic and electric fields as  )()()( rRzZθΘ . It is easily found that θθ mje ±=Θ )( , where m  is called the azimuthal mode number, and zkjezZ ±=)( , where υω /=k  is called the wave number. The function )(rR  is found by solving the following equation  ,01 222=⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎠⎞⎜⎜⎝⎛RrmrdRdrrddrν  (5) with μεβν ′′−= 21k , which is called the radial propagation constant, where β  is the relativistic velocity factor. It is known that the solutions of the differential Eq. (5) are the modified Bessel functions of mth order and argument rν , called ( )rIm ν  and ( )rKm ν . The source charge density ρ  has to be specified at this stage to obtain the exact forms of the azimuthal and axial functions )(θΘ  and )(zZ . A macro-particle of charge eNQ b=  is assumed to move along the pipe (in the z-direction) with an offset ar =  in the 0=ϑ  direction and with velocity cβυ =  (equal to the bunch velocity cbb βυ = ). Neglecting betatron and synchrotron oscillations, assuming that the beam is not affected by the wake field, and using the azimuthal Fourier decomposition of the charge, the charge density can be written [11] ( ) ( )( ) ( ) ( ),1cos;,,0 01∑∞=+−−+=m mmm tzaramPtzr υδδδπϑϑρ  (6) where mm aQP =  is the mth multipole moment, and 10 =mδ  if m = 0, 0 if m ∫ 0. Proceeding to the frequency domain and considering only one mode m, it is seen that the charge density mρ  is proportional to zkjem −)(cos ϑ , which leads to the following longitudinal components of the electric and magnetic fields (reinserting the time dependence) ( ) ( ) ( )[ ],)(sin 21 rKCrICemH mmzktjz ννθ ω += −  (7) ( ) ( ) ( )[ ] ,)(cos 43 rKCrICemE mmzktjz ννθ ω += −  (8) where 4,3,2,1C  are constants to be determined by field matching, as all field strength components have to be matched, i.e. in the absence of surface charges and currents the four tangential field strengths have to be continuous (matching of the radial components is redundant). Finally, using Maxwell equations in a source-free region, the transverse field components can be deduced from the longitudinal ones (with HZGrr0= , where 0Z  is the free-space impedance)  ,0020 ⎟⎟⎠⎞⎜⎜⎝⎛+′=rdEdrGmkjE zzr μβν (9)  ,0020 ⎟⎟⎠⎞⎜⎜⎝⎛′+−=rdGdrEmkjE zz μβνθ  (10)  ,0020 ⎟⎟⎠⎞⎜⎜⎝⎛+′=rdGdrEmkjG zzr εβν (11)  ,0020 ⎟⎟⎠⎞⎜⎜⎝⎛′+=rdEdrGmkjG zz εβνθ  (12) where )(cos/// 000 ϑϑϑ mGGEEEE rrzz ===  and )(sin/// 000 ϑϑϑ mGGGGEE zzrr === . LONGITUDINAL IMPEDANCE Considering only the term m = 0 (longitudinal monopole), the charge density, in the frequency domain, is given by  ( ) ( ) .2;,0zkjearaQzr −−= δυπωρ  (13) The longitudinal source electric field in the vacuum region between the beam and the vacuum pipe ( bra ≤≤ , where b is the inner beam pipe radius) has been computed in Ref. [12] and is given by  ( )( ) ( ) ( )[ ] ,2;,0TM00220)(zkjszeuIuKsIQjzrE−−= αγυεπωω (14) where 2/12 )1( −−= βγ  is the relativistic mass factor, γ/aks = , γ/rku = , and TMα  is the unknown parameter which has to be found by field matching at the layers’s boundaries. The total (i.e. resistive-wall plus space-charge) longitudinal impedance is given by  ( ) ( ) ,;,1 )(// zkjsz ezaEdzQZ ∫∞+∞−−= ωω  (15) which, using Eqs (14), leads to (with L the length of the resistive object)    ( ) ( ) ( ) ( )[ ].2 0TM02200// sIsKsILjZ αγυεπωω −−=  (16)  1 1000 1.μ 106 1.μ 109 1.μ1012f @HzD1.μ 10-81.μ 10-60.00010.01Zêêên@WD» Im @Zêê ê nD » -> Zotter» Im @Zêê ê nD » -> H&NRe @Zêê ê nD -> ZotterRe @Zêê ê nD -> H&N Figure 1: Longitudinal impedance for the case of a (single-layer) 1 m long (round) LHC graphite collimator with radius b = 2 mm and conductivity 115- m10 −−Ω=σ . The resistive-wall impedance is obtained by subtracting from the total impedance of Eq. (16) the impedance obtained with a perfect conductor at br = , i.e. when ( ) ( )1010TM / xIxK=α  with γ/1 bkx = , which yields the space-charge impedance  ( ) ( ) ( )( )( )( ) .2 10100022020SC// ⎥⎦⎤⎢⎣⎡−−=xIxKsIsKsILjZγυεπωω  (17) The present formalism can be used for any number of layers of the vacuum pipe [13]. Only the result for a single layer extending up to infinity is given here ( ) ( )( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ),//22010201020101020101022020RW//⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡′′−′′′−′−×−=γενγενγυεπωωkxKxIxKxIkxKxKxKxKxIxKsILjZ (18) where bx ν=2 . The longitudinal impedance (divided by revffn /= ) for the case of a LHC collimator is shown in Fig. 1, revealing a good agreement with Ref. [10]. TRANSVERSE IMPEDANCE Considering only the term m = 1 (transverse dipole), the charge density, in the frequency domain, is given by  ( ) ( ) .cos;,, 211 zkjearaPzr −−= ϑδυπωϑρ  (19) The longitudinal source-field components (for bra ≤≤ ) have been computed in Ref. [12] and are given by  ( ) ( )( ) ( )[ ] ,cos;,,1TM112201)(zkjszeuIuKsIaPjzrE−−×=αϑγυεπωωϑ (20)  ( ) ( )( ) ,sin;,,1TE12201)(zkjszeuIsIaPjzrG−×=αϑγυεπωωϑ (21) where the unknown parameters TMα  and TEα  have to be found by field matching at the layers’s boundaries. The total horizontal impedance is given by  ( ) [ ],;,2,;,2, )()(11zkjsrbszkjybxxezaBzaEdzPjeBEdzPjZ∫∫∞+∞−∞+∞−⎥⎦⎤⎢⎣⎡ ⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛−=−=ωπυωπυωϑ(22) which, using Eqs. (10) and (11), yields    ( ) ( ) ( ) ( )[ ] .1TM122 10 sIsKasIZLjZx αγβπω −−=  (23) The space-charge impedance is obtained with a perfect conductor at br = , i.e. when ( ) ( )1111TM / xIxK=α , and is given by  ( ) ( ) ( )( )( )( ) .11111122210SC ⎥⎦⎤⎢⎣⎡−−=xIxKsIsKasIZLjZx γβπω  (24) The present formalism can also be used for any number of layers of the vacuum pipe [7]. Only the result for a single layer extending up to infinity is given here  ( ) ( ) ( ) ( )( )( )( )( )( )( )( )( )( ) ( ) ( )( )( )( )( )( )( )( )./212111112121111122121211222121111111111111222111210RW⎪⎪⎪⎭⎪⎪⎪⎬⎫⎪⎪⎪⎩⎪⎪⎪⎨⎧⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛ ′′−′×⎟⎟⎠⎞⎜⎜⎝⎛ ′′−′−−×⎟⎟⎠⎞⎜⎜⎝⎛ ′′−′×⎟⎟⎠⎞⎜⎜⎝⎛ ′−′=xKxKkxIxIxKxKkxIxIxxxkxxIaxKxKkxIxIxKxKxIxIxxxKsIZLjZ xενγμνγβνγγπμνγνγβω(25) The transverse impedance of a LHC graphite collimator is shown in Fig. 2, in good agreement with Ref. [10].  1 1000 1.μ 106 1.μ 109 1.μ 1012f @HzD100100001.μ 1061.μ 108Z x@WêmD» Im @ZxD » -> Zotter» Im @ZxD » -> H&NRe @ZxD -> ZotterRe @ZxD -> H&N Figure 2: Transverse impedance for the case of a LHC graphite collimator (see Fig. 1). CONCLUSION A general formalism to compute both longitudinal and transverse impedances of an infinitely long multi-layer circular beam pipe has been presented. Three frequency regimes are found [8], in agreement with Ref. [10] in the particular case of a (single-layer) LHC collimator. Comparisons with Ref. [2] can be found in Refs. [7,8]. REFERENCES [1] J. Laslett, K. Neil, A. Sessler, Rev. Sci. Instr. (1965). [2] A. Burov et al., Proc. EPAC’02, Paris, 2002. [3] H. Tsutsui, LHC Project Note 318, 2003. [4] L.Vos, CERN-AB-2003-005 ABP, 2003. [5] F. Caspers et al., Proc. EPAC’04, Lucerne, 2004. [6] Al-Khateeb et al., Proc. EPAC’06, Edinburgh, 2006. [7] B. Zotter, CERN-AB-2005-043, 2005. [8] E. Métral, CERN-AB-2005-084, 2005. [9] E. Métral, Proc. PAC’2005, Knoxville, 2005. [10] Henry and Napoly, CLIC Note 142-1991, 1991. [11] T. Suzuki, KEK Preprint 98-127, 1998. [12] R.L. Gluckstern, CERN 2000-011, 2000. [13] B. Salvant et al., to be published. 

Resistive-Wall Impedance of an Infinitely long Multi-Layer Cylindrical Beam Pipe

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