LONGITUDINAL EFFECTS AND FOCUSING IN SPACE-CHARGE DOMINATED BEAMS. John Richardson Harris

Transcription

1 ONGITUDINA EFFECTS AND FOCUSING IN SPACE-CHARGE DOMINATED BEAMS by John Richardson Harris Thesis subitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillent of the requireents for the degree of Master of Science Advisory Coittee: Professor Patrick G. O'Shea, Chair Professor Victor. Granatstein Professor Martin Reiser

2 ABSTRACT Title of Thesis: ONGITUDINA EFFECTS AND FOCUSING IN SPACE-CHARGE DOMINATED BEAMS Degree candidate: John Richardson Harris Degree and year: Master of Science, Thesis directed by: Professor Patrick G. O'Shea Departent of Electrical and Coputer Engineering The purpose of this thesis is to investigate longitudinal effects in space-charge doinated beas, and to begin design of a longitudinal focusing syste for the University of Maryland Electron Ring (UMER). The longitudinal envelope equation is introduced and used to develop a longitudinal intensity paraeter which is analogous to the transverse intensity paraeter. After solving for the free-expansion longitudinal envelope, the electric field necessary to properly focus a parabolic bea in a periodic longitudinal focusing lattice is deterined. The cold fluid odel for space-charge doinated beas is then introduced, and results stated for the free expansion of a rectangular bea pulse. A ore general approach to find the electric field needed in a periodic longitudinal focusing lattice is then developed. This approach is applicable, within certain assuptions, to any bea profile for which the particle velocity and line charge density can be deterined through theory or siulation. The drift copression focusing schee discussed heavily in the literature is explored and probles with this ethod are noted. Soe discrepancies and areas where further work is needed are also addressed.

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4 PREFACE The original version of this thesis was subitted to the University of Maryland in printed for in. This version was assebled fro the original source files, but ay contain inor differences as copared to the official version on file with the University. J.R. Harris May 7, 6

5 iii ACKNOWEGEMENTS I would like to thank y coittee for aking this thesis and defense possible. In addition, I would like to thank Professor Patrick O'Shea and Professor Martin Reiser for their guidance during the past year, Dr. Agust Valfells for valuable discussions on theory, and Dr. Irving Haber and Dr. Rai Kishek for assistance with siulations.

6 iv TABE OF CONTENTS ist of Tables ist of Figures v vi 1. Introduction 1. ongitudinal Envelope Equation Introduction. 6.. ongitudinal Intensity Paraeter Solutions of the ongitudinal Envelope Equation ongitudinal Focusing for Parabolic Beas iitations 8 3. One-Diensional Cold Fluid Model Introduction Governing Equations Rectangular Bea Expansion and the Method of Characteristics Coherent Energy Spread Discrepancies ongitudinal Focusing in UMER: A ore general approach Introduction ongitudinal Focusing Drift Copression Additional Effects ongitudinal Experients. 54

7 v 6.1. Introduction UMER Facility and Diagnostics ongitudinal Experients not requiring Induction Gaps ongitudinal Experients requiring Induction Gaps Conclusion. 6 References. 64

8 vi IST OF TABES 1. ongitudinal Intensity Paraeter for UMER. 11. Coparison of ongitudinal and Transverse Relations. 17

9 vii IST OF FIGURES 1. Space Charge Intensity Paraeter (χ). 3. University of Maryland Electron Ring (UMER) ongitudinal Space Charge Intensity Paraeter (χ ) ongitudinal Envelope for a Contracting Bea. 5. Bea Undergoing Contraction and Foration Siultaneously. 6. Expansion of Parabolic Bea in UMER. 7. ongitudinal Focusing attice Electric Field for Focusing Parabolic Bunch Fluid Analogy for Expansion of Rectangular Electron Bea ine Charge Density and Velocity in Expanding Rectangular Bea Conceptual ayout of Induction Gap in the Rest Frae of the Bea ongitudinal Focusing Voltages for Rectangular Bea. 49

10 1 1. Introduction. The behavior of charged particle beas and the focusing ethods needed to control the depend in part on bea intensity. ow intensity beas are characteried by low space-charge density, and their response to external focusing is governed priarily by theral effects (eittance). The transport of eittance doinated beas in focusing systes is siilar to the transport of light through an optical focusing syste. High intensity beas are characteried by high space-charge density, and the electric forces between the particles in the bea govern its behavior in focusing systes. Space-charge doinated beas behave like a plasa, with pronounced collective and nonlinear effects. The intensity paraeter (χ) has been introduced to provide a easure of bea intensity[1]. This paraeter is the ratio of the transverse space-charge force to the transverse focusing force in a bea, and is given by K χ =, (1) k R where K is the generalied perveance of the bea, k is the betatron oscillation wave nuber in the absence of space charge, and R is the transverse bea envelope. The generalied perveance is directly related to the space-charge density in the bea. The value of χ ranges fro ero (eittance doinated) to one (space-charge doinated); it cannot exceed one, as the space-charge forces would exceed the focusing forces and the bea would no longer be confined by the focusing syste. Although ost research has been done on eittance doinated beas, any current and planned applications require space-charge doinated beas. These

11 applications include high power icrowave sources, free-electron lasers, and heavy-ion inertial confineent fusion drivers (HIF)[-5]. The University of Maryland Electron Ring (UMER), a space-charge doinated bea transport syste, is currently under construction, and will be used to iprove the understanding of space-charge doinated beas. Although the UMER bea is coposed of electrons, it has been designed so that it will behave in a siilar anner as the beas of heavy ions such as bisuth that will be needed for planned HIF achines[6]. As such, UMER is a scale odel which allows research to be done in support of the HIF progra in a university laboratory setting at considerably less expense than would be involved with the actual construction of a HIF driver. A second, related goal for UMER is the investigation of space-charge wave propagation along the direction of travel of intense beas. These longitudinal waves were first studied by D.X. Wang at Maryland in the early 199's, but this research was liited by the length of the bealine available at the tie[7]. Because UMER is a circular achine, the distance over which the bea can propagate is not liited by its physical sie. This will allow longitudinal waves to be studied for longer periods of tie than were possible in previous achines. UMER also has iproved diagnostics which will allow these waves to be studied in ore detail than was possible in the past[8-1]. Although UMER is designed to access the very intense region that will be needed for HIF drivers, it can also be tuned across a wide range of χ. This allows it to access the eittance doinated region and the crossover region near χ =.5, where both eittance and space charge ust be considered.

12 Eittance Doinated Space-Charge Doinated 1. Existing Machines ω p / ω 1.1 ( ) ( ) TD χ DPF χ k / k. UMER χ 1.1 Space-Charge Intensity Paraeter (χ) Fig. 1. Space Charge Intensity Paraeter (χ). This graph relates the intensity paraeter for transverse bea physics with the tune depression (k/k ) and the plasa frequency (ω p ). Beas with χ <. 5 are considered eittance doinated, while beas with.5 < χ 1 are considered space charge doinated. Existing rings are eittance doinated, and operate in the range to the left of the dashed line. UMER is designed to operate across a wide range of intensity paraeter values, fro approxiately χ =. to alost χ = 1. The intended operating range for UMER extends to the right of the dashdot line.

13 Fig.. The University of Maryland Electron Ring (UMER)[11]. 4

14 5 Most of the work done on focusing in UMER has been concentrated on its transverse focusing syste. However, space charge will also drive an expansion of the bea along its axis of travel. In order to control this longitudinal expansion, a longitudinal focusing syste ust be ipleented. This syste will use induction gaps to apply an electric field to the bea along its direction of travel, which will copress the bea longitudinally. The purpose of this thesis is to investigate longitudinal effects in space-charge doinated beas, and to begin design of a longitudinal focusing syste for UMER. The longitudinal envelope equation will be introduced and used to develop a longitudinal intensity paraeter which is the longitudinal analog to the intensity paraeter of eq. (1). After solving for the free-expansion longitudinal envelope, the electric field necessary to properly focus a parabolic bea in a periodic longitudinal focusing lattice will be deterined. The cold fluid odel for space-charge doinated beas will then be introduced, and results stated for the free expansion of a rectangular bea pulse. A ore general approach to find the electric field needed in a periodic longitudinal focusing lattice will then be developed. This approach is applicable, within certain assuptions, to any bea profile for which the particle velocity and line charge density can be deterined through theory or siulation. The drift copression focusing schee discussed heavily in the literature will be explored and probles with this ethod will be noted. Soe discrepancies and areas where further work is needed will also be addressed.

15 6. ongitudinal Envelope Equation..1. Introduction. One approach to investigating longitudinal effects in charged-particle beas is through the longitudinal envelope equation[1] ~ ~ ~ K ε ~ 5 5 ~ + κ =, () 3 where ~ is the RMS half length of the bea, κ is the longitudinal focusing function, K is the longitudinal generalied perveance, and ε ~ is the unnoralied RMS longitudinal eittance. The generalied longitudinal perveance is given by K = 3 gn q 5 β γ 4 πε c (3) where N is the total nuber of particles in the bunch, q is the charge of the electron, ε is the perittivity of free space, is the ass of the electron, c is the speed of light, and β and γ are the relativistic factors. The focusing function κ can be related to the applied electric field, and ay be a function of location and tie. The unnoralied RMS eittance is given by [ ] 1/ ~ ε =. (4) The geoetry factor g is given approxiately by b g 1 + ln, where b is the radius a of the bea pipe and a is the radius of the bea[7,1]. Note that both the RMS halflength ~ and the true half-length are easured fro the centroid of the bea in the bea rest frae. The distance traveled by the bea centroid through the laboratory frae is denoted s, and pried quantities indicate a derivative with respect to s.

16 7 Although β and γ appear throughout this thesis, UMER is strictly nonrelativistic, with a axiu β of.. The presence of β or γ in an equation should not be taken to iply that the equation is correct for relativistic beas. The longitudinal envelope equation was first derived by. Sith[13], with additional work being done by Neuffer[14] and Sacherer[15]. Their derivations of the equation were directly applicable only to beas with unifor, parabolic, or Gaussian distributions. Although the beas of interest to UMER are generally not unifor, parabolic, or Gaussian, the parabolic bea can be taken as an equivalent line charge density for a bea with the sae nuber of particles, the sae eittance, and the sae RMS length[16]. Although the longitudinal envelope equation serves the sae role in longitudinal dynaics that the transverse envelope equation serves in transverse dynaics, certain differences should be noted: 1. The longitudinal perveance is not diensionless as in the transverse case, but rather has diensions of eters. This is also reflected in the extra factor of length in the denoinator of the space-charge ter in the longitudinal envelope equation as copared to the transverse envelope equation.. The relations between RMS and non-rms quantities are different. For the longitudinal envelope equation = ~ 5 and ε = 5 ~ ε, while for the transverse envelope equation x = ~ x and ε = 4 ~ ε x.

17 8 3. The angle referred to by the longitudinal eittance does not exist in real space, since v is parallel to the local direction of travel s v. For transverse eittance, the angle does exist in real space since v x is perpendicular to the direction of travel of the bea... ongitudinal Intensity Paraeter. Before actually solving the longitudinal envelope equation for specific cases, it is instructive to look in ore detail at this equation. The first ter, ~, gives the "acceleration" of the bea edge. Driving this acceleration are three ters: κ ~ is a focusing ter, which tends to ake ~ increasingly negative, resulting in a bea contraction; and K 5 5 ~ and ~ ε 3 ~, which due to their signs tend to ake ~ positive, driving the bea expansion. For beas which are strongly space-charge doinated, the eittance ter can be neglected, while for beas which are strongly eittance doinated, the space-charge (perveance) ter can be neglected. In order to deterine whether the bea is space-charge doinated or eittance doinated, a longitudinal intensity paraeter can be introduced. This paraeter is the longitudinal analog to χ. For consistency with the transverse case the non-rms longitudinal envelope equation[1] is used: K ε + κ =. (5) 3 This non-rms equation describes the envelope of the extree edge (half-length) of the bea,, and can be derived fro the RMS equation by use of the relations between RMS and non-rms quantities. Since the transverse intensity paraeter is the ratio

18 9 between the space-charge force and the focusing force for a atched bea, we take = to obtain the longitudinal atched bea envelope equation: k K ε = 3. (6) Here, the focusing function κ has been replaced by the ero-current synchrotron wave nuber k to indicate that focusing is continuous for a atched bea. In general, the focusing function κ varies throughout the focusing syste. Then by analogy with χ, we take χ longitudinal space charge force = longitudinal focusing K χ =. (7) k 3 Since the goal is to obtain a single nuber to describe the overall behavior of the bea, unifor focusing is assued and χ is rewritten in ters of generalied longitudinal perveance, bea half-length, and longitudinal eittance, using eq. (6). χ = K K ε + 3 K χ =. (8) K + ε ongitudinal eittance (unnoralied effective eittance) can be rewritten in ters of longitudinal energy spread, which can be easured[9,1,17].: ε = 5 ~ ε

19 1 ε = ~ ε 5 βγ n 3 ε 5 = 3 βγ 3 ~ γ k BT c 1/ ε = 5 βγ ~ γ ~ βec E 3 ~ E 5 γc ε 5 = 3 βγ γ β ~ ε 5 E = 3 β γ 5 c. (9) Using this result in eq. (8) gives χ =. (1) K K β γ ~ E c Table 1 shows that, while UMER's transverse intensity χ can be adjusted across a wide range of values, fro space-charge doinated to eittance doinated, the longitudinal behavior of UMER will always be space-charge doinated for practical operating paraeters. The transverse intensity paraeter is particularly useful because it can be related to other paraeters used in bea and plasa physics through siple expressions. This is also true for the longitudinal intensity paraeter χ as derived above. For exaple, has the sae relationship to the longitudinal tune depression that the transverse intensity χ

20 11 Case 1 Case Case3 Case4 Case5 Current 1 A 1 A 1 A 1 A 1 A Pulse 1 ns 7 ns 7 ns 7 ns 7 ns ength Energy 1 ev 1 ev 1 ev 5 ev 1 ev Spread χ ε ' T 53.6 K 53.6 K 53.6 K K K Table 1. ongitudinal Intensity Paraeter for UMER. In Cases 1,, and 3, the bea is alost totally space charge doinated. Cases 4 and 5 cannot be realied with UMER, but were included to show exaples of operating paraeters which would result in a longitudinal intensity different fro χ 1. Note that the perpendicular teperature T is K, which shows that the bea will not norally be in theral equilibriu between its transverse and longitudinal properties. Theral equilibriu is approxiately achieved in Case 4. Operating paraeters assued for UMER were: β =., bea radius 1 c, transverse noralied effective eittance 1 µ.

21 1 paraeter χ has to the transverse tune depression, naely k k = 1 χ (11) in the longitudinal case. The synchrotron wave nuber with space charge ( k ) and the synchrotron wave nuber without space charge ( k ) are related by k K = k (11a) 3 where k κ =. For the transverse case, the betatron wave nubers with and without space charge ( k and k ) are used. Betatron and synchrotron oscillations are analogous, with the forer referring to transverse otion while the latter refers to longitudinal otion. Eq. (11a) is derived fro the longitudinal envelope equation by following the sae procedure used to derive the transverse tune depression fro the transverse envelope equation. A second transverse bea paraeter, the plasa frequency ω, is related to the p transverse intensity paraeter through the relation ω ω p = χ. (1) The transverse ero-current betatron frequency ω = k v relates the ero-current betatron wave nuber ( k ) to the bea velocity. A longitudinal plasa frequency ω can also p be defined. A relationship between the longitudinal intensity paraeter and the longitudinal plasa frequency can be written which is siilar to that between the transverse intensity paraeter and the transverse plasa frequency. However, because

22 13 the dependence of the space charge ter on position is different longitudinally than transversely, the analogous longitudinal equation is slightly different. For transverse otion of a particle in a bea[1], qes & x = ω p x =, (13) 3 γ where x is the location of the particle, & x& is the particle's acceleration, and E is the s electric field due to space charge experienced by the particle. This takes into account relativistic effects which reduce the force between electric charges. For longitudinal otion, postulate qes & = ω p =. (14) 3 γ The longitudinal electric field due to space charge is norally taken as E s g = 4πε γ λ, (15) and for a parabolic bea the line charge density λ is given by[1], λ ( ) = λ 1, (16) where λ is the peak line charge density and is the location in the bea. Cobining eqs. (14), (15), and (16), so = gq & ω = p λ, (17) 5 4πε γ qgλ ω p =. (18) 5 γ 4πε

23 14 But the average line charge density can be written in ters of the full length (twice the half length) of the bea and the nuber of particles as λ ~ = qn. The relationship between the peak line charge density and the RMS average line charge density can be shown to be ~ 4 λ = λ. The plasa frequency can then be rewritten as 3 ω 3 gnq 3 v p = c β γ πε 6, or 5 K v ω p =. (19) 6 3 This is siilar to a for of the expression for the transverse plasa frequency Kv ω p =, () r except for the cubed dependence on position (needed because transverse perveance is diensionless while longitudinal perveance has units of eters) and the factor of 5. 6 The longitudinal version of eq. (1) can be found by dividing eq. (19) by the ero-current synchrotron frequency ω = k v and using the definition of χ fro eq. (7): ω ω p 5 = χ.91χ. (1) 6.3. Solutions of the ongitudinal Envelope Equation. We now proceed to solve the longitudinal envelope equation for certain cases. An approxiate solution to the stationary equation ( = ) has been found previously[18] to be

24 15 ε k 3/ 3/ K + k 1/ 3. () However, the design of UMER, with longitudinal focusing applied at only three points, eans that the bea will not be stationary. To solve for the nonstationary case, we first rewrite the longitudinal envelope equation (eq. ) as d ~ ds K = 5 5 ~ ~ ε + ~ κ, (3) 3 ~ where the derivative has been ade explicit. This equation cannot be integrated iediately. Instead, we ultiply both sides by d ~ [19]. ds d ~ d ~ d ~ d d ~ d ~ K ~ ε = = + ~ ds ds ds ds ds ds 5 5 ~ ~ κ 3 (4) ~ ~ d ~ d ~ K ~ ε d = + 3 ~ ds ds ~ 5 5 ~ ~ κ ~ d ~ d~ ds K = ~ ~ ε ~ κ ~ ~ ~ ~ d + ds ~, s (5) In this equation, ~ is the initial RMS length of the bea, and s is the initial location of the bea center, usually taken as s =. For space-charge doinated beas with no focusing (free expansion), both the eittance and focusing ters in eq. (5) can be

25 Eittance Doinated Space-Charge Doinated k /k.8 ( ) ( ) TD χ DPF χ.6 UMER.4. ω p /ω χ ongitudinal Intensity Paraeter (χ ) 1.1 Fig. 3. ongitudinal Space Charge Intensity Paraeter (χ ). This graph relates the intensity paraeter for longitudinal bea physics with the longitudinal tune depression (k /k ) and the longitudinal plasa frequency (ω p ). Beas with χ <. 5 are considered eittance doinated, while beas with.5 < χ 1 are considered spacecharge doinated. Space-charge forces doinate the longitudinal physics of UMER for all practical operating paraeters. The operating range for UMER is indicated by the arrows at the extree right of the graph.

26 17 Transverse Envelope Equation ongitudinal Envelope Equation 3 = + R R K R k R ε x 3 = + K ε κ Transverse Intensity Paraeter ongitudinal Intensity Paraeter k R K = χ c E K K k K 3 3 ~ 5 + = = γ β χ Transv. Intensity and Tune Depression ong. Intensity and Tune Depression = 1 χ k k k k = 1 χ Transv. Intensity and Plasa Frequency ong. Intensity and Plasa Frequency χ ω ω = p p χ χ ω ω = Table. Coparison of longitudinal and transverse relations.

27 18 neglected, and it can be rewritten as 1 5 ~ 5 ~ c K ds d + =, (6) where the square of the initial rate of expansion, ~ ~ s ds d and the initial value of the space charge ter 5 ~ 5 K have been sued to for the constant 1 c. The non-rms half length is ore intuitive, and the need to use RMS values decreases when the eittance is ignored. Therefore, eq. (6) can be rewritten as, + + = s ds d K K ds d. (6a) Taking the square root of eq. (6a) gives, + + = ± s ds d K K ds d, (7) which can be directly integrated to find ) ( s : K c K c c c K c K c s s ln / ± ± + =. (8) Note that the rates of expansion,s ds d and ds d in eq. (6a) are squared, destroying soe inforation about the bea. To counteract this, the ± sign is introduced when the

28 19 d square root is taken in eq. (7). In eq. (8), the upper sign is taken when, and ds the lower sign is taken when d ds. d Referring to Fig. 4, note that = only when the bea is at a longitudinal ds "waist," or local iniu length. By assuing this condition in eq. (6a), and entering the full expression for c, an expression for the non-rms iniu half length of a 1 space-charge doinated bea undergoing contraction in the absence of external longitudinal focusing is found to be K =. (9) w K d + ds, s The longitudinal envelope calculated in eq. (8) for the UMER bea undergoing free expansion is shown in Fig. 6. Note that the independent variable in eq. (8) is the distance traveled by the bea center, s, and the dependent variable is the non-rms halflength of the bea,, so when d ds, ds d, and the graphing progra will not plot the curve in its entirety. However, the curves in these regions can be found by siply extrapolating the plotted curves. Note also that the length referred to in Fig. 6 is the non-rms full length, which is greater than the RMS half-length ~ by a factor of 5.

29 Fig. 4. ongitudinal envelope for a bea with initial half-length and initial slope d ds, and with a "waist" of half-length when the bea center is located at w s. w Fig. 5. This figure shows a bea which is in the process of foration but which has an initial contraction ( d < ) due to a velocity tilt iposed at the cathode. ds K represents the location of the cathode. The length of the overall line represents the length of the bea which is taken into account by the theory. Although only the portions of the lines to the right of the cathode actually exist, this is not accounted for by the theory. Thus, it is possible to discuss a bea which is contracting even though it is in the process of creation at the cathode, and the overall length of the bea is increasing.

30 1.4. ongitudinal Focusing for Parabolic Beas. The longitudinal envelope equation can also be used to consider a bea where longitudinal focusing is present. For this, we return to eq. (5). If the bea is spacecharge doinated, the eittance ter can be neglected, and eq. (5) can be rewritten as d ~ ds K = 5 5 ~ ~ κ + c, (3) where K ~ ~ 5 ~ d c = + κ +. (31) 5 ~ ds, s Integration can be perfored on eq. (3) to find the longitudinal envelope in the for s ( ~ ) : s = s + ~ ~ K ~ κ 5 5 ~ ~ ~ 4 ± + c. (3) The choice of sign depends on whether the bea is undergoing contraction or expansion. κ could If the electric field applied at each gap was known as a function of tie, ( s t), be deterined, and the longitudinal envelope found directly fro eq. (3). However, since the goal is to deterine the fields needed, the use of eq. (3) is not the ost efficient ethod. The geoetry of the UMER focusing lattice, which was decided early in its developent, poses a constraint on the design of the longitudinal focusing syste (Fig. 7). The longitudinal focusing lattice consists of three locations in the ring which are available for use by the induction gaps. These locations are equally spaced along the ring, with the

31 15 Total Bea ength () 5 τ c β 1 ength [] s 1 ( ) Nuber of Revolutions in UMER 1 1 Nuber of Turns Fig. 6. Expansion of Parabolic Bea, showing that the UMER bea would copletely fill the ring after the sixth revolution. Solid curve is full bea length, dotted line is initial full length of bea, and dash-dot line is the circuference of UMER. Note that the expansion becoes linear past about the seventh revolution. Current is 1 A, pulse length is 1 ns. Injector length is neglected.

32 Fig. 7. ongitudinal Focusing attice, showing location of cathode K, induction gaps IG1 and IG, idpoint between induction gaps M, and longitudinal envelope at each location 3

33 4 first location being about downstrea of the gun, and with a spacing of about 4 between each location in the ring. This particular geoetry is useful because its syetry can be exploited in siplifying the design of the longitudinal focusing syste, and because the results presented below are applicable to any space-charge doinated bea transport syste with siilar syetry. At the cathode (K in Fig. 7), the bea has its initial length ~ and divergence ~. Between K and the first induction gap (IG1), a distance of about, the bea has undergone free expansion as given by eqs. (7) and (8), arriving at IG1 with new length ~ and divergence ~. In the thin lens approxiation, the length of IG1 is very sall, 1 1 but not ero. Accordingly, the bea eerges with length ~ = ~ + δ ~, (33) 1 where δ is sall but not ero, and with divergence ~ given by eq. (3). If the focusing effects of IG1 have totally overcoe the space-charge-driven expansion of the bea, the bea will begin to contract, arriving at the idpoint between the gaps (M) with a new length ~ and divergence ~ 3. If ~ 3 ~ 3 = and ~ 3 = ~, and if the internal structure, entropy, etc., of the bea have reained approxiately unchanged, the envelope fro M to IG should be identical to the envelope fro K to IG1. The fluidanalogy equations which govern changes in the bea are tie-reversible[7,], so we expect that we can produce conditions at M which are identical to those at K provided that ~ ~ 1 = and ~ = ~. Since the forer condition is approxiately fulfilled siply 1 by the short length of the induction gap, the proble is to find the focusing strength at IG1 which will result in ~ = ~. Fro eqs. (3) and (31), 1 1

34 5 ~ K ~ ~ ~ ~ K = ~ κ + κ +. (34) The negative sign was chosen because the bea is undergoing focusing, and therefore ~ < ~. For ~ = ~, K 5 ~ Cobining eqs. (33) and (35) gives Taking the liit of eq. (36) as δ gives ~ K ~ 5 5 ~ κ + + κ 1 =. (35) 1 K ~ 3 ~ = κ 3 ~ 1 κ 1 δ κ δ. (36) K = 5 5 ~ 3 1 κ, (37) the approxiate focusing strength needed to achieve a stable longitudinal focusing lattice for a parabolic bea in UMER as shown in Fig. 7. A critical step in this calculation was the assuption that the length of the first induction gap was sall but not ero. If its length were ero, the bea would be inside the gap for no period of tie, eaning that no focusing would occur, and the bea properties before and after the first induction gap would be identical. Since the gap length is not ero, the gap is able to apply focusing forces to the bea, causing it to begin to contract. However, since the gap length is sall, the bea length will only have changed by a very sall aount, δ, during its transit of the gap. Thus eq. (35) is a eaningful, nontrivial expression allowing us to calculate the focusing constant in the liit of the thin lens approxiation.

35 6 However, this does not take into account the relative lengths of the induction gap and the focusing period. To account for this, eq. 37 ust be ultiplied by the ratio of the focusing period to the gap length: K period κ = ~. (37a) gap Although this does not arise naturally out of the above treatent, it ust be included to obtain the correct results. Fro the expression for the focusing constant in eq. (37a), the actual electric field strength needed to properly focus a parabolic bea in UMER can be calculated. The focusing constant itself is defined in ters of the applied electric field gradient in, E [1]: a κ qe = a. (38) 3 c β γ In this case the prie denotes a derivative with respect to, the distance in the bea frae where = at the bea center. The applied electric field along can also be written in ters of E : a Solving for E in eq. (38) and using eq. (39), a dea E = E d = d. (39) a a d E 3 c β γ = κ d. (4) q a Fro the relation between the focusing constant and the longitudinal perveance in eq. (37),

36 7 E a = 3 c β γ q K 5 5 ~ 3 1 period gap d. (41) Since ~ is the actual RMS half length of the bea before it enters the first induction gap, 1 it is not included in the integration, which yields E a = 3 c β γ q K period gap 3 1, (4) where ~ 1 = 5 is the non-rms half length of the bea before it enters the first 1 induction gap, is the location in the bea at which the field E is applied, and a = is the center of the bea. Eq. (4) can be further described in ters of fundaental quantities by using the definition of longitudinal perveance fro eq. (3): E a = 3 gnq period 4πε γ. (43) gap 3 1 The applied focusing field can be written in ters of the electric field due to space charge, E s g 4πε γ λ, (44) where g is a geoetry factor. Eq. (44) is only exact for certain distributions[1], but is approxiately true for any distribution in the special case where λ is sall[1]. The geoetry factor g depends on bea radius, bea pipe radius, bea length, and location in the bea[1]. The exact value of g ust be calculated nuerically. For beas whose length is uch greater than the bea pipe radius, an average value for g is given approxiately by

37 8 b g α + ln, (45) a where a is the bea radius, b is the bea pipe radius, and α is a constant. The value of α is generally given as one[1], but other values have been proposed. Where bea variation is slow[,3], near the center of a long bea[1], or where the longitudinal electric field is being easured on the bea surface[7], α has been given as ero. When g is averaged over the entire bea cross section, α has been given as.67[1] or.5[7,4,5]. Other authors indicate that g, and therefore α, is independent of radial position[,3]. An experient carried out by D.X. Wang at the University of Maryland found α =.1±. 16 [7]. These values are clearly inconsistent, and additional work is needed to deterine the best value for α. Using eqs. (43) and (44), and assuing the parabolic line charge density of eq. (16), the applied field can be related to the space-charge field by E a 3Nq [ E ] period. (46) s 4λ 1 gap Note that eqs. (4) and (43) are linear in, which is expected since the bea pulse shape is parabolic. Also note that the applied field needed to properly focus a parabolic bea in the UMER lattice is proportional to the longitudinal perveance (and therefore increases with space charge), and depends (slowly) on the ratio of bea diaeter to bea pipe diaeter through the geoetry factor g..5. iitations. Although the longitudinal envelope equation is directly useful in describing the behavior of parabolic beas, and in describing soe fundaental behavior of

38 9 nonequilibriu 1 equivalent beas, it does have liitations. The longitudinal envelope equation does not directly give detailed inforation about the velocity distribution and line charge density in non-parabolic beas, which are necessary to design focusing systes for those beas. Also, the longitudinal envelope equation is not the best forat for describing the propagation of longitudinal space-charge waves. 1 The equilibriu distribution is the Boltann profile, which is parabolic for ero longitudinal teperature and Gaussian for high longitudinal teperature. This distribution retains its shape during expansion[16].

39 Fig. 8. Applied electric field for focusing of parabolic bea in stable UMER longitudinal focusing lattice. 3

40 31 3. One-Diensional Cold Fluid Model Introduction. The longitudinal envelope equation is a frequently used odel for the longitudinal behavior of space-charge doinated beas. However, this odel has liitations and is not useful in all cases. An alternate odel, the cold-fluid odel, akes use of the fact that the equations governing the behavior of space-charge doinated beas are siilar to those governing cold, copressible fluids. One application of this odel which is particularly iportant is to the expansion of an initially rectangular bea. This line charge profile is useful because its flat top could allow better easureents of space charge waves[6] and could provide the constant ipedance necessary for induction linacs[16]. In this section the governing equations are introduced and results are given for the special case of an expanding, initially rectangular bea. 3.. Governing Equations. For a space-charge doinated bea undergoing free expansion, it is possible to write two equations which govern the behavior of the particles in the bea. The first is the continuity equation ρ r r + J =, (47) t which relates the volue charge density ρ and the current density J v, and which can be derived directly fro Maxwell's Equations[7]. If all behavior being considered is strictly longitudinal (along ẑ ) and if the bea has a constant cross-sectional area A, the line charge density λ = Aρ, and the continuity equation can be rewritten in its onediensional for

41 3 since r r J = ρv. λ + vλ =, (48) t The longitudinal force exerted on a particle in a space-charge doinated bea undergoing free expansion is strictly due to space charge. This field is taken as E s g 4πε γ λ. (49) The acceleration of particles in the bea has two physical origins: a stationary ter due to variation of the flow as particles pass through regions with different properties; dv v d dv and a nonstationary ter which describes changes in the flow at any given dt location[8]. Thus F v v e = & = + v = E 3 s. (5) t γ Taking into account the approxiate expression for space charge field given in eq. (49), v v + v t qg 4πε γ 5 λ. (51) This equation is known as the oentu equation[7,4]. The oentu equation and the one-diensional continuity equation together fully describe the particle flow in onediensional cold beas in ost cases. In copressible fluid flow, the oentu equation is replaced by the analogous Euler equation v v + v t 1 p, (5) ρ

42 33 where p is pressure and ρ is volue ass density. Bernoulli's equations for copressible and incopressible flow can be derived fro this equation[8] Rectangular Bea Expansion and the Method of Characteristics. One ethod of finding solutions to the oentu and continuity equations which was first developed for supersonic gas flow and which has been used by the bea physics counity for soe tie is the ethod of characteristics[9-31]. In the ethod of characteristics, increental changes in the fluid propagate through it as waves. One application of the ethod of characteristics to beas is to the evolution of an initially rectangular bea pulse. This solution is given elsewhere[7,4], and will not be repeated in depth here, although the underlying physical behavior will be discussed in general ters and its results will be given below. For longitudinal focusing, it is iportant to understand the source of the bea expansion. Consider the edge of a rectangular bea. Inside the bea, far fro the edge, the electric fields experienced by a charge tend to cancel, since there are opposite contributions fro particles to the left and right of the particle under consideration. At the edge, there will be a net field, since the contributions of charges to one side are not balanced by contributions of charges fro the other side. This results in a net force on charges near the edge of the bea, causing the to accelerate outward fro the bea, causing the bea to expand. Although this picture provides a rough understanding of the physics of bea expansion, it does not iediately give a nuerical description of the expansion. For this quantitative understanding, we turn to the fluid analogy and the phenoenon of cavitation. Consider an infinitely long pipe, filled with a fluid which is confined to

43 34 by a assless piston at = (Fig. 9). At t =, this phanto piston is oved towards the right at infinite speed. Since the fluid consists of real, assive particles, it can only adjust to the new configuration at finite speed. The fluid will adjust by producing two waves, one which travels into the body of the gas at the speed of sound in the gas, and the other which travels outward into the vacuu at a axiu speed, called the escape speed. The speed of sound in a charged particle bea is given by[7,] Zqgλ c =, (53) 5 4πε γ where Z is the charge state of the particles in the bea, q is the fundaental charge, and is the ass of the particles in the bea. For a 1 A electron bea with g =.8 and β =., 6 c = The escape speed is given by[3] s u c = k 1 ax or u ax = c k 1, (54) where k = for an electron bea[]. Thus, the escape speed, also the speed of the expanding edge, is u edge = c. (55) This treatent is not valid at very low pressures, but it is appropriate for beas like that produced in UMER, since the fluid analogy of space-charge doinated beas are cold, high-pressure gasses.

44 35 This situation is identical to that of a bea which has just been produced at the cathode. In that case, the flat top will shrink fro each end with the speed of sound, while the bea will expand outward fro each end with twice the speed of sound. Further explanation of the behavior of an expanding, initially rectangular spacecharge doinated bea can be found by using the full ethod of characteristics to find λ (, t) and v (, t). When this is done, the following results are obtained: Zone I (Dead Zone) λ = λ (56a) v = v = (56b) 1 Zone II (Rarefaction Zone) λ(, ) λ 3 3 t = ± (56c) tc Zeg v(, t) = λ (56d) 3 t 4πε Zone III (Vacuu) no fluid present In eqs. (56), λ is the initial line charge density of the rectangular pulse, v is the initial velocity (in the bea frae) of the particles in the bea, which is taken to be ero, is the location in the bea with = at the bea center, is the initial location of the front or rear edge of the bea, and t is the tie easured fro t = when the phanto piston is reoved. The ± and signs refer to the fact that there are two rarefaction ones, one fored by the erosion of the flat top fro the front edge of the bea, and one fored by the erosion of the flat top fro the rear edge of the bea. The upper sign is used when the erosion is fro the rear edge, and the lower sign is used if the erosion is fro the front edge. The equations presented above are only valid so long as the two

45 Fig. 9. Fluid Analogy for expansion of rectangular electron bea. Gas is initially at rest, confined to <. At t =, assless piston is reoved to the right at infinite speed. Shockwaves propagate into the gas at the speed of sound c, and into the vacuu at escape speed c e, foring a rarefaction one where the copressible fluid is present but at lower density than in the undisturbed region to the left of the left going shockwave. 36

46 37 rarefaction ones have not connected. At this point, the flat top has been eliinated, and the bea is "all edges." A cusp will occur at the iddle of the bea where the slope of the line charge density abruptly changes. After this tie, the shock waves which caused the flat top to erode begin to overlap, and the resulting nonlinear equations cannot be solved exactly. An approxiate solution is given elsewhere[7,]. However, for the purposes of establishing a stable longitudinal focusing lattice, it is desirable to prevent the bea fro reaching the cusp point, for reasons addressed in section 4.3. For a 1 A electron bea with g =. 8 and β =., and an initial full length of 6 the cusp will occur after 1.16 µs. By this tie the bea center will have traveled 69.6, and the new full length of the bea will be Coherent Energy Spread. Coherent energy spread, the difference in kinetic energy between the fastest particle and the slowest particle in the bea, is an iportant consideration in the design of focusing systes. In the expanding rectangular pulse, the particles at the extree head and extree tail are expanding away fro the center of the bea (in the bea frae) at the escape velocity c. The energies of particles at the bea center, at the extree head, and at the extree tail can be calculated nonrelativistically in the laboratory frae as and 1 T ( cβ ) center =, 1 T ( ) head = cβ + c,

47 38 T tail 1 = ( cβ c ). Nonrelativistically, the kinetic energy of the center particle in an electron bea traveling at β =., easured in the laboratory frae, is 1. kev. If the initial bea current is 1 A, and g =. 8, the energy of the particle at the extree head of the bea, corresponding to its escape velocity, will be 1.1 kev, while the energy of the particle at the extree tail of the bea will be 8.5 kev, for an extree coherent energy spread of 3.6 kev. This calculation was repeated for an experient currently being perfored by Yupeng Cui at the University of Maryland. In that experient, the noinal electron bea energy is 5 kev, the actual bea center energy is kev, the bea current is 135 A, and the geoetry factor is approxiately 3.7. The speed of sound in the bea, calculated using eq. (53), is The kinetic energy of the particle at the s extree head of the bea, T head, is 7.14 kev. Siulations perfored by Yupeng Cui[3] using the WARP 3D siulation code indicate that the kinetic energy of the particle at the extree head is 7.19 kev, which indicates only a 1.34% difference between the one-diensional theory and the three-diensional siulation. The particles at the extree head and extree tail of the bea will never increase their velocity. Since the expansion speed, and therefore the coherent energy spread, cannot be changed without changing the initial line charge density, it is ore proper to think about the nuber of particles in the edges as opposed to the extree coherent energy spread. Due to the transverse focusing syste, particles whose energies are different fro the bea's design energy will be isatched. For particles whose energy falls within a certain range, their trajectories will siply differ fro the design orbit of

48 39 the achine. If their energies fall outside that range, their trajectories will be so different fro the design orbit of the achine that they will be lost to the wall. Because there will always be soe particles traveling at the escape speed, if the escape speed is higher than the critical energy at which a particle is isfocused into the wall, there will be soe particles lost fro the bea. However, the nuber of particles lost by the bea will depend on how any have achieved velocities in excess of the critical velocity, which depends directly on how long the bea has been propagating. Therefore, although particle loss ay not be avoidable, it is possible to liit the nuber of particles lost by applying longitudinal focusing to liit the length of the edge regions, and therefore the nuber of particles with an energy above the critical energy Discrepancies. Soe siulations and experients have been conducted to deterine whether a rectangular, space-charge doinated bea erodes according to the results of the cold fluid odel, notably by D.X. Wang at the University of Maryland[7], and by A. Faltens at awrence Berkeley aboratory[4]. Experiental data produced by Faltens and Wang agrees well with siulations perfored by Wang. Recent siulations, which were perfored using the WARP siulation code for space-charge doinated beas, are also in good agreeent. These siulations and experients uniforly show a line charge density which is qualitatively different fro, but quantitatively siilar to, the line charge density deterined fro the cold fluid odel. Specifically, siulations and experients indicate that the transition fro the flat top to the edge is not abrupt as shown in Fig. 1, but ore gradual. The agnitude of the discrepancy is sall enough that it was not even entioned by Wang and Faltens, but the fact that it appears consistently in two

49 4 independent experients and two independent siulations suggests it is ore than ere experiental error. Although the source of this rounding is not yet understood, several assuptions of the cold fluid odel are not exactly correct and ay be to blae. For exaple, the space charge field given in eq. (44) is only correct for a slowly-varying line charge, which is not the case early in the evolution of a rectangular bea pulse. In addition, the cold fluid odel assues that all effects are one-diensional while the actual bea is a three-diensional object. This discrepancy is currently under investigation.

50 Fig. 1 (a - d). ine charge density (left) and electron velocity in the bea frae (right) for the front half of the bea, easured at s = (top) and s = (botto). Graphs at correspond to initial conditions at cathode, while graphs at correspond to conditions at first induction gap. Bea paraeters: 1 A current, 1 ns initial full pulse length, β =., g =

51 Fig. 1 (e - h). ine charge density (left) and electron velocity in the bea frae (right) for the front half of the bea, easured at s = 14 (top) and s = 6 (botto). Graphs at 14 correspond to conditions after injection and one revolution without focusing, while graphs at 6 correspond to conditions after injection and two revolutions without focusing. Bea paraeters: 1 A current, 1 ns initial full pulse length, β =., g =.86. 4

52 43 4. ongitudinal Focusing in UMER: A ore general approach Introduction. The one-diensional cold fluid odel is particularly useful for designing longitudinal focusing systes because it gives detailed inforation about the velocities of particles in the bea and the line charge density of the bea as the bea travels through the transport syste. This inforation, along with the periodicity of the longitudinal focusing syste in UMER, can be exploited to calculate the necessary longitudinal focusing field in a ore general way than was presented in connection with the longitudinal envelope equation. The ethod described below has two advantages: as input it requires only v () and λ () at the location of the gap, which can be found through any eans, be it theory, siulation, or experient; and it is applicable to any space-charge doinated bea transport syste exhibiting a longitudinal focusing syste with periodicity siilar to UMER. 4.. ongitudinal Focusing. As the bea expands fro its initial length at the cathode, its line charge and the velocity distribution of its particles will change. For soe initial conditions, such as the rectangular bea, these properties can be calculated. For other initial conditions, it ay be difficult to calculate these properties, and siulations ust be used. In a longitudinal focusing lattice (Fig. 7), this expansion will proceed until the first induction gap has been reached, at which point soe electric field, which ay vary in tie, is applied to the bea. Because the oentu and continuity equations are tie reversible[7,], it should in principle be possible to reverse the bea's expansion by reversing the velocity (in the bea frae) of every particle in the bea by applying a carefully tailored electric

53 44 field at the first induction gap. This converts the bea's expansion into contraction. Because this contraction is the irror iage of the bea's earlier expansion, it will reach a waist where the bea properties are identical to those at the cathode (assuing entropy does not increase). Thus the expansion fro this waist to the second induction gap is identical to the expansion fro the cathode to the first gap. Each expansion, each contraction, and each induction gap pulse will be identical if the velocity reversal is done perfectly. The task is now to deterine what electric field should be applied to reverse the velocity of each particle in the bea. Assue a bea of half-length, centered at the h origin in its rest frae. A planar diode, with a hole in its center, will serve as a odel for the induction gap. The diode has a gap length l. An electric field E (t), which is unifor in space but variable in tie, can be applied across the gap. The entire diode is traveling with velocity cβ in the rest frae of the bea. As the diode oves past the bea, each point in the bea is exposed to the field in the gap ( E (t) ) for l seconds. cβ E (t) varies slowly enough that the field applied to any particle is approxiately constant during its passage through the gap. Because the bea consists of electrons, an ipulse I is applied to each particle as it passes through the gap: I = F t = q l cβ l (. (57) cβ cβ h [ E ] = q E ) E = p = v net s Note that this equation is nonrelativistic, so only the particles' velocities are affected, not their effective asses. The location of the gap ( = h cβt ) in the bea frae has been used to describe the applied electric field in ters of spatial coordinates. E s ( ) is the