Guo, W M

Reiser, M

Wang, D X

Wang, J G

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Particle Accelerators, 1992, Vols. 37-38, pp. 181-189Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United KingdomTHEORETICAL STUDY OF THE LONGITUDINALINSTABILITY AND PROPOSED EXPERIMENTtJ. G. WANG, M. REISER, W. M. GUO and D. X. WANGLaboratory for Plasma Research and Department of Electrical Engineering,University of Maryland, College Park, Maryland 20742.( Received 3 January 1991)A theoretical model for the longitudinal instability in a transport pipe with general wall impedance ispresented. The results show that a capacitive wall tends to stabilize the beam. Design features for an.experimental study of the instability with an electron beam in a pure resistive-wall environment will bediscussed, including the design parameters, setup and components for the experiment.INTRODUCTIONThe induction linac is being studied in the USA as a driver for Heavy Ion InertialFusion. When the heavy ions are accelerated by the induction gaps, the beam seesa gap impedance along the accelerator and transport channels. Theoretical studiespredict that there is a resistive-wall instability which is detrimental to the beam.The analysis of the instability in this paper takes into account the general wallimpedance of the transport pipe. It predicts that the instability could be reduced bya capacitive wall and enhanced by an inductive wall. It also shows that the temporalgrowth rate of the resistive-wall instability is inversely proportional to the squareroot of the mass of the particles. An experiment to study the instability with heavyions would be costly because of the large mass of the ions which requires a longdistance to observe the effect.We plan to study experimentally the longitudinal instability with electrons.Specifically, an electron gun has been designed and constructed to produce a beamof 5 to 50 ns in width, 2.5 keY in energy, and a maximum of 160 rnA in current. Aninduction linac following the gun will impart a small head-to-tail energy spread tothe beam to balance beam expansion in the longitudinal direction. The beam is furthermatched to a resistive-wall channel of a few meters in length with uniform solenoidalmagnetic field. Various diagnostics will be used to determine the instability in thistransport experiment. Comparison with simulation could provide useful informationfor heavy ion induction linacs.t Research supported by the U.S. Department of Energy.181182 1. G. WANG et ale2 BEAM LONGITUDINAL INSTABILITY WITH GENERAL WALLIMPEDANCEThe longitudinal beam dynamics can be described by the linearized cold one-dimensional fluid model, which consists of the continuity equation and momentumtransfer equation 1,2,3:(1)where A and v are the line charge density and the particle velocity, the subscriptso and 1 representing the unperturbed and perturbed quantities, respectively, Ez isthe induced longitudinal electrical field by the a.c. component of the beam current,and 11 = elm denotes the ratio of the charge and mass of the charged particles. Fora nonrelativistic beam under the long wavelength condition, the field Ez can becalculated as4(2)where Ew is the field on the pipe wall of the transport channel, 80 is the permittivityof free space, and g is a geometric factor given byb 2g = 2 In - +-a 3'(3)with a and b as the beam radius and pipe radius, respectively. In the derivationof Eq. (3), a uniform beam profile is assumed and the average beam self-field overthe whole beam cross section is taken into account.In general, the pipe wall has an impedance per unit length, given in the frequencydomain byZw = R(w) + iX(w), (4)where R(w) and X(w) are the real and imaginary parts of the complex impedance.The field Ew is generated by the perturbed beam image current on the wall and canbe expressed byX(w ) JAE = - J *R( ) + *2 __0 _1w /~1 V Wo V ,W o Jz(5)where W o = kvo is the characteristic frequency of the instability. The velocity v*with which the perturbation propagates should be very close to the mean speed ofparticles and is approximated as Vo in the following analysis. It is not difficult toverify the validity of Eq. (5) with simple circuit models, while a derivation of thisformula is left in a forthcoming paper. Therefore, the total longitudinal field (2) actingon the beam isSTUDY OF LONGITUDINAL INSTABILITY 183(6)(g 2 X(Wo)) OA 1Ez ~ - -- - Vo -- -- - A1VoR(wo)·41C80 W o OZEquation (1) along with Eq. (6) can be solved by using a Laplace transform in timeand a Fourier transform in space; that leads to the dispersion relation(7)In the parameter range of interest, the dispersion equation D(s, k) == 0 may be solvedapproximately to give rise to the perturbed wave frequency and the temporal growthrate(8)(9)This equation is correct only when k is real as implicitly defined by the Fouriertransform in space. Equation (8) shows that the slow wave grows at the expense ofthe beam energy. For a pure resistive wall, in the low-frequency approximation whereR(wo) == Rand X(wo) == 0, the slow-wave growth rate is~ ~ (lJA041C80)1/2Wi~ voR .2 gIf the pipe wall is inductive, e.g., Z == R + iwoL, one gets(10)(11 )Thus, the addition of an inductive component increases the growth rate in comparisonwith Eq. (9). For a capacitive wall modeled by a circuit of Rand C in parallel, Eq.(8) can be rewritten as1 R ( ryAo )1/2Wi ~ 2vo 1 + w6 R2C2 _g_ + v'6 R2C . .41C8o 1 + w6R2C2As expected, the growth rate of the slow wave is reduced when a capacitance is added.This result agrees with Reference 1, which states that, in general, the capacity reducesgrowth rates, compared with the case of pure resistance by lowering the impedanceas frequency increases. Reference 5 analyses the resistive instability growth rates in1. G. WANG et al.1841.41.2..-..Cllt8 1.0-::.~~ 0.8....=~.c....~ 0.6et.'~.~ 0.4....=~~0.20.00Capacitive3 4 5 6Relative Time Constant (~/~o )FIGURE 1 Relative growth rate Wi/Wi. R vs. the relative time constant r/ro where Wi is the growth ratefrom Eqs. (9)-(11); Wi. R is specified as the growth rate for a pure resistive wall; r = RC for the capacitivewall, or r = L/R for the inductive wall; and ro = l/wo. It is assumed that Vo = O.lc, R = 100 Q/m, andW o = 2n x 107 s -1 in this example.the modules represented by a parallel R, L, C circuit. Its results can also be recoveredfrom Eq. (7).To illustrate the results from Eqs. (9)-(11), Figure 1 shows the relative growth rateversus the time constant of the wall impedance. In this example the growth rate fora pure resistive wall is set to unity. The beam is supposed to have a velocity of fJ == 0.1.The pipe wall resistance is 100 !l/m and the characteristic frequency is assumed tobe 10 MHz. The lower branch of the plot is for the capacitive wall with r/ro == Rewawhile the upper branch is for the inductive wall with r/ro == woL/R. The growth ratefor the capacitive wall decreases rapidly when the capacitance is increased. Thisdependence is not sensitive to the beam parameters. By contrast, the growth rate foran inductive wall is highly beam-velocity dependent.3 DESIGN STUDY OF THE RESISTIVE-WALL INSTABILITYEXPERIMENTEquation (8) shows that the temporal growth rate of the resistive-wall instability isinversely proportional to the square root of the mass of the beam particles. AnSTUDY OF LONGITUDINAL INSTABILITY 185experiment to study the instability with heavy ions would be costly because of thelarge mass of the ions which requires a long distance to observe the effect. Withelectrons, the experiment could be done at a small scale and much less cost.For a pure resistive wall, the phase velocity of the slow wave in the beam frameand the growth rate can be derived from Eq. (8) as(12)(13)(14)where K is the generalized perveance and, for non-relativistic electrons, is given by4 I(A)K = 1.515 x 10 V 3 /2(V) 'with I and V as the beam current and energy equivalent voltage, respectively. Thee-fold instability growth length in the beam frame is determined by919 = .2n8ovoRFor a beam pulse of the width T, the number of e-folds in the growth from thebeam head to tail is2 2n80n = Rvo -- T.9The number of e-folds per unit channel length is given by(15)(16)N = G~n2e~Y/4C:l/2Y/2R,which is proportional to the wall resistance, and to the square root of the beamcurrent, and is inversely proportional to the fourth root of the beam voltage. For aperturbation starting from the beam head at the channel entrance and ending at thebeam tail at the channel exit, the relationship between the pulse length T and thechannel length L is given by(17)In order to observe the instability in a reasonable short distance, an electronbeam has been designed with the following parameters: beam pulse width of 5 ns,beam voltage of 2.5 keY, and beam current of around 100 rnA. This corresponds toabout 15 em of beam pulse length in space. Assuming a resistance of 10 kQjm and9 = 2, Eqs. (14)-(16) yield 19 ~ 12.2 ern, n ~ 1.2 and N ~ 0.91. A transport channelof 1.3 meters in length would satisfy Eq. (17). These parameters will be scaled in theexperiment.186 1. G. WANG et al.Control and data takingr - - - - -- ---,ILens Lens Induction LensI Electron DiagnosticI gun- --ChainberI 2 module 3I TRANSPORTL ELECfRON BEAM INJEcrOR _-.J- -- - -- --------CHANNELFIGURE 2 Schematic diagram of the resistive-wall instability experiment.A schematic diagram of the experimental setup is shown in Figure 2. An electronbeam injector, originally designed for longitudinal pulse compression experiments, isemployed as the beam pulse source. The injector consists of a variable-perveancegridded electron gun, three matching lenses and an induction acceleration module.The electron gun produces a beam pulse with the desired parameters. The matchinglenses confine the beam in the transverse direction. The induction acceleration moduleimparts a small head-to-tail energy spread to the beam to balance beam expansionin the longitudinal direction due to space charge. The beam from the injector isguided into a resistive-wall channel of a few meters in length with a uniform solenoidalmagnetic field to provide transverse focusing. The end diagnostic chamber will pickup all the necessary information to determine the instability in this transportexperiment. The components for the injector have been constructed and extensivelytested. The assembling of the injector system is underway. The coating of the transportchannel with resistive material is in progress. The various diagnostic equipments areto be built. The design and performance characteristics of the major components ofthe system are described briefly as follows:3.1 Electron GunThe detailed information about the gun design and performance characteristics canbe found in Reference 6. This is a variable-perveance gridded gun with Piercegeometry electrodes. The gun perveance can vary from 0.22 x 10- 6 to 1.35 X10- 6 AV- 3/ 2 by adjusting the A-K gap. The maximum beam energy from the gunis 10 keY and the beam current can be as high as 160 rnA at an anode voltage of2.5 kV. The beam current is controlled by a fast grid-cathode pulse which is adjustablein length from a few nanosecond up to hundreds of nanoseconds. A 5 ns grid-cathodepulse is shown in Figure 3, where one still can see a flat top of the pulse and a smallrisetime of less than 1 ns. The beam pulse shape should resemble the grid-cathodepulse waveform. The grid-cathode pulse can be modulated with a perturbation toinitiate the beam instability.STUDY OF LONGITUDINAL INSTABILITY 187FIGURE 3 Example of the grid-cathode pulse, which shows 5 ns width and less 1 ns risetime.3.2 Induction Acceleration ModuleReference 7 gives a detailed description of the performance analysis and testing resultsof this device. Originally designed for pulse compression experiments, the compactinduction acceleration module produces a quadratic time-varying voltage on its gapduring about 50 ns. The gap voltage as a function of time from the test is shown inFigure 4. This gap voltage waveform is determined by the modulator consisting of6~----------------""5605040302010o..._...c:;r.- ----.r---__- -,...--......,-----.r----Iot (ns)Figure 4 Induction gap voltage vs. time in the first 50 ns where the data points were from themeasurement and the smooth curve was an ideal fitting to (2 curve.188 1. G. WANG et al.a few inductances and capacitances. It is easy to change this waveform as one desires.For the resistive-wall instability experiment with a beam pulse of 5 ns, the inductionmodule will only impart a small energy difference from the beam head to tail tobalance the longitudinal expansion. The induction module should be synchronizedwith the electron gun within 1 ns during operation.3.3 Resistive- Wall PipeProduction of a resistive-wall pipe follows the work of Reference 8. Specifically, aglass tube of 3.8 cm in diameter and a couple of meters in length in the first phaseof the experiment will be coated with tin oxide. The design goal for the resistance isa few thousand ohms per meter. The tube is located inside and coaxial with a longsolenoid which produces a uniform magnetic focusing field of around 100 gauss. Thisfield provides forces in transverse direction to balance space charge forces. Anidentical pipe with good conductivity will also be employed for comparison study.3.4 DiagnosticsThe diagnostics in the experiment includes accurate measurement of the beam currentwaveform, beam profile, time-resolved beam energy, etc. An existing Faraday cup isto be improved to achieve fast response: less than 1 ns in risetime. A fast Rogoskicoil will be used to make non-destructive beam current measurement at the inputand output of the transport channel. A time-resolved spectrum analyzer is underwayfor design and construction.4 SUMMARYThe cold one-dimensional fluid model has been used to analyze the longitudinalinstability in a beam transport channel with general wall impedances. The temporalgrowth rate of the instability can be dramatically reduced by a capacitive componentof wall impedance along the transport channel. By contrast, the inductive wallcomponent would enhance the instability.Based on the result of this analysis, an experiment to study the resistive-wallinstability with an electron beam has been designed. An electron beam injectorproducing the desired beam parameters has been constructed and its componentshave been extensively tested. Manufacturing of a resistive wall and preparation ofthe diagnostic tools are underway. The experimental results will be reported elsewherein the near future.REFERENCES1. E. P. Lee and L. Smith, in Proceedings of the 1990 Linear Accelerator Conference (Albuquerque, NM,1990).2. E. P. Lee, Proceedings of the 1981 Linear Accelerator Conference, Santa Fe, NM, October 19-23, 1981.3. J. Bisognano, I. Haber, L. Smith, and A. Sternlieb, IEEE Trans. Nucl. Sci. NS-28, 3 (June 1981).4. A. Hofmann, CERN report 77-13, p. 139 (CERN, Geneva, 1977).STUDY OF LONGITUDINAL INSTABILITY 1895. L. Smith, Lawrence Berkeley Laboratory internal note HIFAR-Note-217 (1988).6. J. G. Wang, E. Boggasch, P. Haldman, D. Kehne, M. Reiser, T. Shea, and D. X. Wang, IEEE Trans.Electron Devices 37 (12) (December 1990), pp. 2622-2628.7. J. G. Wang, D. X. Wang, E. Boggasch, D. Kehne, M. Reiser, and T. Shea, Nucl. Instrum. Meth. A 301,(1991) pp. 19-26. .8. C. K. Birdsall, G. R. Brewer, and A. V. Haeff, Proc. I.R.E. 41, 7 (July 1953).

Theoretical study of the longitudinal instability and proposed experiment

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