Comparison of the longitudinal coupling impedance from different source terms

ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 593 (2008) 171– 176

Contents lists available at ScienceDirect

Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Comparison of the longitudinal coupling impedance from different source terms A.M. Al-Khateeb , R.W. Hasse, O. Boine-Frankenheim FAIR-Accelerator Theory Group, GSI-Darmstadt, Planckstr. 1, 64291 Darmstadt, Germany

a r t i c l e in fo

abstract

Article history: Received 4 April 2008 Received in revised form 5 May 2008 Accepted 11 May 2008 Available online 22 May 2008

The longitudinal coupling impedance and the transmission coefficient resulting from a thin ring and from a uniform disk are obtained analytically for a resistive cylindrical beam-pipe of finite wall thickness. The impedances are derived and then compared with the well-known corresponding expression for perturbations on a uniform, coasting beam [A. Al-Khateeb, O. Boine-Frankenheim, R.W. Hasse, I. Hofmann, Phys. Rev. E 71 (2005) 026501]. The transmission coefficients from both sources are found to be exactly the same. Differences do appear in the expressions for the electromagnetic fields within the beam region, and therefore leading to different coupling impedances. By applying the results to parameters relevant for the SIS-18 synchrotron at GSI, it is found that the formula from the ring source underestimates the space-charge impedance at all beam energies and it shows a noticeable deviation from the disk formula for all frequencies. Although their mathematical expressions are different, resistive-wall impedances from the two sources are found to be numerically equal. The spacecharge impedances become equal asymptotically only in the so-called ultra-relativistic limit. & 2008 Elsevier B.V. All rights reserved.

Keywords: Coupling impedance Resistive-wall impedance

1. Introduction Reliable impedance calculations are an important issue for the design and optimization of ring accelerators for high currents. Especially in the regime of low and medium beam energies and for low frequencies there still exist relevant differences in the impedances obtained from different approaches (see e.g. [1,2]). This regime is important for the upgrade of the existing SIS-18 heavy ion synchrotron at GSI and for the design of the new synchrotrons as part of the FAIR project [3]. An important issue is the representation of the source term in Maxwell’s equations. At low energies it is important to account for the transverse beam profile. A homogeneously charged disk is used in many studies in order to model a longitudinal perturbation on the beam (e.g. in Refs. [2,4]). For the calculation of the longitudinal coupling impedance at ultra-relativistic energies the source term is usually reduced to a thin ring charge [5]. This source term is often employed in impedance calculations for arbitrary beam energies (see e.g. [6]). In some limiting cases, like low or high frequencies, the differences between source terms can be expressed in the form of multiplicative g-factors, cf. the textbooks [7–9] and others [10,11].

 Corresponding author. Permanent address: Department of Physics, Yarmouk University, Irbid, Jordan. E-mail address: [email protected] (A.M. Al-Khateeb).

0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.05.011

A treatment for arbitrary frequencies, however, was missing. The present work will shed some light on the coupling impedance and transmission coefficient from two different source terms, namely, a thin ring and uniform disk propagating down a resistive cylindrical beam-pipe of finite wall thickness. The paper is organized as follows: In Section 2, source terms from a uniform disk and a thin ring in the wave equations for the electromagnetic fields are presented. The coupling impedance and transmission coefficient for a ring are obtained in Section 3. In Section 4, we present the coupling impedance and transmission coefficient for the disk. In Section 5 we apply our results to the SIS-18 heavy ion synchrotron. Finally we present our conclusions in Section 6.

2. Model equations The general wave equations satisfied by the magnetic induction ~ B and electric field ~ E in a conducting medium of conductivity S, permittivity 0, and permeability m0 are obtained from Faradays and Amperes laws [12–14], namely, r 2~ Bð~ r; tÞ  m0 0

Bð~ r; tÞ q2~ q~ Bð~ r; tÞ ~  ~jð~ ¼ m0 r r; tÞ  m0 S 2 qt qt

(1)

r2~ Eð~ r; tÞ  m0 0

~ ~ q2~ q~ Eð~ r; tÞ q~ jð~ r; tÞ rrð r; tÞ Eð~ r; tÞ ¼ m0 þ  m0 S qt qt 0 qt 2

(2)

ARTICLE IN PRESS 172

A.M. Al-Khateeb et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 171–176

where r and ~ j are the external (free) charge and current densities, respectively, which obey the following continuity equation: qrð~ r; tÞ ~ ~ þ r  jð~ r; tÞ ¼ 0. qt

(3)

We assume a source term of total charge Q and transverse charge distribution sðr; yÞ is moving in a cylindrical pipe of radius b with constant longitudinal velocity ~ u ¼ bcz^ along the z axis. The charge and current densities are as follows: Z 2p Z a rð~ r; tÞ ¼ sðr; yÞdðz  vtÞ; Q ¼ sðr; yÞr dr dy (4) 0

(5)

For a uniformly charged disk, the surface charge density distribution in the transverse direction is sd ¼ Q =pa2 [2,15,16]. A thin ring of radius a moving parallel to the z-axis is represented by sr ¼ Q dða  rÞ=2pa [5]. Accordingly, the Fourier time-transformed charge and current densities for both distributions are Q eikz z ; pa2 bc

rr ðr; z; oÞ ¼

Q dða  rÞ ikz z e ; 2pabc

Q ~ jd ðr; z; oÞ ¼ 2 eikz z z^ pa ~j ðr; z; oÞ ¼ Q dða  rÞ eikz z z^ r 2pa

(6)

(7)

where o ¼ kz v has been used and kz is the wave number in the direction of beam propagation. Due to the axial symmetry of the particle beams under consideration, only transverse-magnetic (TM) cylindrical waveguide modes couple to the propagating beams such that Bz ¼ 0. All other field components are obtained from Ez ðr; z; oÞ via Maxwells equations, where Ey ðr; z; oÞ and Br ðr; z; oÞ vanish identically because of axial symmetry and periodicity demands Ez ðr; z; oÞ ¼ Ez ðr; oÞeikz z . Upon Fourier transforming Eq. (2) in time and in transverse space coordinates, and making use of the density and current from Eqs. (6) and (7), respectively, we get the following differential equations for the longitudinal electric field components in the region 0prpa: " # 2 2 d 1 d kz ðdÞ Q kz  (8) þ E ðr; oÞ ¼ i 2 pa 0 g20 bc dr 2 r dr g20 z "



 b m bcS þi 0 Er ðr; z; oÞ c o

(13)

where g in Eq. (12) stands for g0 in the non-conducting regions, and for g in the conducting wall region. In the following sections we will solve the wave equations for Ez for both types of beams and then find the corresponding expressions for the coupling impedance for a resistive beam-pipe of arbitrary wall thickness.

3. Longitudinal coupling impedance for a thin ring source

0

~ jð~ r; tÞ ¼ rð~ r; tÞ~ v ¼ sðr; yÞbcdðz  vtÞz^ .

rd ðr; z; oÞ ¼

By ðr; z; oÞ ¼

# 2 2 d 1 d kz ðrÞ Q kz E ðr; oÞ ¼ i  dða  rÞ þ 2pa 0 g20 bc dr 2 r dr g20 z

(9)

For a point charge Q moving down a beam-pipe with an offset a in the y0 ¼ 0 direction with a constant longitudinal velocity ~ u ¼ bcz^ , and in decomposing the corresponding charge and current densities in terms of multipole moments [7], the lowest monopole moment gives rr ð~ r; tÞ ¼

Q dða  rÞdðz  bctÞ 2pa

(14)

Q ~j ð~ dða  rÞdðz  bctÞbcz^ r r; tÞ ¼ 2pa

(15)

This monopole source has an axially symmetric transverse charge distribution and it represents an infinitesimally thin ring with radius a. Time Fourier-transformed charge and current densities in Eqs. (14) and (15) are already given in Eq. (7). For a thin ring beam moving inside a metallic cylindrical pipe of radius b and finite wall thickness d extending from r ¼ b to r ¼ b þ d with vacuum outside r4b þ d, the longitudinal electric field component EðrÞ z within each region of interest reads, " # 2 d 1d Q kz ðrÞ 2  s dða  rÞ; 0prpa (16) þ 0 Ez ðr; oÞ ¼ i 2pa 0 g20 bc dr 2 r dr "

# 2 d 1d ðrÞ 2  s þ 0 Ez ðr; oÞ ¼ 0; dr 2 r dr # 2 d 1d 2  þ s EzðrÞ ðr; oÞ ¼ 0; dr 2 r dr

aprpb

(17)

bprph ¼ b þ d

(18)

hpro1

(19)

"

"

# 2 d 1d ðrÞ 2  s þ 0 Ez ðr; oÞ ¼ 0; dr 2 r dr

2

where the relativistic factor g2 0 ¼ 1  b was introduced. In the beam-pipe region such that aprpb, which is free of charges, Eqs. (8) and (9) will be used to deduce the electric field in that region by setting their right-hand side to zero. However, within a conducting metallic wall of conductivity S which includes no bulk free charges, we obtain the following equations for the electric field, namely: " # 2 2 d 1 d kz ðd;rÞ  þ (10) E ðr; oÞ ¼ 0 dr 2 r dr g2 z !

2g2 1 1 m So 1 1 (11) ¼ i 02 ! ¼ 2 1  i 2 02 2 g2 g20 g g kz kz ds 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ds ¼ 2=m0 So is the skin depth at frequency o. For the TMmodes with azimuthal symmetry, the electromagnetic field components By ðr; z; oÞ and Er ðr; z; oÞ are needed for matching the solutions at the different interfaces involved in the problem under consideration and are obtained from Ez ðr; z; oÞ via the Maxwell equations as follows: Er ðr; z; oÞ ¼ i

g2 qEz ðr; z; oÞ qr kz

(12)

where we introduced s0 and s such that s0 ¼ kz =g0 and s ¼ kz = g. The overall regular general solution for the electric field EðrÞ z in each region is 8 r A I ðs rÞ; 0proa > > > 1r 0 0 > < A I0 ðs0 rÞ þ Ar K 0 ðs0 rÞ; aorpb 2 3 ðrÞ (20) Ez ðr; oÞ ¼ > bprph ¼ b þ d Ar I ðs rÞ þ Ar5 K 0 ðs rÞ; > > 4r 0 > : A K ðs rÞ; hpro1 6 0 0 where I0 and K 0 are the zero order modified cylindrical Bessel functions of first and second kinds, respectively. Integrating the differential Eq. (16) for EzðrÞ from r ¼ a  d to r ¼ a þ d for vanishingly small d we get the following boundary condition of the discontinuity of qEzðrÞ =qr at r ¼ a, namely: qErpa qErpa kz Q z .  z ¼i qr qr 0 g20 bc 2pa

(21)

From Eq. (21), the continuity of Ez at r ¼ a, and the continuity of Ez and By at r ¼ b and h, we obtain the following six algebraic equations: Ar1 I0 ðs0 aÞ ¼ Ar2 I0 ðs0 aÞ þ Ar3 K 0 ðs0 aÞ

(22)

ARTICLE IN PRESS A.M. Al-Khateeb et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 171–176

Ar2 I1 ðs0 aÞ  Ar3 K 1 ðs0 aÞ  Ar1 I1 ðs0 aÞ ¼ i Ar2 I0 ðs0 bÞ

þ

Ar3 K 0 ðs0 bÞ

¼

Ar4 I0 ðs bÞ

þ

1 Q  iQf 0 0 g0 bc 2pa

Ar5 K 0 ðs bÞ

(23) (24)

  Z Ar2 I1 ðs0 bÞ  Ar3 K 1 ðs0 bÞ ¼ Ar4 I1 ðs bÞ  Ar5 K 1 ðs bÞ

(25)

Ar4 I0 ðs hÞ

(26)

þ

Ar5 K 0 ðs hÞ

¼

Ar6 K 0 ðs0 hÞ

Ar4 I1 ðs hÞ  Ar5 K 1 ðs hÞ ¼ Ar6 ZK 1 ðs0 hÞ

(27)

where Z is defined as follows: Z¼

o0 g0 . i gðS  io0 Þ

(28)

The arbitrary constant A1 needed for the determination of the longitudinal electric field inside the beam region and for the calculation of the longitudinal impedance is obtained by simultaneously solving the system of Eqs. (22)–(27), namely, Ar1 ¼ i

Qkz ½I0 ðs0 aÞ þ HK 0 ðs0 aÞ 2p0 g20 bcH

I1 ðs hÞK 0 ðs0 hÞ þ ZK 1 ðs0 hÞI0 ðs hÞ F¼ K 1 ðs hÞK 0 ðs0 hÞ  ZK 1 ðs0 hÞK 0 ðs hÞ

(29)

(30)

1 I1 ðs bÞ  F K 1 ðs bÞ Z I0 ðs bÞ þ FK 0 ðs bÞ

(31)

H¼

I1 ðs0 bÞ  GI0 ðs0 bÞ . K 1 ðs0 bÞ þ GK 0 ðs0 bÞ

(32)

The longitudinal electric field in the region rpa, on the other hand, is Qkz ½I0 ðs0 aÞ þ HK 0 ðs0 aÞ 2p0 g20 bcH

I0 ðs0 rÞ eikz z .

In terms of G0 , Eq. (34) takes on the following form: Z k;r ðoÞ ¼ i

  nZ 0 2 K 0 ðs0 aÞ ZK 1 ðs0 bÞ  G0 K 0 ðs0 bÞ þ . I ðs aÞ 0 0 I0 ðs0 aÞ ZI1 ðs0 bÞ þ G0 I0 ðs0 bÞ g20 b

(36)

For a perfectly conducting wall we have tanh s d ! 1, Z ! 0, and G0 ! 1. Accordingly, Eq. (36) becomes a pure space-charge impedance associated with a thin ring source, namely, Z ðscÞ k;r ðoÞ ¼ i

  nZ 0 2 K 0 ðs0 aÞ K 0 ðs0 bÞ  . I ðs aÞ 0 I0 ðs0 aÞ I0 ðs0 bÞ g20 b 0

(37)

In the ultra-relativistic limit such that s0 a51 and s0 b51, we have I0 ðs0 aÞ ! 1, I0 ðs0 aÞ ! 1, K 0 ðxÞ ! ðln ðx=2Þ þ ge ÞI0 ðxÞ, where ge ¼ 0:5772 is the Euler constant. Accordingly, Eq. (37) takes on the following form:   nZ 0 b Z ðscÞ . (38) 2 ln k;r ðo; g ! 1Þ ¼ i a 2g20 b Using Eqs. (36) and (37), the resistive-wall impedance is ðscÞ Z ðrwÞ k;r ðoÞ ¼ Z k;h ðoÞ  Z k;h ðoÞ nZ 0 2 I ðs0 aÞ g20 b 0   K 0 ðs0 bÞ ZK 1 ðs0 bÞ  G0 K 0 ðs0 bÞ þ .  I0 ðs0 bÞ ZI1 ðs0 bÞ þ G0 I0 ðs0 bÞ

¼i

G¼

EzðrÞ ðr; z; oÞ ¼  i

173

(33)

(39)

For a perfectly conducting wall, Z ! 0 and G0 ! 1, the resistive-wall impedance of Eq. (39) vanishes identically. In the thick wall limit such that d ! 1 we get G0 ! 1 and, hence, Eq. (39) becomes   nZ 0 2 K 0 ðs0 bÞ ZK 1 ðs0 bÞ  K 0 ðs0 bÞ þ Z ðrwÞ k;r ðoÞ ¼ i 2 I0 ðs0 aÞ I0 ðs0 bÞ ZI1 ðs0 bÞ þ I0 ðs0 bÞ g0 b ¼i

nZ 0 I20 ðs0 aÞ Z . g20 bs0 b I20 ðs0 bÞ 1 þ ZI1 ðs0 bÞ=I0 ðs0 bÞ

(40)

For a thin ring source in a pipe the transmission coefficient defined as the ratio of transmitted to incident fields at r ¼ b is found to be exactly the same expression already reported for a disk source [2], namely,

FK 1 ðs hÞ  I1 ðs hÞ I0 ðs0 bÞ þ HK 0 ðs0 bÞ ZHK 1 ðs0 hÞ I0 ðs bÞ þ FK 0 ðs bÞ rffiffiffi b 1  h s0 bK 1 ðs0 hÞI0 ðs0 bÞ

tr ¼



1 . cosh s d þ ðK 0 ðs0 hÞÞ=ðZK 1 ðs0 hÞÞ sinh s d þ ðI1 ðs0 bÞÞ=ðI0 ðs0 bÞÞðZ sinh s d þ ðK 0 ðs0 hÞÞ=ðK 1 ðs0 hÞÞ cosh s dÞ

We now calculate the corresponding longitudinal coupling impedance as a volume integral over the transverse distribution of the beam as follows: 1

Z

3 d x0~ jr ðr 0 ; z; oÞ EðrÞ ðr 0 ; z; oÞ  ~ Q 2 V beam   nZ 0 K 0 ðs0 aÞ 1 þ ¼ i 2 I20 ðs0 aÞ I0 ðs0 aÞ H g0 b   nZ 0 2 K 0 ðs0 aÞ K 1 ðs0 bÞ þ GK 0 ðs0 bÞ ¼ i 2 I0 ðs0 aÞ þ . I0 ðs0 aÞ I1 ðs0 bÞ  GI0 ðs0 bÞ g0 b

Z k;r ðoÞ ¼ 

(41)

4. Longitudinal coupling impedance for a uniform disk This section presents a review of the calculations of the coupling impedance and the transmission coefficient of a cylindrical pipe of finite wall thickness d for a uniform disk source. The results below are found in Ref. [2]. For TM modes in azimuthally symmetric beam-pipe structures, the general solution for the z-component of the electric field in the four regions involved is [2]

For the case of a very well conducting beam-pipe wall of finite thickness and for ds kz 51, we have the following expressions for G [2]:

8 Q > > Ad1 I0 ðs0 rÞ  i 2 ; > > > pa  0 kz bc > > < d d A2 I0 ðs0 rÞ þ A3 K 0 ðs0 rÞ; EðdÞ z ðr; oÞ ¼ > > Ad I ðs rÞ þ Ad K ðs rÞ; > > 4 0 5 0 > > > d : A6 K 0 ðs0 rÞ;

1 G   G0 ; Z

Requiring the continuity of the tangential field components EðdÞ z and By at r ¼ a, r ¼ b, and r ¼ h ¼ b þ d, we obtain the following

(34)

G0 ¼

K 0 ðs0 hÞ tanh s d þ ZK 1 ðs0 hÞ . K 0 ðs0 hÞ þ ZK 1 ðs0 hÞ tanh s d

(35)

0prpa aprpb

(42)

bprph ¼ b þ d hpro1:

ARTICLE IN PRESS 174

A.M. Al-Khateeb et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 171–176

For a thick wall such that d ! 1 we have G0 ! 1, and therefore, Eq. (52) takes on the following form:

coefficients [2]: Ad1 ¼ i

Q s0 a ½I1 ðs0 aÞ  HK 1 ðs0 aÞ pa2 0 kz bcH

(43) Z ðrwÞ ðoÞ ¼ i k;d

F¼

I1 ðs hÞK 0 ðs0 hÞ þ ZK 1 ðs0 hÞI0 ðs hÞ K 1 ðs hÞK 0 ðs0 hÞ  ZK 1 ðs0 hÞK 0 ðs hÞ

(44)

G¼

1 I1 ðs bÞ  FK 1 ðs bÞ ; Z I0 ðs bÞ þ FK 0 ðs bÞ

(45)

H¼

I1 ðs0 bÞ  GI0 ðs0 bÞ . K 1 ðs0 bÞ þ GK 0 ðs0 bÞ

Note that the parameters F, G, and H above are exactly those which appear in the thin ring expression for Ar1 of Eq. (29). We see that the main difference between the ring and the disk manifests itself through the different mathematical expressions of Ar1 and Ad1 , or equivalently in the electric field expressions within the beam region rpa. The longitudinal electric field in the region rpa is EðdÞ z ðr; z; oÞ ¼ 

iQ eikz z I0 ðs0 rÞ ½I1 ðs0 aÞ  HK 1 ðr s0 aÞ pa0 g0 bcH

iQ eikz z .  2 pa 0 kz bc

(46)

   nZ 0 4g20 1 1  2I1 ðs0 aÞ K 1 ðs0 aÞ  I1 ðs0 aÞ . 2 2 H 2bg kz a2

(47)

The total impedance of Eq. (47) can be written as a superposition of space-charge and resistive-wall parts as follows: ðoÞ þ Z ðrwÞ ðoÞ Z k;d ðoÞ ¼ Z ðscÞ k;d j;d

ðscÞ Z k;d ðoÞ ¼ i

(48)

   nZ 0 4g20 K 1 ðs0 aÞ K 0 ðs0 bÞ 2 þ 1  2I ðs aÞ 0 1 I1 ðs0 aÞ I0 ðs0 bÞ 2bg20 k2z a2

(49)

  nZ 0 8g20 2 K 0 ðs0 bÞ 1 þ I1 ðs0 aÞ 2 2 I0 ðs0 bÞ H 2bg0 kz a2

(50)

Z ðrwÞ ðoÞ ¼ i k;d

where Z ðscÞ ðoÞ is the space-charge impedance and Z ðrwÞ ðoÞ is the k;d k;d resistive-wall impedance. For s0 a51 and s0 b51, the space-charge impedance of a uniform beam in Eq. (49) takes on the following form: nZ 0 4g20 2bg20 k2z a2  nZ 0 i 2 ln 2bg20

Z ðscÞ ðoÞ ¼ i k;d

   K 1 ðs0 aÞ K 0 ðs0 bÞ þ 1  2I21 ðs0 aÞ I1 ðs0 aÞ I0 ðs0 bÞ  b a

If we compare Eq. (53) with (40), we see that the resistive-wall impedances for a thick wall have similar structures. Differences appear only in the factor I20 ðs0 aÞ in Eq. (40) which is replaced by the factor 4I21 ðs0 aÞ=s20 a2 in Eq. (53). Again, in the limit s0 a51 and s0 b51, both resistive-wall expressions become identical. For a uniform disk the transmission coefficient td d is found to be exactly equal to the thin ring transmission coefficient tr . According to Gauss’ integral law, as expected, integral quantities like resistive-wall impedance and transmission factor outside the sources of charge depend only on the total charge and not on the details of the charge distribution within the beam.

To visualize possible differences between impedances from the different source terms, the theory above will be applied to the heavy ion synchrotron SIS-18 at GSI-Darmstadt with a ring circumference of L ¼ 216 m. It has a stainless steel wall at radius b ¼ 10 cm of thickness d  0:3 mm with a conductivity of S ¼ 1:1  106 =O m. In Fig. 1, we plot the space-charge impedances of Eq. (37) and (49) vs. beam velocity b at revolution frequency o0 and beam radius a ¼ 5 cm. The figure show that the impedances from the thin ring source are always lower than those from the uniform disk. This difference is shown in Fig. 2. At the beam reference velocity b ¼ 0:155 and for the design ratio b=a ¼ 2 of SIS-18, we see an impedance difference of about 600 O which corresponds to 25% of the space-charge impedance at this energy. In Fig. 3, we plot the space-charge impedances per harmonic number Z k ðoÞ=n vs. mode frequency f ¼ o=2p, where o ¼ no0 , o0 ¼ bc=R, and n is the harmonic number. Here we fixed the beam velocity at b ¼ 0:155 with a beam size a ¼ 5 cm. For all frequencies, the figure shows an obvious difference between the space-charge impedances resulting from the two source terms. This impedance difference is plotted in Fig. 4. Approximately up to 50 MHz the difference is about 600 O and is nearly constant. It becomes smaller by increasing the frequency until it becomes zero at about 300 MHz. For the higher frequencies above 300 MHz, the thin ring impedance becomes larger than the space-charge

(51)

uniform ring

where the following expansions of I0;1 and K 0;1 for small arguments have been used: K 0 ðxÞ   ln

x  ge , 2

Note that, according to the results in Eqs. (38) and (51), both source terms produce asymptotically the same space-charge impedance in the so-called ultra-relativistic limit. Upon substituting for H, the resistive-wall impedance of Eq. (50) can be written in terms of G0 as follows [2]: ðrwÞ ðoÞ ¼ i Z k;d

¼i

102

10

1

nZ 0 8g20 I21 ðs0 aÞ 1 2bg20 k2z a2 s0 bI0 ðs0 bÞ I1 ðs0 bÞ  GI0 ðs0 bÞ nZ 0 8g20 I21 ðs0 aÞ Z . 2bg20 k2z a2 s0 bI0 ðs0 bÞ ZI1 ðs0 bÞ þ G0 I0 ðs0 bÞ

103 Im (Zsc || (ω0)) [Ω]

1 x  K 0 ðxÞI1 ðxÞ; I1 ðxÞ  ; x 2 I0 ðxÞ  1. K 1 ðxÞ 

(53)

5. Numerical examples

The corresponding longitudinal coupling impedance is [2] Z k;d ðoÞ ¼ i

nZ 0 4I21 ðs0 aÞ Z . bg20 s0 b s20 a2 I20 ðs0 bÞ 1 þ ZI1 ðs0 bÞ=I0 ðs0 bÞ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

 (52)

Fig. 1. Space-charge impedance as a function of beam velocity b at the revolution frequency, beam radius 5 cm.

ARTICLE IN PRESS A.M. Al-Khateeb et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 171–176

175

9000

2500

uniform ring

8000 7000 Im (Z ||sc (ω0)) [Ω]

Im (Zdsc − Zsc r )[Ω]

2000

1500

1000

6000 5000 4000 3000 2000

500 1000

0.1

0.2

0.3

0.4

0.5 

0.6

0.7

0.8

0.05

0.9

0.1

0.15

0.2

0.25

0.3 0.35 a/b

0.4

0.45

0.5

0.55

Fig. 5. Space-charge impedance as a function of beam radius a. Fig. 2. Impedance difference as a function of beam velocity b.

3000 uniform ring

104

 = 0.155  = 0.5  = 0.95

2000

103 Im (Zdsc−Z rsc ) [Ω]

Im (Z sc || (ω) /n) [Ω]

2500

1500 1000 500

102

10

0 10-1

1

102

10

103

f [MHz]

1 0.1

0.15

0.2

0.25

Fig. 3. Space-charge impedance as a function of frequency f.

0.3 0.35 a/b

0.4

0.45

0.5

0.55

Fig. 6. Impedance difference as a function of beam radius a.

1000 800

Interestingly, Fig. 6 shows that the variation of the impedance difference with beam size is constant for fixed beam energy. At the beam reference velocity b ¼ 0:155, the impedance difference is again about 25% of the disk impedance at the same beam energy, as also can be read from Fig. 2.

Im (Zdsc−Zrsc) [Ω]

600 400 200

6. Summary and outlook

0 -200 -400 10-1

1

10 f [MHz]

102

103

Fig. 4. Impedance difference as a function of frequency f.

impedance from the uniform disk. Variation of the SIS-18 spacecharge impedance with beam size at revolution frequency is shown in Fig. 5. We see that the space-charge impedance from the uniform disk is greater than the thin ring source impedance.

In this work we investigated the longitudinal coupling impedances and transmission coefficients from a thin ring source term and from a uniformly charged disk propagating down a resistive cylindrical beam-pipe of finite wall thickness. Results are mainly compared with the corresponding expression of Ref. [2]. Some interesting limiting cases of space-charge and resistive-wall impedances from the two source terms have also been obtained and discussed. Transmission coefficients from both source terms are exactly the same. Differences, however, do appear in the electromagnetic fields within the sources for rpa, and therefore leading to different coupling impedances. For the SIS-18 synchrotron of GSI-Darmstadt it is found that the space-charge impedance from the thin ring source deviates appreciably from the disk formula for

ARTICLE IN PRESS 176

A.M. Al-Khateeb et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 171–176

a wide range of beam energies and mode frequencies. Although their mathematical expressions are different, resistive-wall impedances from the two source terms are found to be numerically equal. The space-charge impedances become equal asymptotically only in the so-called ultra-relativistic limit. The space-charge impedance is caused by the beam distribution only, while the resistive-wall impedance is as well influenced by the wall characteristics outside the sources. These findings, hence, are expected because according to Gauss’ integral law integral quantities like resistive-wall impedance and transmission factor outside the sources of charge depend only on the total charge and not on the details of the charge distribution within the beam.

Acknowledgments A. Al-Khateeb would like to thank the Arab Fund for Economic and Social Development (AFESD, State of Kuwait) for the financial support of his stay at the GSI-Darmstadt (Germany) via their Fellowship Award. He also thanks the FAIR-accelerator theory group of GSI-Darmstadt for its hospitality.

References [1] F. Zimmermann, K. Oide, Phys. Rev. ST Accel. Beams 7 (2004) 044201. [2] A.M. Al-Khateeb, O. Boine-Frankenheim, R.W. Hasse, I. Hofmann, Phys. Rev. E 71 (2005) 026501. [3] An International Accelerator Facility for Beams of Ions and Antiprotons: Conceptual Design Report hhttp://www.gsi.de/GSI-Future/cdr/i. [4] A.M. Al-Khateeb, O. Boine-Frankenheim, I. Hofmann, G. Rumolo, Phys. Rev. E 63 (2001) 026503. [5] R.L. Gluckstern, CERN Yellow Report, CERN 2000-011, 2000. [6] N. Wang, Q. Qin, Phys. Rev. ST Accel. Beams 10 (2007) 111003. [7] A.W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley, New York, 1993. [8] M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994. [9] S.Y. Lee, Accelerator Physics, World Scientific, Singapore, 1999. [10] J.G. Wang, H. Suk, D.X. Wang, M. Reiser, Phys. Rev. Lett. 72 (1994) 2029. [11] R. L. Warnock, Proceedings of 1993. Particle Accelerator Conference, IEEE, Washington, DC, 1993, p. 3378. [12] R.E. Collin, Field Theory of Guided Waves, McGraw-Hill, NY, 1960. [13] R.E. Collin, Foundations of Microwave Engineering, second ed., McGraw-Hill, NY, 1992, pp. 108–111. [14] D.M. Pozar, Microwave Engineering, Addison-Wesley, Reading, MA, 1990 (Chapter 5). [15] B.W. Zotter, S.A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators, World Scientific, Singapore, 1998 (Chapter 6). [16] A.M. Al-Khateeb, R.W. Hasse, O. Boine-Frankenheim, I. Hofmann, Nucl. Instr. and Meth. Phys. Res. A 587 (2008) 236.