Keil, E

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ON THE ENERGY LOSS OF A CHARGED RING PASSING A CORRUGATED CYLINDRICAL WAVEGUIDE E. KEIL CERN 1. Introduction The energy loss of a charged ring passing a sequence of RF accelerating cavities has recently received much attention in connection with the collective acceleration of ions in electron rings[1]. Various models and techniques were employed to represent the RF structure: modal analysis for closed cylindrical cavities [2], [3], [4] and for corrugated cylindrical waveguides [5], Wiener-Hopf techniques for sets of infinite half-planes [6], [7], and diffraction theory for one infinite half-plane[8]. As in[5] we shall use a periodic corrugated waveguide as a model and obtain the energy loss by an analysis of the resonances of this system. However, we do not restrict the nature of the waveguide modes from the beginning, and continue the calculation to very much smaller wavelengths. 2. The Modes in a Periodic Corrugated Cylindrical Waveguide If one only considers the axial velocity of the electrons in the rings, and if one considers them perfectly centered in the waveguide, they only interact with TM waveguide modes which are independent of the azimuthal co-ordinate φ. The energy loss in a closed cavity due to the azimuthal motion of the electrons was calculated by Neil[9]. We use the field expansions given by Walkinshaw and Bell [10]. We only write down Ez; Eρ and Hφ follow from Maxwell's equations. Expansion of Ez in the slots (I): Ez= ∞ Bs[F0(Γsρ)/Fo(Γsa)] cos(2πsz/g) Σ s=0 + ∞ Ds[Fo(Γ'zρ)/Fo(Γ'sa)] sin[(2s-1)πz/g] (1) Σ s=1 where 551 Γs2=|k2-(2πs/g)2|, (2) Γ's2=|k2-[(2s-1)π/g]2|. (3) If k>2πs/g or if k>(2s-1)π/g we define: Fj(Γρ)=Yo(Γb)Jj(Γρ)-Jo(Γb)Yj(Γρ). (4) Otherwise: Fj(Γρ)=Ko(Γb)Ij(Γρ)-(-1)jIo(Γb)Kj(Γρ). (5) In all formulae the time dependence exp(iωt) should be understood. It may be seen that the slot fields satisfy the boundary conditions of a perfectly conducting waveguide at the boundaries drawn in heavy lines in Fig. 1, where also the geometrical parameters a, b, d, g are shown. k=ω/c=2π/free-space wavelength. Expansion of Ez in the axial region (II): EZ(II)= ∞ Am[Go(xmρ)/Go(xma)] exp(—iβmz) (6) Σ m = — ∞ where x2m=|k2-βm2|, (7) βm=βo+2πm/d, (8) βo=k/β-2πl/d, (9) where l is chosen such that -π/d<βo≤π/d. βc= v is the phase velocity of the travelling wave Gj(xρ)=Jj(xρ) for k>βm, (10) Gj(xρ)=Ij(xρ) for k<βm. (11) Imposing the condition that Ez and Hφ be equal on both sides of the boundary ρ=a yields the following infinite homogeneous set of linear equatlons[10]: [CT H C M-ST L S M-U] A=0. (12) Here A is a vector with components Am; H, L and M are diagonal matrices with elements Hss=KsΓsFO(Γsa)/F1(Γsa), (13) Lss=2Γ'sFo(Γ'sa)/F1(Γ'sa), (14) Mmm=(g/dxm)G1(xma)/Go(xma). (15) Here Ko=1 and Ks=2 for s>0. C and S are matrices with elements Cst=(-1)sg βtg sin βtg /[( βtg )2-(πs)2], (16) 2 2 2 552 Sst=-(-1)sig βtg cos βtg /[( βtg )2-( (2s—1)π ) 2]· (17) 2 2 2 2 U is the unit matrix and XT is the transpose of X. The frequencies ωλ where (12) has non-trivial solutions are those where waves travelling with the phase velocity βc propagate in the waveguide. They were calculated by computer. Since the infinities of the determinant of (12) can be given analytically, the computer programme uses a function of frequency in which the infinities are eliminated but which has the same zeros as the determinant. This makes it fairly straightforward to find all zeros in a given frequency interval. A further check is the comparison of the average density of resonances with an analytical formula [11]. 3. The Excitation of a Single Resonance by the Passage of a Charged Ring Following the procedure of Akhiezer, Lyubarskij and Fainberg we introduce the vector potential = - (18) t which we normalize such that ∫ |A|2dv=1/εo (19) V1 where V1 is the volume of a single cell. This implies that the coefficients Am, Bs and Ds are multiplied by the factor N=[1/[εo(I+J+J')]]½ (20) where the integrals are defined as follows: I= πa2d +∞ A2m [ G21(xma) ± 1 β2m 2 G1(xma) ]. (21) Σ c2 x2m G2o(xma) k2xma Go(xma) m=-∞ Upper sign for Gj given by (10), lower sign for Gj given by (11). J= πg ∞ B2s Κs[ b2 F21(Γsb) — a2 ( F12(Γsa) ± 1) Σ c2 Γ2s 2 F2o(Γsa) 2 F20(Γsa) s=0 ± a2 ( gπs )2 2 F1(Γsa) ), (22) 2 gk Γs aF0(Γsa) Ko=2 and Ks=1 for s>0. J'= πg ∞ D2S [ b2 F12(Γ'sb) a2 ( F12(a) ± 1) Σ c2 Γ'2s 2 F20(Γ'sa) 2 F2o(Γ'sa) s=1 ± a2 ( (2s-1)π )2 2 F1(Γ'sa) ] (23) 2 gk Γ'sa Fo(Γ'sa) 553 Upper sign for Fj given by (4), lawer sign for Fj given by (5). We expand the vector potential in the presence of the electron rings in terms of the resonances of the empty waveguide. ()= Σ qλ(t)Aλ() (24) λ and find that the field co-ordinate qλ(t) for the λ-th mode obeys the equation: +ω2λqλ= 1 dv (25) Nc VN where Nc is the number of cavities and VN is their volume. is the current density of the charged ring whose only non-vanishing component is taken to be: jz(ρ, φ, z, t) = H( h -|z-vt|) Qv 2 2 H(ρ-R1)H(R2-ρ) (26) 2π R22-R21 h H(x) is Heaviside function with H(x)=1 for x≥0, and H(x)=0 for x<0. Q is the ring charge, v its velocity, h its axial dimension, and R1 and R2 its smaller and bigger radius, respectively. The solution of (25) with qλ= = 0 is: qλ= +∞ A'm [eiΩmt- 1 ( Ωm +1)eiωλt+ 1 ( Ωm —1)e-iωλt] (27) Σ ω2λ-Ω2m 2 ωλ 2 ωλ m=-∞ where A'm=-Qvi Am( sin βmh )[ 2[R2G1(xmR2)-R1G1(xmR1)] ] (28) 2 Νcωλ βmh xm(R22-R12)Go(xma) 2 and Ωm=βmv. The energy radiated per unit time into the λ-th mode becomes: Iλ= dQ2V2 { | -|( sin βmh )2[ 2[R2G1(xmR2)-R1G1(xmR1)] ]2 Σ |Am|2 2 4ω2λ dωλ -v βmh (xm(R22-R21)Go(xma) m=m dβo 2 + ( sin βmh )2[ 2[R2G1(xmR2)-R1G1(xmR1)] ]2}· (29) Σ |Am|2 2 2 |dωλ + v | βmh xm(R22-R12)Go(xma) m=m" dβo 2 Here, the first sum is to be taken over all m' with ωλ — Ωm,=0, and the second sum over all m" with ωλ+=0. With our choice of travelling waves (9) the first resonance condition takes the form m'=l. In order to fulfil the other resonance icondition ωλ= = 0 we 554 have to replace βo by-βo and m by-m. It has been verified that changing the sign of βo leaves the resonant frequencies unchanged and reverses the order of the Am. Hence the second sum is just the same as the first one. If we further assume that dωλ/dβo«v we find for the energy radiated per unit time: Iλ= dQ2v |A1|2( sin β1h )2[ 2[R2G1(x1R2)-R1G1(x1R1)] ]. (30) 2 2ω2λ β1h/2 x1(R23-R12|Go(x1a) The energy radiated into the λ-th mode in a single cell is Uλ=Iλd/v. 4. Results Fig. 2 shows a typical example of the frequency spectrum of the radiated energy when electron rings travel along a waveguide with rather shallow corrugations. Tabs. I and II summarise the influence of the waveguide dimensions, and of the ring dimension and velocity expressed in terms of γ=[1—(V/C)2]-½. A completely different set of waveguide shapes and the resulting energy loss spectrum are shown in Fig. 3. The waveguide is now loaded with closely spaced infinitely thin irises. In cylindrical geometry this is a close analogy to the infinite half-planes studied in two-dimensional models [6] [7]. The dimensions are chosen such that the approximate formula [5] U= Q2 g (31) 4πεo 2a2 should Just be valid. The effect of variations in the waveguide dimensions is shown in Tab. III. In the last column the scaling law (31) was applied to the radiated energy. It may be seen that this formula gives an excellent description of the dependence on the waveguide dimensions. However, the total enengy radiated is underestimated by about a factor of three. Again, the ring dimensions and γ have hardly any influence on the radiation loss. 5. Conclusions Field expansion into Fourier series and matching of all field components at the common boundary surface are shown to be a suitable method for calculating the energy radiated by electrons travelling in a cylindrical corrugated waveguide. In practice, this method is limited to wavelengths longer than a few millimetres for reasons of computer time. The energy losses are calculated for several waveguide dimensions; they are compared with published results [5] for one particular set of wave-555 guide dimensions. In all cases, the electron ring dimensions and their velocity have no significant influence on the energy loss. Table I Radiated Energy U for Various Waveguide Dimensions N=1013 electrons γ=20 Wavelength ≥4 π mm Minor ring dimensions (R2—R1)h=1×1 mm2 Average distance of ring from iris a—(R1+R2)/2=5 mm Hole radius a[mm] Tube radius b[mm] Cell length d[mm] Slot width g[mm] Radiated Energy U [ keV ] electron cell 53.14 83.14 153.14 73.14 820 53.14 166.28 153.14 73.14 860 53.14 166.28 153.14 36.57 650 53.14 166.28 76.57 36.57 26.57 166.28 153.14 73.14 2230 Table II Radiatied Energy U in KeV/electron/cell, for Various Ring Dimensions and γ's N=1013 electrons Wavelength≥3π mm Hole radius a=53.14 mm Cell length d=153.14 mm Tube radius b=83.14 mm Slot width g=73.14 mm Minor ring dimensions (R2—R1)h=1×1 mm2 γ Average distance of ring from iris a — R1+R2 2 1 mm 2 mm 5 mm 10 mm 5 1080 980 780 610 20 960 940 910 870 50 900 890 880 Table III Radiated Energy U for Waveguide with Infinitely Thin Irises N=1013 electrons γ=50 Wavelength≥3π mm Minor ring dimensions (R2—R1)h=1×1 mm2 Average distance of ring from iris a—(R1+R2)/2=5 mm Hole radius a[mm] Tube radius b[mm] Cell length d=g[mm] Radlited Energy U Radiated energy scaled with (31) [ keV ] electron cell 50 250 20 190 1.00 40 250 20 290 0.98 60 250 20 130 0.99 200 20 180 0.95 50 300 20 190 1.00 50 250 10 100 0.95 50 250 30 280 0.98 556 Fig. 1. Geometry of corrugated cylindrical waveguide and electron ring 557 Fig. 2. Energy loss spectrum at γ=20. Waveguide dimensions a=53.14 mm, b=83. 4 mm, d=.53.14 mm, g=73.14 mm; ring dimensions R1=47.64 mm, R2=48.64 mm, h=1 mm. 558 Fig. 3. Energy loss spectrum at γ=50. Waveguide dimensions a=50 mm, b=250 mm, d=g=20mm; ring dimensions R1- 5 mm, R2=45 5 mm, h=1 mm. 559 REFERENCES 1. V. I. Veksier et al., Proc. Int. Conf. High Energy Acc., Cambridge, Mass., p. 289, 1967. 2. O. A. Kolpakov and V. I. Kotov, Sov. Phys.-Techn. Phys. 9, 1072, 1965. 3. P. L. Morton and V. K. Neil, UCRL-18103, p. 365, 1968. 4. B. S. Levine and A. M. Sessler, Proc. 1969 Particle Acc. Conf., Washington D. C. (to be published). 5. Ο. Α. Kolpakov, V. I. Kotov and Om-San-Kha, Sov. Phys.- Techn Phys, 10, 18, 1965. 6. G. V. Voskresenskii and B. M. Bolotovskii, Sov. Phys.- Doki. 9, 459 196. 7. A. M. Sessler, private communication. 8. J. D. Lawson, Rutherford High Energy Lab. /M144, 1968. 9. V. K. Neil, private communication. 10. W. Walkinshaw and J. S. Bell, A. E. R. E. G/R 675, 1951. 11. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, p. 432, Interscience, New York, 1953. 560 

On the energy loss of a charged ring passing a corrugated cylindrical waveguide

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