Space charge fields in azimuthally symmetric beams: integrated Green's function approach
aa r X i v : . [ phy s i c s . acc - ph ] F e b Space charge fields in azimuthally symmetric beams: integrated Green’s functionapproach
Petr M. Anisimov ∗ and Nikolai A. Yampolsky Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA (Dated: February 3, 2021)Electromagnetic fields induced by the space charge in relativistic beams play an important rolein Accelerator Physics. They lead to emittance growth, slice energy change, and the microbunchinginstability. Typically, these effects are modeled numerically since simple description exists only inthe limits of large- or small-scale current variations. In this paper we consider an axially symmetriccharged beam inside a round pipe and find the solution of the space charge problem that is validin the full range of current variations. We express the solution for the field components in termsof Green’s functions, which are fully determined by just a single function. We then find thatthis function is an on-axis potential from a charged disk in a round pipe, with transverse chargedensity ρ ⊥ ( r ), and it has a compact analythical expression. We finally provide an integrated Green’sfunction based approach for efficient numerical evaluation in the case when the transverse chargedensity stays the same along the beam. INTRODUCTION
The space charge effect is a basic collective phe-nomenon in Accelerator Physics that plays an impor-tant role both in electron and proton machines. In highcurrent regimes, self-generated electromagnetic fields be-come so strong that they lead to significant emittancegrowth [8, 11, 13, 15], slice energy change, and the mi-crobunching instability [12], which is especially detrimen-tal in the context of free electron laser linacs [5, 12]. Inthe case of x-ray free electron lasers with periodic en-hancement of electron peak current [14, 22, 24], stronglongitudinal current modulation may result in large spacecharge forces which, in turn, may limit the performanceof these schemes.Compact analytical expressions for the space chargeinduced fields are currently available in the limits of ei-ther large- or small-scale current variations. The firstlimit provides a local description [7] while the secondlimit requires an impedance based description [20] thatis non-local as it depends on the Fourier spectrum of thecurrent. This latter approach allows for semi-analyticaltreatment and is implemented in the Elegant numericalcode [4] in order to include longitudinal space charge ef-fects on a beam axis.An alternative approach adopted in many numericalcodes, such as OPAL [2], Astra [9], and Parmela [3]. Thisapproach uses a Poison solver in order to find the spacecharge induced fields. However it is very time consum-ing and a semi-analytic description for the space chargeinduced fields applicable in the full range of current vari-ations is highly desired.In this paper we provide a semi-analytical descriptionof the space charge problem in the case of an axially sym-metric beam in a round pipe. The derived expressionsfor components of the induced field are valid in the fullrange of current variations. They also can be efficientlyevaluated by the method of Integrated Green’s functions b adz
FIG. 1. An axially symmetric electron beam in a perfectlyconducting pipe of radius a . A disk section of length dz andthe transverese charge density ρ ⊥ ( r, z ), where z is a coordi-nate along the beam in the beam reference frame, is shown. (IGF).We present our approach in Section , where we findthat the Green’s function for a charged disk in a roundpipe fully determines the components of the inducedfields. In Section , we express the Green’s function interms of an on-axis potential from a charged disk in around pipe and find a compact analytical approximationfor this potential. Section uses the compact analyticalexpression for the Green’s function in order to present thefield components in a form suitable for the IGF approach.In Section we suggest how to improve a semi-analyticaldescription of the space charge fields in numerical codessuch as Elegant by providing a step-by-step instructionsof IGF approach with our Green’s function. We finallysummarize our findings in the conclusion. BASIC EQUATIONS FOR THE SPACE CHARGEFIELDS
The most general approach considers an electron beamwith charge density ρ ( r ) that travels on axis of a perfectlyconducting pipe of radius a at a speed v z (see Fig. 1).It starts in the beam frame, R , where the electrostaticpotential induced by the space charge, ̺ ( R ), is a solutionof the Poisson equation:Φ( R ) = 14 πε Z ̺ ( R ′ )Γ( R | R ′ ) d R ′ , (1)where Γ( R | R ′ ) is the Green’s function for the Laplace’sequation inside a perfectly conducting pipe. The corre-sponding Green’s function is derived in Appendix as anexpansion in terms of radial-azimuthal eignenfunctionsthat can be used in systems without translational sym-metry.In this paper, we will consider axially symmetric den-sities in the lab frame, r , of the form ρ ( r ) = ρ ⊥ ( r ) λ ( z )with normalization R ρ ⊥ ( r ′ ) d r ′ = 1. In this case, theelectrostatic potential in the beam frame is expressed asΦ( R, Z ) = 14 πε Z ∞−∞ Λ( Z ′ ) G ( R, Z − Z ′ ) dZ ′ , (2)where Λ( Z ′ ) = γ − λ ( Z ′ /γ ) and G ( R, Z − Z ′ ) = ∞ X n =1 c n J ( µ ,n R/a ) aµ ,n J ( µ ,n ) e − µ ,n | Z − Z ′ | /a , (3)with c n = R ρ ⊥ ( r ′ ) J ( µ ,n r ′ /a ) d r ′ that has to be indi-vidually calculated for different transverse density pro-files. In the particular case of c n = 1, one recovers aresult for an electrostatic potential of a point charge onthe axis of a perfectly conducting pipe (see Eq. (27) inRef. [6]).The electrostatic potential in the beam reference frameresults in the electric field with longitudinal and trans-verse components that have the following expressions inthe lab frame E z ( r, z ) = − πε γ Z ∞−∞ λ ( z ′ ) ∂∂z g ( r, z − z ′ ) dz ′ , (4) E r ( r, z ) = − γ πε Z ∞−∞ λ ( z ′ ) ∂∂r g ( r, z − z ′ ) dz ′ , (5)in terms of transformed to the lab frame Green’s func-tion, g ( r, z − z ′ ) = G ( r, γz − γz ′ ). Additionally, inthe lab frame, there is also an azimuthal magnetic field B φ ( r, z ) = v z c E r ( r, z ), which reduces the overall trans-verse space charge force acting on charged particles inthe beam by γ − . GREEN’S FUNCTION ANALYSIS
In the previous section, we have reduced the problemof space charge induced fields to the finding Green’s func-tion, G ( R, Z − Z ′ ), — the electrostatic potential from acharge distribution, ̺ ( R ) = ρ ⊥ ( R ) δ ( Z − Z ′ ). Due to theaxial symmetry of the charge distribution, one can use G H , Z - Z ' L - - - - È Z - Z' È(cid:144) a FIG. 2. (Color online) An on-axis Green’s function, G (0 , Z − Z ′ ), in case of uniform transverse distribution for differentvalues of b/a = 0.1 (dashed red), 0.01 (dot dashed green) and0.001 (solid blue). The short-range behavior, | Z − Z ′ | ≪ a ,corresponds to the potential from a uniformly charged disk ofradius b in free space. As distance increases, | Z − Z ′ | ≫ b ,the on-axis Green’s function becomes the potential of a pointcharge in a pipe. Transformation to the lab frame substitutes Z → γz thus scaling the transitions points to a/γ and b/γ correspondingly. the Poisson representation [18] in order to the potentialoutside the axis of symmetry once the potential on theaxis has been found: G ( R, Z − Z ′ ) = 12 π Z π G (0 , Z − Z ′ + iR cos φ ) dφ, Z > Z ′ , (6)where G (0 , Z − Z ′ ) = ∞ X n =1 c n aµ ,n J ( µ ,n ) e − µ ,n | Z − Z ′ | /a . (7)Sometimes, however, the integral cannot be taken analyt-ically or potential has to be found near the axis. There-fore, we propose an alternative approach here based onthe formal separation of variables: G ( R, Z − Z ′ ) ≡ J (cid:18) R ∂∂Z (cid:19) G (0 , Z − Z ′ ) , Z = Z ′ , (8)where a function of a derivative is defined via its Tay-lor series expansion [16]. As a consequence of our ap-proach, one can confirm, based on a formal substitu-tion x → R ∂∂Z into the Bessel differential equation, (cid:16) x d dx + x ddx + x (cid:17) J ( x ) = 0, that the derived Green’sfunction is indeed a solution of the Laplace equation, (cid:16) ∂ ∂R + R ∂∂R + ∂ ∂Z (cid:17) G ( R, Z − Z ′ ) = 0.As the transformed Green’s function is g ( r, z − z ′ ) = G ( r, γz − γz ′ ), we have just reduced the space chargeproblem to finding electrostatic potential on the pipeaxis, G (0 , Z − Z ′ ), which is shown in Fig. 2, in the caseof a uniform transverse distribution, ρ ⊥ ( r < b ) = 1 /πb , G H , Z - Z ' L à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à - - È Z - Z' È(cid:144) a FIG. 3. (Color online) The on-axis Green’s function, G (0 , z − z ′ ) in the case of uniform transverse charge density with b/a = 0 .
01 (red squares). The near (dashed blue) and far(dot-dashed green) zone contributions are plotted separately.The boundaries of these zones | Z − Z ′ | = µ − , a and | Z − Z ′ | = b are marked by the vertical dashed lines. Transformation tothe lab frame substitutes Z → γz thus scaling the transitionspoints to µ − , a/γ and b/γ correspondingly. for different values of b/a . It shows that there are twodistinct regions with different behaviors of the Green’sfunction. The first region corresponds to the far zone, | Z − Z ′ | ≫ b , where the Green’s function is just a po-tential of a point charge in a pipe. The second regioncorresponds to the vicinity of the source, | Z − Z ′ | ≪ a ,in which case the free space approximation can be used.Both of these regions allow for accurate and compactanalytical expressions for G (0 , Z − Z ′ ). Furthermore,we can be combined these expressions together in a sin-gle analytical expression due to a broad overlap region, b ≪ | Z − Z ′ | ≪ a , which corresponds to a point chargein a free space.In the first region, | Z − Z ′ | ≫ b , the sum in Eq. (7)converges rapidly due to vanishing exponential factorsexp( − µ ,n | Z − Z ′ | /a ). The actual number of terms con-tributing to the sum can be estimated as µ ,n | Z − Z ′ | /a ∼
1. Under this condition, c n ≃
1, resulting in the expres-sion for the on-axis potential from a point charge on thepipe axis [6]. A compact form representation for the on-axis Green’s function in the far zone is hence a sum ofthe geometric series: G (0 , Z − Z ′ ) ≈ a πe − µ , | Z − Z ′ | /a − e − π | Z − Z ′ | /a , | Z − Z ′ | ≫ b, (9)where we have used that µ ,n − µ , ≈ π ( n −
1) and µ ,n J ( µ ,n ) ≈ /π .The second region is in the vicinity of the source, | Z − Z ′ | ≪ a . Hence, one can ignore a boundary ef-fect of the pipe surface but has to account for an actualtransverse charge density, ρ ⊥ ( r ). In the absence of theboundary effect the convergence of the sum in Eq. (7) is G H , Z - Z ' L à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à D H % L - - - È Z - Z' È(cid:144) a - - - È Z - Z' È(cid:144) a FIG. 4. (Color online) The on-axis Green’s function, redsquares, and its analythical approximation, blue line, for theuniform transverse charge density with b/a = 0 .
01. The insetshows the relative error, ∆, between the exact and analyti-cal representations of the on-axis response function. The reddashed line corresponds to b/a = 0 .
1, while the green dot-dashed and solid orange lines correspond to b/a = 0 .
05 and b/a = 0 .
01 respectively. defined by c n rather than by the exponent. Hence, onecan replace summation with integration. We chose theintegration variable to be x = µ ,n with the Jacobian ofthis transformation dn/dx ≃ /π : G (0 , Z − Z ′ ) ≈ a Z d r ′ ρ ⊥ ( r ′ ) Z ∞ J ( xr ′ /a ) e − x | Z − Z ′ | /a dx. (10)Carrying out one integration results in the following com-pact form representation of the on-axis Green’s function G (0 , Z − Z ′ ) ≈ Z d r ′ ρ ⊥ ( r ′ ) p ( Z − Z ′ ) + r ′ , | Z − Z ′ | ≪ a, (11)which is indeed the on-axis potential from a charged diskin free space. The Table I provides examples of this po-tential for different transverse charge distributions.So far, we have found two analytical representations forthe on-axis Green’s function that describe two physicallimits. Figure 3 compares these approximate representa-tions given by Eqs. (9) and (11) with values of the on-axisGreen’s function given in Eq. (7). It shows that approx-imate expressions describe the on-axis Green’s functionwell within their applicability regions. We note that thereis an overlap region where two physical limits have com-mon domain of mutual applicability, b ≪ | Z − Z ′ | ≪ a .This region is well defined when the beam size is suffi-ciently smaller than the pipe radius, b < a , and corre-spond to the case where the charge distribution can bealready approximated by a point charge yet the presenceof pipe walls can be still ignored. In this region, bothexpressions result in the same asymptotic form for theon-axis Green’s function and coincide with the expres-sion for the point charge potential in free space: G (0 , Z − Z ′ ) = 1 | Z − Z ′ | , b ≪ | Z − Z ′ | ≪ a. (12)This allows matching of the two analytical representa- tions as their product divided by their mutual asymptoticexpression. As a result, the on-axis Green’s function ina perfectly conducting pipe can be approximated with: G (0 , Z − Z ′ ) ≈ | Z − Z ′ | a πe − µ , | Z − Z ′ | /a − e − π | Z − Z ′ | /a Z d r ′ ρ ⊥ ( r ′ ) p ( Z − Z ′ ) + r ′ , (13) ρ ⊥ ( r ) b R d r ′ ρ ⊥ ( r ′ ) √ ( Z − Z ′ ) + r ′ πb , r ≤ b x rms b | ∆ Z | + √ ∆ Z +12 πb (cid:16) − r b (cid:17) , r ≤ b √ x rms b Z / | ∆ Z | + ( ∆ Z +1 ) / +3 | ∆ Z | / πb e − r /b , r ≤ ∞ √ x rms √ πb Erfc ( | ∆ Z | ) e ∆ Z TABLE I. The transverse density distributions with their rmssizes and the corresponding on-axis Green’s functions in thevicinity of the sources with ∆ Z = ( Z − Z ′ ) /b . for b < a and | Z − Z ′ | ∈ (0 , ∞ ).The main advantage of using the compact form ex-pression is its simplicity. In contrast, evaluation of anexact Green’s function described with Eq. (7) requiressummation of a large number of terms and takes a sig-nificant amount of time. Figure 4 shows the comparisonbetween these two approaches for the uniform transversecharge distribution with b/a = 0 .
01. The inset showsthe relative error of using the approximate Green’s func-tion instead of exact. The absolute error is the largest at Z = Z ′ but does not exceed 0 . /a based on a simplemethod of summing Bessel series proposed by Greenwoodin Ref. [10]. THE INTEGRATED GREEN’S FUNCTIONAPPROACH
In the previous section, we have provided a compactform for the Green’s function that describes an axiallysymmetric space charge problem. Using this result, thelongitudinal space charge field takes on the following formfor the translationally invariant transverse distributions E z ( r, z ) = 14 πε γ Z ∞−∞ λ ( z ′ ) ∂∂z ′ g z ( r, z − z ′ ) dz ′ , (14)with g z ( r, z − z ′ ) = J (cid:16) rγ ∂∂z (cid:17) g (0 , z − z ′ ). The opera-tional form for the Green’s function suggested in Eq. 8also allows for a similar representation for the transverse component of the space charge induced field E r ( r, z ) = 14 πε Z ∞−∞ λ ( z ′ ) ∂∂z ′ g r ( r, z − z ′ ) dz ′ , (15)with g r ( r, z − z ′ ) = − J (cid:16) rγ ∂∂z (cid:17) g (0 , z − z ′ ) [17]. This solu-tion for the space charge problem allows for efficient nu-merical evaluation based on the Integrated Green’s func-tion approach [1, 19, 21].For the purposes of providing a recipe for numericalevaluation of the Eqs. 14 and 15, we will use the Inte-grated Green’s function approach in a constant functionbasis. This approach approximates the charge density as λ ( z ′ ) = ( λ j +1 + λ j ) / z ′ ∈ [ z j , z j +1 ] for z j = jh , and arrives to thefollowing discrete convolution E r,z ( z i ) = hA r,z X j λ j w i − jr,z , (16)with normalization constants A r = 1 / πε and A z = A r /γ .The Green’s functions derived in this paper behavedifferently near the source. Namely, the longitudinalGreen’s function, g z ( r, z − z ′ ), is a continuous function of z , while the radial Green’s function, g r ( r, z − z ′ ), has adiscontinuity at z = z ′ . Taking this into account, leadsto the following integrated Green’s function for the lon-gitudinal component w z ( ζ ) = g z ( r, ζ − h ) − g z ( r, ζ + h )2 h , (17)with ζ = h ( i − j ); and for the radial component: w r ( ζ ) = g r ( r, ζ − h ) − g r ( r, ζ + h )2 h + 1 h g r ( r, − ) δ ζ, ( − h, ,h ) , (18)where w r ( h ) uses the Green’s function defined for z ≥ z ′ in order to evaluate g r ( r, w r ( − h ) and g r ( r, − )use z ≤ z ′ definition. DISCUSSION
Equation 13 is the main result of the paper that hasallowed us to express the space charge induced fields interms of the Green’s functions that have simple analyth-ical representation. Equations 14 and 15 with the corre-sponding Green’s function are the next important resultof this paper. These expressions for the components ofthe field can be efficiently evaluated using the IntegratedGreen’s functions, Eqs. 17 and 18.It is often a case that the transverse beam size is asmallest scale of the problem and that only the spacecharge induced fields within the beam are of interest.Thus, the particle accelerator codes, similar to Elegant,include only the effects of the space charge induced fieldsat r = 0. X-ray free electron lasers often operate ina space charge dominated regime and a more diligenttreatment of the space charge induced fields is in order.In what follows we will illustrate the steps required toobtain the off-axis behavior for the components of thespace charge induces field.The radial component of the space charge inducedfield, evaluated on z i = ih grid, is equal to E r ( z i ) = hA r P j λ j w i − jr , where λ j is a space charge distributionon the same grid and A r = 1 / πε . Equation 18 definesthe integrated Green’s function, w r ( ζ ), in terms of g r ( r, z − z ′ ) = − r ∂γ∂z g (0 , z − z ′ ) , (19)where we have kept the leading term in the Taylor seriesexpansion of J ( x ) for the purpose of illustration only.In order to evaluate the derivation of the on-axis Green’sfunction, we can apply the logic used in deriving the an-alythical representation of the on-axis Green’s functionitself and arrive to the following expression: ∂∂Z G (0 , Z − Z ′ ) ≈ | Z − Z ′ | a ∂∂Z πe − µ , | Z − Z ′ | /a − e − π | Z − Z ′ | /a ! ×× Z d r ′ ρ ⊥ ( r ′ )[( Z − Z ′ ) + r ′ ] / . (20)Evaluation of the longitudinal component of the spacecharge induced field follows the same steps but uses A z = A r /γ and w z ( ζ ) as defined in Eq. 17 in terms of g z ( r, z − z ′ ) = (cid:18) − r ∂ γ ∂z (cid:19) g (0 , z − z ′ ) , (21)where we have kept the first two leading terms in the Tay-lor series expansion of J ( x ) for the purpose of illustrationonly. The second term requires the second derivative ofthe on-axis Green’s function that has the following form: ∂ ∂Z G (0 , Z − Z ′ ) ≈ | Z − Z ′ | a ∂ ∂Z πe − µ , | Z − Z ′ | /a − e − π | Z − Z ′ | /a ! ×× Z d r ′ ρ ⊥ ( r ′ ) (cid:2) Z − Z ′ ) − r ′ (cid:3) [( Z − Z ′ ) + r ′ ] / . (22)From the provided illustration, the solution of thespace charge problem at r = 0 requires a single evaluation of a discreet convolution. An additional convolution pro-vides a linear contribution to the radial component of thespace charge induced field. The third convolution can bealso carried out in order to find a next order correction tothe longitudinal component of the space charge inducedfield. One can apply the illustrated approach in order tofind the next order corrections with computational effortthat scales linearly with number of corrections. CONCLUSIONS
We have studied the space charge problem for the beamwith axially symmetric charge distribution in a smoothperfectly conducting pipe. We have develop a Green’sfunction-based description for the space charge inducedfields that is valid in the full range of current variationsand applicable to beams with varying beam radius. TheGreen’s function discussed in the paper corresponds tothe electrostatic potential of a charged disk in a pipe andis completely defined by its behavior on the pipe axis dueto axial symmetry.We have found a compact analytical approximation forthe on-axis Green’s function that allows for analyticalcalculation of the Green’s function off axis. Having acompact representation of the on-axis behavior of the re-sponse function can improve semi-analytical codes, suchas Elegant, as it offers a significant advantage over nu-merical evaluation of the exact solution.Finally, we have provided a detail prescription basedon the Integrated Green’s function approach for efficientnumerical evaluation of the fields in the case of transla-tional symmetry. This approach describes transverse aswell as longitudinal component of the space charge in-duced field and scales linearly with the number of radialcorrections.
ACKNOWLEDGEMENTS
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Green’s function
The Green’s function for the Laplace’s equation insidea perfectly conducting pipe must satisfy the followingequation ∇ Γ( R | R ′ ) = − πδ ( R − R ′ ) (23)with the boundary condition Γ( R = a ) = 0 and Γ( Z →±∞ ) = 0.To find a Green’s function, it is common to expand theGreen’s function in terms of axial eigenfunction solutionsof the Laplace’s problem in cylindrical coordinates, e ikZ .Here, however, the radial eigenfunction expansion is usedin order to allow for beam size variation along the beam.ThusΓ( R | R ′ ) = ∞ X m = −∞ ∞ X n =1 G m,n ( Z | R ′ ) J m ( µ m,n R/a ) e im Φ , (24)where J m is the ordinary Bessel’s function of order m and µ m,n is defined as a solution of J m ( µ m,n ) = 0. Sub-stituting this expansion in 23 we find that ∇ Γ( R | R ′ ) = ∞ X m = −∞ ∞ X n =1 (cid:20) d G m,n ( Z | R ′ ) dZ −− µ m,n a G m,n ( Z | R ′ ) J m ( µ m,n R/a ) e im Φ . (25)With the help of the orthogonality properties for theBessel’s function, we can now obtain that d G m,n ( Z | R ′ ) dZ − µ m,n a G m,n ( Z | R ′ ) = − a J m +1 ( µ m,n ) J m ( µ m,n R ′ /a ) e − im Φ ′ δ ( Z − Z ′ ) . (26)Then, if we look at the points where z = z ′ , (26) be-comes a simple eigen function problem with a well knownsolution G m,n ( Z | R ′ ) = g m,n ( R ′ , Φ ′ ) e − µ m,n | Z − Z ′ | /a . (27)The condition for the discontinuity of the first derivativeat Z = Z ′ yields that g m,n ( R ′ , Φ ′ ) = 2 aµ m,n J m +1 ( µ m,n ) J m ( µ m,n R ′ /a ) e − im Φ ′ . (28)Finally, the Green’s function for axially symmetriccharge distributions isΓ( R, Z | R ′ , Z ′ ) = ∞ X n =1 aµ ,n J ( µ ,n ) × (29) × J ( µ ,n R/a ) J ( µ ,n R ′ /a ) e − µ ,n | Z − Z ′ | /a , (30) Comparison with previous results
Let us apply the results of the paper for analyticalanalysis of the space charge problem. In what follows,we will use Eq. 14 in order to derive a previously knownexpressions for the longitudinal component of the chargeinduced electric field in long- [7] and short-scale [20] cur-rent variation limits.The short-scale current variations commonly arise dueto microbunching instability that is driven by the longi-tudinal component of the space charge induced electricfield. It is common to assume a beam with a circular crosssection of radius b and a constant transverse density pro-file. In the case that observation point is located on-axis( r = 0), one defines an impedance (per unit length) Z ( k )as: E z ( k ) = − Z ( k ) I ( k ) , (31)where I ( k ) is the Fourier component of the current I ( z ) = cλ ( z ), with λ ( k ) = (2 π ) − R ∞−∞ λ ( z ) e − ikz dz . Theimpedance that has been implemented in Elegant [4] tosimulate space charge effect on a beam in a drift space[12] is Z ( k ) = iπε ckb (cid:20) − kbγ K (cid:18) kbγ (cid:19)(cid:21) , (32)where K is the modified Bessel function of the secondkind [20].The longitudinal space charge impedance in Eq. 32 isvalid in the short-wavelength limit, k → ∞ , where theeffect of boundaries can be neglected. The impedancethat includes the effect of boundaries has the followinglong-wavelength limit: Z ( k →
0) = i πε c kγ (cid:20) − (cid:18) ba (cid:19)(cid:21) , (33)according to Ref. [23]. This impedance corresponds tothe on-axis case of another well-known expression for the longitudinal component of the space charge induced elec-tric field [7]: E z ( r, z ) ≈ − πε cγ dI ( z ) dz (cid:20) − r b − (cid:18) ba (cid:19)(cid:21) . (34) Long-scale current variation limit
We begin our comparison with a long-scale currentvariation limit. According to Eq. 34, the longitudi-nal component of the space charge induced electric fieldis proportional to the current derivative. IntegratingEq. 14 by parts and assuming that charge derivative, dλ ( z ) /dz = c − dI ( z ) /dz , is constant on a scale δz ≫ a/γ , we obtain the following expression for the longi-tudinal component of the space charge induced electricfield: E z ( r, z ) = − πε cγ dI ( z ) dz Z ∞−∞ G ( r, Z ′ ) dZ ′ , (35)where the Green’s function is G ( r, Z ′ ) = G (0 , Z ′ ) − r ∂ ∂Z ′ G (0 , Z ′ ).The r -term for the longitudinal component of thespace charge induced field is proportional to the integralof the second derivative of the on-axis Green’s function,which is equal to − ∂∂Z ′ G (0 , Z ′ → + ). Let us recall herethat the on-axis Green’s function is the electrostatic po-tential of a charge disk and thus its negative derivative isequal to electric field. Using E Z (0 , z → + ) = 2 πρ ⊥ (0)for the uniform transverse charge distribution, one ob-tains that the value of coefficient is 4 /b .The integral of the on-axis Green’s function for theuniform transverse charge distribution is Z ∞−∞ G (0 , Z ′ ) dz ′ = 1 x ∞ X n =1 µ ,n J ( µ ,n ) J ( µ ,n x ) , (36)with x = b/a . Thus, in order to show the equivalencewith Eq. 34, one has to show that x (1 − x ) = ∞ X n =1 µ ,n J ( µ ,n ) J ( µ ,n x ) , (37)which is the case of Dini expansion of the function.The Dini expansion of a function defined on the in-terval x ∈ [0 ,
1] with the following boundary condition f (1) + f ′ (1) = 0 is f ( x ) = P ∞ n =1 c n J ( µ ,n x ) with thecoefficients c n = 2 J ( µ ,n ) Z xf ( x ) J ( µ ,n x ) dx. (38)Based on recursion relations J ( x ) = − dJ ( x ) /dx and J ( x ) = x − d [ xJ ( x )] /dx , one can show that the expan-sion coefficients of f ( x ) = x (1 − x ) are indeed equalto c n = 8 /µ ,n J ( µ ,n ) and prove Eq. (37). Thus, weconclude that the Green’s function based approach re-produces the well-known result for the longitudinal com-ponent of the space charge induced field in the limit ofa long-scale current variation and constant beam radius[7]. Short-range current variation
The Green’s function description, presented in this pa-per provides an alternative expression for the longitudi-nal space charge impedance: Z ( k ) = ikcγ Z ∞−∞ e − ikz ′ /γ Φ (0 , , z ′ ) dz ′ , (39)and thus is determined by the Fourier spectrum of theon-axis response function. Evaluation of the Fourier spectrum of the on-axis response function according toEq. (8), leads to the following representation of the lon-gitudinal space charge impedance:ˆ Z lsc ( k ) = iπε ckb " ba ∞ X n =1 A n µ ,n γ /k a J (cid:18) µ ,n ba (cid:19) , (40)where the sum is due to the presence of the pipe.The free space approximation used to describe the limitof short-range current variations ignores the presence ofthe pipe. Thus it replaces the discrete modes spectruminside the pipe with the continuous spectrum of the freespace. This reflects the fact that a large number of trans-verse modes contribute to the space charge filed. In thiscase consecutive terms in the sum in Eq. (40) are close toeach other and the sum can be replaced with the integral.Transition from the sum to an integral over x = bµ ,n /a has the following Jacobian dn/dx = a/πb . The resultingintegral on an interval x ∈ [0 , ∞∞