Increasing the intensity of an induction accelerator and reduction of the beam breakup instability

A 7 cm cathode has been deployed for use on a 3.8 MV, 80 ns (FWHM) Blumlein, to increase the extracted electron current from the nominal 1.7 to 2.9 kA. The intense relativistic electron bunch is accelerated and transported through a nested solenoid and ferrite induction core lattice consisting of 64 elements, exiting the accelerator with a nominal energy of 19.8 MeV. The principal objective of these experiments is to quantify the space-charge limitations on the beam quality, its coupling with the beam breakup (BBU) instability, and provide an independent validation of the BBU theory in a higher current regime, I > 2 kA. Time resolved centroid measurements indicate a reduction in BBU > 10 × with simply a 50% increase in the average B-field used to transport the beam through the accelerator. A qualitative comparison of experimental and calculated results are presented, which include time resolved current density distributions, radial BBU amplitude relative to the calculated beam envelope, and frequency analyzed BBU amplitude with different accelerator lattice tunes.


I. INTRODUCTION
Relativistic electron beams used to study fundamental nuclear physics or provide intense sources of photons are challenged with instabilities to overcome when increasing the intensity of the beam [1][2][3][4]. One of particular interest is the beam breakup (BBU) instability which manifests itself as a transverse magnetic coupling to destroy the beam quality. BBU was first observed in the 1960s [5] and reported in detail by Stanford Linear Accelerator Center scientists in 1968 [6]. Shortly after its discovery, BBU was studied for the first time in detail on the Experimental Test Accelerator and Advanced Test Accelerator linear induction accelerator facilities [7][8][9][10]. A 10% reduction in beam current was observed on the Experimental Test Accelerator after acceleration through eight cells with rf oscillations on the beam envelope as large as 1 cm [7]. Initially the Advanced Test Accelerator only transported 10 kA up to 15 MeV and lost 85% of the beam current at 50 MeV for vacuum transport which was later improved upon with laser ion guiding [9,10].
BBU has also been observed in the rf linac community at CEBAF, the Jefferson Lab FEL and the KEARI facility [1,2,11,12]. References [1,2] experience multipass multibunch BBU due to a transverse magnetic dipole higher order mode (HOM) excited by the misaligned beam bunch as it passes through the accelerator cavity. HOM in 2.15 GHz range limited CEBAF currents ∼40 μA. After reducing the Q of the HOM through a variation in the transport matrix, threshold currents > 100 μA were achievable. Lower energy 352 MHz and 1.3 GHz rf cavities experience 446-520 MHz HOM limiting threshold currents to < 10 μA [11,12]. In each case BBU is suppressed through rf beam focusing using a TE HOM thereby increasing the threshold current to > 1 mA.
The first axis of the dual-axis radiography for hydrodynamic testing (DARHT) facility is exploring the limitations of increasing the intensity of the electron beam for future radiographic capabilities. DARHT Axis-I is unique for these studies because it is relatively simple to change the size of the cathode emission size to increase or reduce the total current and therefore change the space charge of the beam while holding everything else constant. In order to effectively increase the intensity of the beam, the BBU instability must be quantitatively understood and effectively eliminated. Beam position monitors (BPMs) provided time resolved centroid and BBU measurements.
As a facility BBU was first studied at DARHT analytically in 1991 [13] and experimentally on the integrated test stand which consisted of the Axis-I injector and one cell block consisting of eight induction cells [14]. This experiment scratched the surface, because BBU grows exponentially as the number of acceleration gaps is increased. BBU has also been studied extensively on DARHT Axis-II [3,4,15] and only recently has been explored in detail on the full scale DARHT Axis-I even though the accelerator has been running since 1999. The results presented provide an independent validation of the BBU theory in a higher current regime, I > 2 kA, and a successful demonstration Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. of the reduction of BBU and full transport through 64 cells without disruption of the beam quality.

II. BEAM BREAKUP INSTABILITY
As stated above in the Introduction, BBU manifests itself as a transverse magnetic coupling to destroy the beam quality. The existing TM 0n0 modes in the induction cell cavities interact with the misaligned beam, placing a time dependent transverse magnetic dipole kick on the beam as it passes through the acceleration gap. The time-dependent dipole kick places an rf oscillation on the beam envelope, breaking up the distribution and causing eventual loss of beam current.
The BBU growth along the accelerator is characterized by the equation below [16]: where ξ is the measured BBU amplitude at a given location and ξ o is the measured BBU amplitude at the entrance of the accelerator. The amplitude decreases with acceleration to 1=2 power and increases exponentially with the maximum growth factor, Γ m : where c is the speed of light, I b is the beam current, N g is the number of gaps, Z ⊥ is the transverse impedance of the induction cell cavity, and h1=Bi is the average of the inverse magnetic field strength. The transverse magnetic mode couples more strongly to the beam as the current and the number of acceleration gaps increases, increasing the rf oscillation on the beam for a fixed cell design and transverse impedance. The frequency band coupled to the beam in the measurements presented below in Sec. IV (FWHM ∼50 MHz at 750-800 MHz) is slightly smaller compared to the modes measured with a drive rod (300-900 MHz) by Refs. [13,21,24]. Aside from designing the cell cavity to minimize the transverse impedance, reducing the number of acceleration gaps and increasing the transport magnetic field are the only ways to vacuum transport high current beams through an accelerator. The accuracy of the BBU measurements described below is evaluated by the time of the BBU to reach maximum Γ m , t, and resonate the induction cell cavity: where Q is the quality factor of the cavity, and ω is the radial BBU frequency. The calculated saturation time of the BBU is tabulated for the separate tunes below in Sec. IV.

III. EXPERIMENTAL SETUP
The experimental configuration used to study the BBU instability was the DARHT Axis-1 linear induction accelerator ( Fig. 1). The accelerator is composed of a 4 MV Blumlein injector [17,18] and 32 Blumleins [19,20] used to drive a total of 64 induction cells (two cells per Blumlein). The linear induction accelerator is broken up into eight cell blocks consisting of eight ferrite induction cores in each cell block for a total of 64 induction cells. Each induction cell has the ability to impart 250 keV of energy into the beam for a total of 16 MeV in addition to the 3.8 MeV acquired in the diode.

A. BPMs
The transported beam current and centroid is monitored by BPMs at the end of each cell block and internal to each cell block, so there is a BPM every four cells with axial spacings between 185-224 cm. The BPMs consist of eight B-dots, or inductive monitors, oriented azimuthally every 45°. There are four position B-dots, one top and bottom for AEy measurements and one left and right for AEx measurements. There are four more oriented at 45°relative to the position B-dots that are used for current averaging over the cross section and as additional position measurements. The B-dots are simply a type-N coaxial feedthrough with the center conductor soldered to an aluminum tab machined out of the inner cross section of the flange they are all housed in. The B-dots pick up the inductive image current as the beam head and tail pass by the B-dot. The amplitude of the signal is dependent on the proximity of the beam relative to the B-dot. A perfectly centered beam will have equal signals on each B-dot in a BPM housing, assuming each BPM has the same impedance.
The AEx and AEy B-dots in each BPM were used to measure the BBU amplitude throughout the accelerator. This was done utilizing two methods; in the first method we used a hardware integrator (resistive and capacitive circuit) to measure the integrated position offset. This signal was normalized to the position offset in x and y and then the BBU amplitude in mm is calculated. One disadvantage of this method is the integrator effectively reduces the amplitude of the frequency spectrum. Evidence of this effect is shown below in Sec. IV where a comparison of the frequency spectrum is made between the two methods. The second method requires an unintegrated Δx and Δy signal which was sampled up to 8 GHz. A fast Fourier transform was applied to the signal over AE50 ns in addition to the pulse length of the beam. The frequency spectrum was then integrated from 600-900 MHz to determine the BBU intensity.

B. Induction cells
The Axis-I induction cell design is described in Ref. [21]. Each cell consists of a ferrite induction core that is driven with an oil-insulated transmission line. The transverse impedance for the cells is calculated with the formula, first derived by Ref. [22], where Q is the quality factor of the cell cavity at the resonant frequency, B y dz is transverse magnetic field component which imparts change in the transverse momentum to the particles as they traverse the acceleration gap, dz, ω o is the resonant frequency, and U is the stored energy in the cell. This formulation indicates the importance of reducing the transverse magnetic field and damping the Q value of the resonant frequency to minimize Z ⊥ . Another formulation of the transverse impedance from Refs. [13,21,23] with respect to the acceleration gap is where Z o is the impedance of free space (377 Ω), η is a function of the surface impedance of the cavity wall, which ranges from 0.7-2, g is the accelerating gap width in the induction cell (19 mm), and b is the beam pipe radius (73 mm) yielding Z ⊥ ¼ 745.9η Ω=m, so the transverse impedance ranges from 522 to 1491 Ω=m. References [13,21,24] measured 400-1200 Ω=m over frequency ranges of 300-900 MHz on the final cell cavity design.

IV. TRANSPORT AND BBU MEASUREMENTS
The initial conditions of the beam extracted from the diode and injected into the accelerator lattice are shown in Fig. 2. The voltage in the diode is monitored by a flush mounted coaxial E-dot aligned axially with the edge of the cathode shroud. The E-dot is mounted on the surface of the vacuum tank, ∼1 m radially from the centerline of the diode. The E-dot picks up the capacitive charge voltage as voltage rises and falls on the cathode shroud. The integrated voltage waveform is shown in Fig. 2. The extracted beam current measured at BPM02, 82.4 cm downstream of the cathode face, for the 5 cm cathode is ∼1.7 kA and for the 7 cm cathode a current of ∼2.9 kA is yielded (Fig. 2). The calculated beam envelope transported to the diagnostic location at z ¼ 167 cm with the first transport magnet current set to 200 A is shown in Fig. 3(a) and the beam envelope transported through the diode for the 2.9 kA, 3.8 MV beam is shown in Fig. 3(b). Both envelopes were calculated using the TRAK electron-gun design code [25,26]. More details of the diode physics are described in Refs. [27,28].
We began tuning the beam envelope for the 2.9 kA beam using the nominal tune for the 1.7 kA beam and the initial envelope conditions measured from a sweep of the first transport magnet (Fig. 3). The 2 rms beam size, a ¼ 2σ, is the second moment of the integrated Jðx; yÞ distribution from the measured optical transition radiation (OTR) profile at z ¼ 167 cm [ Fig. 3 target. These measurements were also compared with calculations made with the TRAK electron-gun design code and provided the initial conditions listed in Table I for transport magnet settings of 190 and 200 A. When examining Fig. 3(d) it is evident there is disagreement where the beam reaches a waist and thereafter. In the experiment the beam reaches a waist of 5.5 mm with a magnet current of 222 A (776 G); however in the TRAK calculation a slightly smaller waist of 3.1 mm is achieved with a magnet current of 225 A (787 G). However, the TRAK calculation does not include two beam-target interaction effects: the first is shorting out of the radial electric fields, which will further reduce the beam size, and the second is beam induced target heating and gas desorption [29]. The second effect will cause migrating gas desorbed off the surface of the target to be quickly ionized by the intense electron beam and the ions will backstream into the beam potential, reducing the beam space charge and overfocusing the electrons upstream of the target. This effect will increase the measured spot size at the target for magnet currents ranging from 220-230 A. When the beam on target is > 1 cm, target heating and gas desorption becomes negligible and does not explain the discrepancy in the measured spot sizes for magnet currents > 230 A. Looking at the data more closely, it is evident that an increase in the emittance beyond the focus, due to thermalization of the beam, would explain the difference and a fit to the data indicates a 22.5% emittance growth.
We set the first magnet to 190 A for all of the tunes except our final tune. Initially it was assumed because of the high space charge (K ¼ 6 × 10 −4 ) of the beam coming out of the gun we could not converge the beam envelope too steeply because it would overfocus in the low or field-free regions between cell blocks. K is the  dimensionless perveance, or the ratio of the space-charge forces to the inertial forces of the beam and is defined as where e is the fundamental electron charge, I is the electron beam current, ϵ o is the permittivity of free space, and m e is the electron mass. The first attempt at tuning the 2.9 kA beam, Tune 4, is shown in Fig. 4, where with a gradual increase of the magnetic field from 360 G at the beginning of the first cell block to > 800 G at the end of the second cell block we were able to gradually converge the beam envelope down to ∼13 mm. The envelopes shown in Fig. 4(a) were calculated using the XTR code [30,31] and were initiated with the initial conditions measured and calculated at z ¼ 167 cm above in Fig. 3 and Table I. Evidence of the space-charge force quickly increasing the 2.9 kA beam radius between the cells is shown at z ¼ 650 cm in Fig. 4(a). Initially only three magnets were changed in cells 8, 9, and 10 to tune the envelope of the 2.9 kA beam from the nominal 1.7 kA tune; this is evident for the hBi calculated in Fig. 4(b) where there is only a slight difference after cells 5-8 (z ¼ 572 cm) and cells 9-12 (z ¼ 796 cm). After initially attempting Tune 4 and examining Eqs. (1) and (2) it was expected that the final BBU amplitude should increase by expð2.9=1.7Þ ∼ 5.5 with everything else held constant and assuming a similar initial BBU amplitude, ξ o , for both cases. However, after quickly investigating the signals on the downstream BPMs it was apparent that there was a substantial amount of BBU, which manifests itself as rf superimposed on the beam envelope. The top row of Fig. 5 shows the BBU on the raw signal of the 2.9 kA beam is ∼20× higher than the 1.7 kA beam. After performing a fast Fourier transform over the 200 ns window of interest the frequency spectrum indicates nearly a ∼40× increase in BBU. Both data sets indicate the BBU spectrum ranges from 700-850 MHz. The two cases shown here are a single shot representation for a data set composed of at least five shots; these illustrate the median BBU amplitude.
Through a more detailed analysis of the integrated frequency spectrum at the end of each cell block it is apparent that the average BBU amplitude for the 2.9 kA tune was > 10× higher than measured with the 1.7 kA beam [ Fig. 6(a)]. BBU is negligible for the nominal 1.7 kA tune until BPM07 (z ¼ 980 cm) and the average measured initial value for five shots is hξi ¼ 0.044AE 0.032. There is a fluctuation in the BBU amplitude along the accelerator at BPM07 for the 1.7 kA tune. This is most likely due to the coupling of a misaligned beam to the cell cavities in cell block 2 (CB2). Slight offsets > 1 mm in the beam centroid enhance the BBU amplitude. Reducing the beam centroid offsets helps mitigate the BBU growth; the physics of these effects are explored in more detail in Ref. [32]. The BBU amplitude for Tune 4 matches the 1.7 kA tune at BPM07 at the end of CB2 (z ¼ 980 cm) and then increases ∼11× the amplitude of the 1.7 kA beam at BPM09, through the third cell block. The error bars in Fig. 6(a) indicate the shot to shot variation of the BBU amplitude which ranges from 10%-60% depending on the amplitude and location. Over the length of the accelerator BBU grows 500× for the 1.7 kA beam and > 10 3 × for Tune 4, leading to a factor of ∼28× higher BBU amplitude for the 2.9 kA beam at BPM 20 (z ¼ 3.6 m) (Fig. 6). Direct comparison of the BBU amplitude for the 1.7 kA beam with the 2.9 kA beyond BPM07 beam clearly shows ξ 2.9 =ξ 1.7 ≫ expð2.9=1.7Þ ∼ 5.5. Applying a least squares fit to both data sets in Fig. 6(b) we are able to back out the slope of the exponential BBU growth and determine a measurement of the transverse impedance of the accelerator cells. The slopes are slightly different; the 1.7 kA data set yields 1514 Ω=m and the 2.9 kA data set yields 1121 Ω=m. Each are close to the measured and calculated cavity values mentioned above in Sec. III B.
These values for Z ⊥ and the final hBi at the end of the accelerator lattice from Fig. 4 are used to estimate the maximum BBU growth factor, Γ m , from Eq. (2) and are tabulated in Table II. The 1.7 kA data set yields 6.95 and the 2.9 kA data set yields 8.85, indicating > 20% increase in growth rate. The measured and calculated Q values for the induction cells from Refs. [13,14,24] range from 3-6. The saturation time of Γ m , from Eq. (3), is calculated for the peak frequency range of the BBU amplitude (700-850 MHz). The minimum saturation time for the 2.9 kA tune, assuming Q ¼ 3 and 850 MHz as the main harmonic, yields 9.9 ns and the maximum saturation time, assuming Q ¼ 6 and 700 MHz, yields 24 ns (Table II). The saturation time for the 1.7 kA tune ranges from 7.8-18.9 ns. Examining the BBU signal in the top row of Fig. 5(b) we see that the BBU onset time is slightly after the beam head near 275 ns and it reaches its peak for Δx in 10 ns and its peak for Δy in 15 ns. Each of these is within the calculated saturation time for Γ m and less than the extracted pulse length, clarifying the accuracy of these measurements.  A better tune with increased hBi along the length of the accelerator was required because of this substantial growth in the BBU amplitude. The hBi in four cells of each cell block for each of the successive tunes and their corresponding envelopes calculated in XTR are shown in Fig. 7. Each tune iteration brought the beam envelope down more steeply by increasing the hBi at the end of the accelerator, where the BBU amplitude was most apparent. Eventually we had to work our way upstream and begin tuning from the first cell block, in Tune 6, because of the lack of suppression.
BBU at the upstream end of the accelerator is negligible for Tunes 5-7; it did not begin to become apparent until BPM07 (z ¼ 980 cm) as shown in Fig. 8(a). A reduction in hξi of ∼5× is evident at BPMs 17-20 (z > 30 m) in Fig. 8(a) over tune iterations from 4-6. The BBU growth along the accelerator for each successive tune is shown in Fig. 8(b). The initial BBU amplitude, ξ o , is different for each tune contributing to the staggering of each curve. In addition the final product of INh1=Bi decreases for each successive tune as expected, from 237 A=G for Tune 4 to 194 A=G for Tune 6; an increase in hBi of 20%. Tune 7, which has a slightly reduced hBi compared to Tune 6, is shifted down on the BBU growth curve because its initial amplitude at BPM07, 0.0275 ≫ hξi at BPM07 for Tune 6. Each tune has nearly the same slope on the BBU growth curve [ Fig. 8(b)] indicating the consistency in the transverse impedance of the induction cells in the accelerator lattice. The calculated transverse impedance for each tune, from a least squares fit to the data, is shown in Table II. The average impedance for these four tunes is 1086 AE 26 Ω=m. In addition the final h1=Bi, Γ m , and the minimum and maximum saturation times for Γ m are also included for these four tunes in Table II. It is instructive to show Γ m is reduced 20%, indirectly proportional to the hBi increase from Tune 4 to 6. Also the saturation times for all of the tunes have an average range of 8.9-21.6 ns.
After demonstrating the ability to reduce the BBU amplitude along the accelerator and comparing results from Figs. 6 and 8 a further reduction in the BBU was required to optimize the beam quality and achieve a minimal spot size on target. For the final tune the magnetic field was increased to the highest permissible value while maintaining a constant reduction in the beam envelope without overfocusing the beam between cell blocks (Fig. 9). Initially, a higher field of 700 G (Table I)  used for the final tune in the first transport magnet to reduce the initial radius and increase the convergence angle of the beam as it entered the first cell block as shown in envelope calculations in Fig. 9(b). The space charge of the beam made it difficult to increase the field or reduce the beam any more in size until after the second cell block (z > 10 m). After CB2 the field increased linearly from 1 kG up to 2.3 kG in CB8 and the beam radius was reduced from 12 to 6 mm. This 50% increase in hBi, or 2.3× increase in the final field, for the 2.9 kA beam reduced hξi > 10× from Tune 4 to the final tune, nearly matching the 1.7 kA tune from BPM13 (z ¼ 2.2 m), end of CB5, to the end of CB8 [ Fig. 10(a)]. It is worth noting the slight fluctuation in BBU amplitude at BPM07 (z ¼ 980 cm) for the final 2.9 kA tune and 1.7 kA tune is due beam centroid misalignments as mentioned above. This result demonstrates the importance of increasing hBi for suppression of BBU. The 70% increase in current from 1.7 to 2.9 kA required ∼50% increase in hBi in order to maintain relatively the same BBU amplitude. Figure 10(b) shows the relative growth of the BBU amplitude for each of the tunes. As in Fig. 8(b) the initial BBU amplitude, ξ o , is different for each of the tunes in Fig. 10(b). Again a least squares fit was applied to the data sets in Fig. 10(b) and their transverse impedances are indicated in Table II. They are all slightly different, but each is near the measured and calculated cavity values mentioned above in Sec. III B. The final h1=Bi, Γ m , and minimum and maximum saturation times for these tunes are also shown in Table II; indicating a further reduction in Γ m has been achieved with the final 2.9 kA tune and cavity saturation times < 20 ns.
The alternative method of analyzing the BBU amplitude with respect to the beam size, as mentioned in Sec. III A, was also done. However, prior to examining these results it is important to point out the comparison of the hardware integrated BPM signal versus the raw unintegrated case. This comparison was done for highest BBU amplitude tune, the 2.9 kA Tune 4, at BPM19 (z ¼ 34.3 m), the end of the accelerator lattice, where the BBU amplitude was substantially large. A single shot representation, indicating the median BBU amplitude, from at least five shots of the two sampling methods is shown in the top row of Fig. 11. It is worth noting that the signal amplitude of the hardware integrated case is nearly an order of magnitude less than raw unintegrated case. A fast Fourier transform was applied to the signal over AE50 ns in addition to the pulse length of the beam for both data sets and the frequency spectrum is shown in the bottom row of Fig. 11. In both cases the frequency spectrum ranges from 700-900 MHz however the peak of the unintegrated case is 25× higher. After integrating the frequency spectrum from 600-900 MHz, to determine the BBU intensity, it was evident that the hξi for five shots was 20× higher for the unintegrated case. This indicates the sensitivity of raw unintegrated signals and the importance of using this method for the BBU analysis.
The fluctuation of the beam distribution relative to the beam size, radial BBU amplitude, was examined at the end of the accelerator lattice where the beam distribution was expanding, due to field-free transport, at BPM 22 (z ¼ 37.8 m) for the tunes in Figs. 9 and 10. The 2 rms envelope for each of these tunes at BPM22 is listed in Table III and can be extracted from Fig. 9(b). After normalizing out the beam centroid offsets at BPM22 the BBU motion on the beam is examined in Fig. 12. The standard deviation of the measured fluctuation of the beam distribution is calculated in x and y over the 50 ns window. A single shot representation, indicating the median BBU amplitude, from at least five shots is shown in Fig. 12. This was done for at least five shots for each tune and a tabulation of the results is shown in Table III. The BBU amplitude for Tune 4 is 8.1 AE 4.6 mm or nearly 50% of the beam size, whereas the amplitude for the 1.7 kA tune and the final 2.9 kA tune are about 3% and 5% of the beam size; indicating the importance of having hξi=a ≤ 5%.
One additional comparison to make note of aside from the radial BBU amplitude is the beam current measured at BPM22. A clear indicator that the BBU is rf disruption on the beam distribution is to examine the beam current [ Fig. 13(a)]. The beam current for the 1.7 kA tune and the final 2.9 kA tune are relatively constant throughout the beam pulse, whereas the initial 2.9 kA Tune 4 current is broken up along the pulse with rf. There is not bunching of the beam current for the 2.9 kA Tune 4, but a large rf oscillation on the beam envelope that may result in an aliasing [33]. At this z location (37.8 m) we are > 3.5 m from the nearest induction cell cavity to be picking up rf. The total charge of the two 2.9 kA tunes measured at BPM22 agrees to within 2%. The summary of the radial BBU amplitude along the length of the accelerator is shown in Fig. 13(b). It is important to point out that radial BBU amplitude as summarized in Table III is nearly an order of magnitude higher for the initial 2.9 kA Tune 4 at the end of the accelerator and begins to grow drastically after CB5,  z > 20 m. The radial BBU amplitude for the 1.7 kA tune and the final 2.9 kA tune track pretty closely to one another throughout the whole accelerator. The final numbers at BPM22 in Fig. 13(b) are slightly different than in Table III  because a larger data set was used for Table III.

V. CONCLUSIONS
An increase in the intensity of the DARHT Axis-I beam of 70%, with nearly the same transport lattice, leads to a ∼28× increase in the final BBU amplitude. After several tune iterations, we successfully reduced the BBU amplitude 5× by simply increasing the hBi in the accelerator 20%. However, this was still insufficient; a final tune was developed to increase hBi by 50%, or the B-field at the accelerator exit by 2.3×, to reduce the BBU from Tune 4 by >10×, a comparable level to our nominal 1.7 kA tune.
Comparison of the unintegrated BPM measurements to the hardware integrated BPM measurements indicate a sampling sensitivity of >20× and the importance of using this method for the BBU analysis. These axially dependent BBU measurements indicate the necessity of increasing the hBi in an accelerator lattice to reduce BBU and the difficulty of designing an accelerator lattice for an electron beam with a dimensionless perveance > 10 −4 . These experiments are another independent validation of the theory in Ref. [16], Eqs. (1) and (2), as it applies to high-current linear induction accelerators with strong solenoidal fields. These results lend confidence for the use of this theory for future intense relativistic accelerator facilities, which will continue to be challenged by spacecharge force limits on focusing strength combined with required B to minimize BBU.