using a hybrid-ﬂuid model to simulate the ion-hose

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 8, 114202 (2005)

Using a hybrid-fluid model to simulate the ion-hose instability in long-pulse electron linacs

Bruce E. CarlstenLos Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 3 June 2005; revised manuscript received 3 June 2005; published 30 November 2005)

1098-4402=

A numerical model of the ion-hose instability for long-pulse electron linacs is presented, where the ionmotion is represented by fluid parameters. In order to gain extra numerical stability, the fluid behavior ofthe ions is evolved via particle-in-cell (PIC) techniques. This methodology provides a much fastersimulation than a full PIC calculation, allowing for end-to-end simulations of the ion-hose instability inactual linear accelerator configurations. After the description of the simulation model and some simpletest cases, the instability is analyzed for a variety of nominal accelerator transport conditions. Simulationsof the instability are provided for sections of the DARHT long-pulse accelerator that show differentgrowth regimes of the instability. We find that large-amplitude growth is possible in accelerator andtransport regions lacking uniform external focusing, for electron pulse lengths of 2 � sec and longer.

DOI: 10.1103/PhysRevSTAB.8.114202 PACS numbers: 29.27.�a

I. INTRODUCTION

The ion-hose instability has been identified as a keylimiting factor in long-pulse electron induction linacs [1–3]. This instability is well known, and has been studied indetail for electron beams in induction linacs operating inthe ion-focusing regime (IFR). This instability is similar infunction to the resistive hose instability [4,5], and earlyanalyses of it were based on similar theory [6,7]. Thisinstability was first experimentally observed in 1986 atthe Naval Surface Warfare Center (White Oak) [8] andlater at Sandia National Laboratories in 1988 [9], and thegrowth of this instability had been accurately measured in arecirculating induction accelerator by 1990 [10]. Closelyrelated mechanisms have also been suggested as possiblelimitations in future linear colliders, known as the fast ion-beam instability [11,12], and observed in storage rings,known there as the e-p instability [13,14].

The ion-hose instability arises when an electron beampasses through the residual gas in a beam line. The electronbeam ionizes the gas and can interact with the resultingions electrostatically. For practical parameters, most, if notall, of the electrons formed during the ionization process(the ionization electrons) are expelled from the location ofthe primary electron beam and do not cancel the electricfield of the ions. If the electron beam’s transverse centroiddoes not vary in time, there is no net force on the centroidfrom the ions. However, if the beam’s centroid varies intime, the relative displacement with the ion distributionwill lead to a force deflecting the centroid. If the electronpulse is long enough for several ion oscillations, a para-metric resonance can develop and this mutual force canlead to an exponential growth of the centroid offset. Thefollowing model shows the basic mechanism.

This model is too simplistic to show many features of theion-hose instability, and does not include the primarymechanism of the instability used in benchmarking thealgorithms in the next section. However, this model shares

05=8(11)=114202(23) 11420

features with the key mechanism that dominates the insta-bility for long-pulse electron machines. Consider an elec-tron beam traversing a channel that also contains an ioncolumn, offset transversely. The electrons are attracted tothe ions as they move axially, and execute betatron oscil-lations about the centroid of the ions. The ions are assumedto be fixed axially, but are likewise attracted transversely tothe electrons. If the ions have infinite mass, the betatronmotion of the electrons will be periodic in axial position.However, if the ion mass is sufficiently small (molecularfor times scales on the order of 1 � sec for kiloAmperebeams), the betatron oscillation amplitude can increaseexponentially with distance.

For simplicity, we will assume that both the electronbeam and the ion channel have uniform density that isaxisymmetric about their transverse centroids, and thatthe separation between their centroids is small comparedto their overall radii. The differential equation governingthe motion of the ion centroid xi�z; t� at an axial location zis given by

d2

dt2xi � �!2

i �xi � xb�; (1)

where !i is the ion oscillation frequency and xb�z; t� is thecentroid of the slice of the electron pulse at z at time t.Likewise, the differential equation for the centroid of theelectron slice is given by

d2

dz2xb � �

!2b

v2 �xb � xi�; (2)

where v is the axial beam velocity.We will look for solutions of the form

xb � �xbe�t��z; xi � �xie

�t��z: (3)

This is not the only possible solution to these equations, buthas features relevant to the instability dominating long-pulse electron beams. We will be looking for a steady-state

2-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevSTAB.8.114202

BRUCE E. CARLSTEN Phys. Rev. ST Accel. Beams 8, 114202 (2005)

solution that grows exponentially along the axis. Thesesolutions lead to these equations

�x b�2 � �!2b

v2 � �xb � �xi�; �xi�2 � �!2i � �xi � �xb�:

(4)

These equations have four unknowns— �xb and �xi are ingeneral complex with an unknown phase relation.Eliminating �xb and �xi leads to

0 � �2�2 �!2b

v2 �2 � �2!2i : (5)

Let us also assume that the beam is injected at z � 0 with asinusoidal variation, � � j!. This leads to the solution

�2 �!2!2

b=v2

!2i �!

2 ; (6)

and the oscillation amplitude will grow exponentially if!<!i, or if the oscillation frequency is less than the ionoscillation period.

We will see similar behavior in detailed numerical simu-lations of long-pulse electron accelerators. However, addi-tional effects will lead to saturation of the instability whichare not included in the above simple analysis. Use ofmultiple ion species will lead to damping by the mixingof oscillation phases of the different species. Also, as theparticle oscillation magnitudes exceed the transverse radiiof the beam and ion distributions, the frequency of theoscillations decreases and there is additional phase mixing.For actual accelerators designed for low fractional ioniza-tion, the electrons in the beam are confined by an externalmagnetic field and the oscillations of the ions lead tosaturation of the instability as their oscillation amplitudeapproaches the beam radius.

The ion-hose instability is an important possible limita-tion to emerging, long-pulse electron induction linacs, suchas the 2-� sec long DARHT second-axis accelerator [15].For these types of machines, analytic estimates are insuffi-cient to determine the actual saturated amplitude of theion-hose instability. Particle-in-cell (PIC) calculations arecapable of solving this problem exactly, but simulationsover the entire linac, including acceleration and focusing,are too cumbersome to be practical at this point, as movingwindow frames [16] are irrelevant for long-pulse accelera-tors, because the electron pulse fills the entire length of theaccelerator. Heroic PIC simulations of an accelerator witha length of 50 m using the code LSP were done in [3], butdue to the coarseness of the grid, external focusing andacceleration were smeared and details of the accelerationwere lost. The BUCKSHOT code [17] was written to studythis instability in the IFR. This code uses a gridless PICmodel with an analytic push for the amplitude growth.BUCKSHOT does not include particle acceleration, andeven with its limited model, is not capable of modeling afull accelerator design. In this paper, we introduce a new

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numerical model of this effect that is practical to use forfull accelerator designs, complete with acceleration andarbitrary external focusing. It consists of a reduced hybrid-fluid model for the ion motion, coupled with a slice modelfor the electron beam, and is capable of calculating thetime evolution of the ion-hose instability. The algorithmfor simulating this instability in long-pulse electron linacsruns quickly because (1) the fluid model requires theminimum amount of calculations to determine the iondistribution parameters driving the ion-hose instability,(2) the fractional ionization is very small, thus the ionmotion only depends on the electric fields from the electronslices and not their own self-fields, and (3) we assume thateach electron slice has essentially the same transversedistribution (although with a possible offset), so we canseparate the centroid motion from the calculation of theslice transverse distribution. This last approximation wasverified during the simulations reported in Refs. [3,18] forparameters of interest for long-pulse accelerators.

We describe the numerical model in detail in the nextsection. Next, we include simple test cases verifying cor-rect dynamics for both the ions and the electrons. First, thelinear motion of the ions in the potential well of a uniformelectron beam is found. Next, the linear motion of theelectron beam in the potential well of a uniform ion chan-nel is found. The mechanics in the code coupling the ionmotion to the electron slice motion is benchmarked againstthe well-known short-pulse ion-hose instability mechanismin the ion-focusing regime, which has both analytic andnumerical solutions.

Following that, we present simulations of the ion-hoseinstability for various nominal accelerator transport con-ditions to study the instability growth for the cases of bothaxially uniform focusing and discrete focusing with shortsolenoids. We confirm that the growth is suppressed by acontinuous axial magnetic field as demonstrated in [3], butadditionally see that large-amplitude growth can occur ifperiodic focusing is used. In the final section, we simulatethe ion-hose instability for sections of the DARHT long-pulse accelerator that have periodic focusing, using thenominal vacuum specifications but with an alternativefocusing tune that encourages ion-hose growth. The ion-hose instability is phenomenologically described for boththe upstream region near the injector and the downstreamregion just after the last accelerating induction cell.Considering only a single species of ions (H2O�), large-amplitude growth is found by the end of the 2-� secelectron pulse in these regions of periodic external focus-ing. A larger transported beam size, decreased pressure inthe vacuum vessel, and multiple ion species would tend tosuppress the growth rate and saturated amplitude.

II. SIMULATION MODEL

In this section, we describe the ion-hose instabilitysimulation model. We use a reduced hybrid-fluid model

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USING A HYBRID-FLUID MODEL TO SIMULATE THE . . . Phys. Rev. ST Accel. Beams 8, 114202 (2005)

for the ion motion in order to simplify the calculation andto reduce the required simulation time for modeling prac-tical accelerator designs.

An exact ion-hose calculation can be made in a straight-forward manner using standard PIC techniques. However,the storage and execution time requirements to implementthis technique for a full-length long-pulse accelerator de-sign with the required resolution to resolve the beamemittance evolution are prohibitive [3]. The electronbeam for a long-pulse accelerator will fill the entire lengthof the accelerator, on the order of 50 m for the DARHTlong-pulse machine. In addition, this calculation is three-dimensional. In order to simplify the simulation techniqueso a reasonably accurate simulation can be made on apersonal computer with short execution times (on the orderof an hour on a 2 GHz personal computer), we will maketwo important assumptions. First, we assume that we cannumerically separate the calculation of the transverse cen-troid motion along the electron-beam pulse from the cal-culation of the transverse beam distribution. Second, weassume that the influence of the ions’ electric field isnegligible on their motion (it only contributes collectivelyto the electron slices’ centroids) in comparison to theelectric field from the electrons. Also, we use the factthat only the fluid values of the ions are needed to deter-mine the electron centroid motion, and the full ion distri-bution function does not need to be known.

The induced dipole force (leading to centroid motion) ismore sensitive to the ion density than the higher-orderradial force (leading to variations in the transverse distri-bution), thus the first assumption is satisfied for sufficientlysmall ion densities. It is reasonable to expect that we canseparate the effect of the collective ion space charge on thebeam slices into orders. The lowest order would be thedipole force, the next order would be axisymmetric focus-ing (azimuthal mode number m equaling zero), the nextorder would be a single sinusoidal azimuthal variation infocusing (m � 1), then m � 2, and so on. For small trans-verse offsets, as one sees in induction linacs with externalfocusing, only the first two orders are important. It shouldbe pointed out that the primary purpose for this type ofnumerical model is to determine accelerator parametersthat have unacceptable ionization effects, which includetypically relatively small ion-hose amplitudes and emit-tance growth ratios, so the coupling between orders will besmall. The lowest order will be described in detail in thispaper, and the m � 0 order has been recently described forlong-pulse electron linacs [19]. This is important, since weonly need to resolve the electric field from the ions at thelocation of the electron slice centroids (which is a one-dimensional calculation), and not everywhere (which is atwo-dimensional calculation, and much slower). Also, thetype of electron beams described here are all space-chargedominated, and thus minor emittance growths will not haveappreciable effects on the electron-beam slice distribution.

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The second assumption is clearly valid for fractionalionizations on the order of 10�3 and below. Finally, useof the fluid model can be justified by considering the mo-ments of the Vlasov equation [20]. The zero order momentleads to the equation of continuity,

@n@t� ~r � �n ~u� � 0; (7)

and the first order moment leads to the fluid equation ofmotion,

m�@ ~u@t� � ~u � ~r� ~u

�� e� ~E� ~u� ~B�; (8)

where n is the real-space density function found from thedistribution function by

n �Zf� ~r; ~v�d ~v: (9)

Likewise,

~u �Z~vf�~r; ~v�d ~v (10)

is the average velocity at a point ~r in real space. These fluidequations are exact moments of the Vlasov equation, andsince the ion space-charge field only depends on n (theirvelocities are too small to create a significant magneticfield), the fluid equations are sufficient to fully describe theevolution of the ion space-charge field. This is an importantresult, because it relieves us from explicitly knowing thefull ion distribution function and greatly simplifies thenumerical calculation.

One further step is made in the simulations reportedhere. An instantaneous PIC model is used for finding theevolution of the density and the average velocity instead ofdirectly evolving the continuity equation [Eq. (7)] and thefluid equation for the flow of momentum [Eq. (8)] on afinite-difference grid. This approach is still consistent—the importance of Eqs. (7) and (8) is that they show thatonly density and average velocity components are neededto evolve the density accurately, and that indeed any twodistributions with the same discretized real-space densitiesand average velocities will evolve the same. This is apowerful concept, because it allows us to substitute anyspecific distribution function at any time, as long as it hasthe same density and average velocity profiles as theoriginal distribution function. So at each time step, wewill substitute a simple PIC distribution model with parti-cles at the grid intersection points for the distributionfunction, which allows us to numerically evolve the distri-bution in an easy and straightforward manner. After evolv-ing this distribution, new density and average velocitycomponents are found on the grid, providing the parame-ters for the next time step. It should be pointed out thatbecause the fluid equations fully describe the beam density,

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the space-charge fields to any order in m can be found inthis manner.

Using these assumptions, we can create a simulationmodel of an accelerator including features required forincluding the ion-hose instability. We will numericallymodel the long-pulse electron beam as a series of shortslices. We will evolve the transverse distribution of the firstslice, and assume the transverse distribution of all otherslices are the same. We will follow the centroid of eachslice individually. Numerically, this is done by stepping thebeam axially, and then evolving the interaction between theions and the electron-beam slices at that axial location,finding the centroid of each electron slice and the iondistribution as a function of time as the rest of the beampasses that axial location.

An additional important consideration for using thisprocedure is understanding the behavior of the ionizedelectrons in the presence of an external axial magneticfield that may be present in order to confine the electronbeam. This field has the potential of also restricting theelectrons’ transport to the beam pipe wall, and may de-crease the net ionization. We can study the motion of theionized electrons by numerically integrating their nonrela-tivistic equation of motion

mddt� _x; _y; _z� �

eI

2�c"0a2 �x; y; 0� � e� _x; _y; _z� � �0; 0; B0�

(11)

FIG. 1. One cyclotron orbit of electrons originated at 15 evenly spfor the parameters of (a) beam current 2 kA, focusing field 0.2 T, bearadius 1 mm, and (c) beam current 0.2 A, focusing field 0.02 T, beam

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inside the electron beam and

mddt� _x; _y; _z� �

eI

2�c"0�x2 � y2�

�x; y; 0� � e� _x; _y; _z�

� �0; 0; B0� (12)

outside the beam, where I is the beam current, a is thebeam edge radius, and B0 is the applied external axial field,for a uniform density, space-charge dominated beam.Inspection of these equations of motion show that thenormalized trajectories r=a scale as I=�a2B2

0�. To verifythis scaling, we plot in Fig. 1 trajectories for ionizedelectrons, created at 15 evenly spaced positions in the drivebeam, for a beam current, external field, and beam radius of(a) 2 kA, 0.2 T, 1 cm, (b) 20 A, 0.2 T, 1 mm, and (c) 0.2 A,0.02 T, 1 mm, respectively. All cases have the same valueof I=�a2B2

0�, and show the same normalized trajectories.The trajectories are plotted for one cyclotron period, whichvaries for the different trajectories and between cases. Notethat the cyclotron periods are short relative to long-pulseelectron beams (� sec time scales), and since the ionizedparticles are created continuously, we can consider thedensity of these trajectories in a time-averaged sense.Averaged over time, 48% of the ionized electrons are out-side the drive beam for all three cases, so we can assumethat the effective ionization of the gas is only 52% of thetotal ionization. (This is somewhat of an underestimatebecause additionally as the ion channel moves transversely,

aced different initial locations within the primary electron beamm radius 1 cm, (b) beam current 20 A, focusing field 0.2 T, beamradius 1 mm. On average, 48% of the current is outside the beam.

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more of the ionization electrons will be lost, but this is agood estimate for small ion centroid displacements.)

In Fig. 2, we plot the fraction of the ionization electronsoutside the uniform density, space-charge limited beam, asa function of the scaling parameter I=�a2B2

0�. As the fieldvanishes (and the ionization electrons are not bound), thefraction of ionization electrons that stay within the primarybeam vanishes, as expected. For values of I=�a2B2

0� greaterthan 10, the error in the net ionization from simply usingthe ionization rate equation is 10% or less. The finalassumption of this model is that the scaling parameterI=�a2B2

0� is relatively large (at least a few, and typicallygreater than ten), which is valid for typical cases of interest,which we will verify for the test cases.

A. Specific simulation mechanism

The accelerator simulations were performed by the codeSLICE [21]. The SLICE code self-consistently pushes parti-cles in a single axial slice of the beam using the Lorentzforce equation, including the beam’s self-radial electric,axial magnetic, and azimuthal magnetic fields and allexternal fields to fourth order in radius. The code usesthe long-beam approximation, in which Gauss’s law isused for the radial electric field and Ampere’s law isused for the azimuthal and axial magnetic fields. A collec-tion of particles (typically 4000 in these simulations) isused to describe the beam at each axial position. As agroup, these particles are stepped axially, with a 1 mmstep size.

With the separation assumption, the transverse distribu-tion only needs to be calculated for one axial slice of thebeam pulse, but the centroid position needs to be calculatedalong the entire beam pulse at every axial location. To dothis, the electron-beam pulse is divided into separate slices.The total transverse distribution is calculated for the firstslice, and is assumed to be the same (except for centroidposition) for all the following slices. Ions are produced

FIG. 2. Fraction of ionization electrons outside primary beam,as a function of the scaling parameter I=�a2B2

0�.

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from each slice from primary impact ionization, given bythe rate equation [22]:

dnidt� ngne�v; (13)

where ni is the ion density, ng is the initial gas density(assumed to be uniform over radial distances), ne is theelectron bunch density (which is a function of time at agiven axial and radial position), � is the cross section, andv is the velocity of the electron beam (which we willassume to be the speed of light, c). Numerically, the crosssection can be modified to account for a loss of ionizationcaused by a relatively small scaling factor I=�a2B2

0�. For anelectron-beam slice of temporal length �, the increase inthe fractional ionization is then given by

�nine� ng�v� � ��sec�Pg�torr�9:783�1026� cm�2

(14)

where Pg is the gas pressure in torr.The numerical procedure is to first step the beam axially,

and then to perform the ion-related calculations for allbeam slices at that axial location. The ions generated byprimary impact ionization are assumed to be axially fixed,but transversely mobile. The ion distribution is calculatedafter each slice, on a transverse grid of typical size 51 by 51grid lines. Only the ion density and average velocity is keptat each grid intersection point, as these parameters fullydefine the ion fluid. For the first slice, there is no ambiguity.Ion density is assigned to each node in accordance with therate equation [Eq. (13)] and the local electron-beam den-sity. An ion velocity of zero is assigned to each gridintersection point. Next, the force on the slice centroidfrom this ion density is calculated. (For the first slice, thereis no net dipole force as SLICE assumes each slice isaxisymmetric about its centroid position.) Finally, the iondistribution is evolved under the influence of that electron-beam slice’s electric field. For the ions just from the firstslice, this force is axisymmetric, and the ion centroid doesnot move. The fluid density and velocity can be evolvedusing standard finite-difference techniques, using the con-vective form of the Lorentz force law [Eq. (8)] to evolvethe fluid velocity and the continuity equation [Eq. (7)] toevolve the fluid density. However, the electron-beam den-sity is often hard edged, leading to a discontinuity in theion distribution. In order to add extra stability, the fluidmodel parameters are evolved in a PIC manner instead, aswas described above. At each grid intersection point, weassign a superion with the total ion density correspondingto all the ion density within a grid square centered at thatintersection point, and with velocity equal to the fluidvelocity at that point. We let these superions move forthe time period corresponding to the slice length, andafterwards assign new ion densities and average velocitiesto the grid intersection points. This procedure is then

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FIG. 3. (Color) Initial electron-beam centroid offsets, versusslice number. The red line is the horizontal offset and the blueline is the vertical offset.


followed for all the successive electron pulse slices. For theadditional ionization due to new electron slices, additionalion density is assigned to the grid intersection points whilekeeping the total ion momentum constant. For successiveslices, the ion and slice centroids will diverge, leading toboth slice and ion dipole forces.

This procedure greatly simplifies the calculation.Because each electron slice itself stays axisymmetric aboutits centroid, the resulting force on each ion is simple to find(it is essentially a one-dimensional calculation and scaleslinearly with the number of electrons followed in thesimulation). Though the ion distribution can become verydistorted, the calculation of the force on only the electronslice centroids also scales linearly with the number of gridintersection points, and is likewise very fast. With the fluidmodel for the ions, the number of ions that are needed touse in both the ion evolution step and the slice centroidevolution step is equal to the total number of grid inter-section points, and never increases. Without the fluidmodel, the number of ions followed would linearly in-crease with the slices, and this calculation would becomeprohibitively slow. Likewise, without the assumption ofseparating the electron transverse distribution from thecentroid positions, calculating the ion motion would re-quire a complex two-dimensional field solver, with execu-tion time scaling more than linear with the number ofelectrons followed in the simulation.

Another important assumption is that the secondaryelectrons do not appreciably effect the ionization. This isnot obvious, because they are transversely accelerated bythe primary beam up to (and then beyond) the peak value ofthe ionization cross section as a function of electron en-ergy. To estimate this effect, we use the ionization formula[Eq. (13)], with a nominal maximum cross section of 3��10�20� m2 (see, for example, Ref. [23] for argon) at anenergy of 100 eV (velocity of 0.02 c), and a time of1 � sec . Note that the electron density ne is equal to theion density ni times a coupling factor �, which cannotexceed unity (and which is typically about 0.1 fromFig. 1). This leads to a fractional increase in the ionizationfrom secondary impact of about �ni=ni � �5:6�10�4�over a time of 1 � sec , which is negligible. Note thateven if the secondary electron velocity is increased to c,the fractional ionization increase is about 2%, even ne-glecting the drop in ionization cross section at higherelectron energy. We can conclude that the secondary ion-ization from the secondary electrons is negligible.

Comparing the speed of the hybrid calculation with amulitparticle PIC simulation, the hybrid model can simu-late a 50-m long accelerator with a 2 � sec -long electronwith an axial resolution of 1 mm in about an hour, whereasa mulitparticle PIC simulation requires on the order of 10 hfor an equivalent geometry (including acceleration and arealistic magnetic field profile), but with only a 5-cm axialresolution. Overall with similar resolution, the hybrid

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model yields a calculation well over 2 orders of magnitudefaster. The cost in accuracy of this faster calculation is theapproximation that all electron slices in the primary beamhave the same transverse distribution (although potentiallyoffset), but which has been verified to be a minor effect forthe types of simulations we are interested [18].

III. TEST CASES

At this point it is important to simulate some simple testcases to verify the operation of this ion-hose model. Thetwo algorithms that we want to validate are the centroidevolution algorithm and the ion evolution algorithm. Forthese tests, we start with a 60 nsec electron pulse separatedinto 30 slices of 2 nsec apiece. We assume that the cent-roids of all slices are offset only along the vertical axis, in ahalf sinusoid, shown in Fig. 3. The maximum verticaloffset is 1 cm. The electron-beam current is 4 kA, withan edge radius of 3 cm. The magnetic field required toobtain balanced flow at 2 MeV is about 0.037 T. We assignan initial horizontal velocity to the centroids given by

vx;cent �ycenteBm�

; (15)

which will balance the focusing magnetic force with thecentrifugal force. The evolution of the horizontal andvertical centroid positions is shown in Fig. 4, and the totalradius of the centroids is shown in Fig. 5, as a function ofaxial position, which is numerically stable (the ripple isdue to a slight mismatch in the applied magnetic field).

The next test case will be for the ion density oscillatingin the potential well of the electron distribution. For thiscase, the electron slice centroids are assumed to be per-fectly aligned, and an initial ion distribution is establishedoffset 1 cm vertically. The ionization process is turned offso we can see the ion centroid evolution in the field from

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FIG. 6. (Color) Ion centroid motion in the electron beam’spotential well. The red line is the horizontal offset and theblue line is the vertical offset.

FIG. 4. (Color) Electron-beam centroid position for corkscrew-ing case. The red line is the horizontal offset and the blue line isthe vertical offset.


the electron beam. We see the ion centroid does executesinusoidal motion in Fig. 6.

The transverse ion motion obeys this harmonic equation

�x ion � �xioneI

mi2"0c�a2 ; (16)

which, for these parameters, leads to an oscillation periodof 40 nsec for hydrogen ions. With 2 nsec per slice, this isin excellent agreement with Fig. 6.

The next test case we will look at is the evolution of anelectron slice centroid initially offset 1 cm horizontally, ina uniform ion channel centered about the origin. Thecentroid should obey this equation

�x cent � �xcenteI

me�2"0c�a2 �cng�; (17)

where the ion channel was created by primary impactionization from a slice of the beam of length � into a

FIG. 5. . Electron centroid position for corkscrewing case,showing total radial offset.

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residual gas. For a slice beam current of 4 mA (in orderto turn off the focusing magnetic field) and length 2 nsec,and a pressure of 10�7 torr and cross section of 8�10�9 m2, this leads to a betatron wavelength of 1.54 m,again in excellent agreement with the calculation, shown inFig. 7. It should be pointed out that an absurdly high crosssection was needed in order to get a reasonable betatronperiod for the case of such a low current and short ioniza-tion time.

The final test case we will present will be for short-pulse,high-fractional ionization. This instability has been studiedin great detail for short-pulse transport in the ion-focusingregime (IFR) [22,24]. The instability has an analytic de-scription in this regime and several analytic and numericalchecks can be made. We will benchmark our routineagainst the simplest analytic test case, that of anelectron-beam offset to an initially preionized channel,with equal charge density as that of the beam. The electron

FIG. 7. (Color) Electron-beam slice centroid betatron motion inan ion channel. The red line is the horizontal offset and the blueline is the vertical offset.

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FIG. 8. (Color) Evolution of both the ion centroid and rmsradius, for the test case using only the electrons’ space chargeto check the ion distribution calculation. The red line is for theion centroid and the blue line is for the ion rms radius.


beam and ion channel are of uniform density and of equalradii, and the offset is a small fraction of the beam and ionchannel radii.

The initial electron slices in the beam will oscillate inthe ion channel with simple betatron motion (as shown inFig. 7), because the relatively heavy ions will take time tomove. Since all electron slices follow nearly the same path,the ions at each axial location will be accelerated withessentially uniform force toward the betatron oscillationposition at that axial location. The deformed ion channelwill have the same betatron period as the slice centroid,with lesser amplitude.

For this case, we will assume that the fractional ioniza-tion is unity. The following simple model will describe thekey instability mechanism for this regime. The net force onthe electron slice centroid obeys this second-order differ-ential equation:

�x b �eI

me�2"0c�a2 �xi � xb�: (18)

With the assumption that xi � "xb where " is a (complex)scalar, this has solution

xb � x0ejk�1�"�1=2z; (19)

where k2 � eIme�2"0c�a2 .

Note that this solution has a longer betatron period thanwithout the ion motion, since " is mostly positive real.

Because of inertia, the ion channel cannot increase itsoscillation period as fast as the electron slices, and theeffective ion oscillation period is shorter than that for theslices. This implies

" � "r � j"i; (20)

where "r and "i are both real and positive. Expanding thesolution for the slice centroid, we find

xb � x0ejk�1�"r�1=2zek"iz=2�1�"r�: (21)

Since "i is real, there is exponential growth in the magni-tude of the betatron oscillations.

There are two key elements leading to this amplitudegrowth. First, the ion centroid oscillation increases theelectron slice centroid betatron period. Once the electronslices fall behind in phase relative to the ion centroids, theamplitude of their betatron oscillations increasesexponentially.

Because of the large electric field from the ions (fullfractional ionization), this effect can only be modeled bySLICE with an additional feature added. With full fractionalionization, both the electrons and ions are mostly in elec-trostatic equilibrium relative to their rms radii. For theelectrons, we simulate this by only doing a one-dimensional push (in the axial direction), ensuring thatthe original transverse electron distribution is kept.Calculating the true electrostatic self-fields from the ionsis two-dimensional, and very time consuming. Instead, we

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assume that the ion distribution is axisymmetric and uni-form about its centroid, and add in an analytic approxima-tion for its own self-field based only on the actual iondistribution rms radius. In the following simulations, thefirst electron slice will provide an ionized channel centeredabout x � 2 mm, with no additional ionization from suc-cessive electron slices. All successive electron slices willbe injected on axis. The 1-kA electron beam is 2 cm inradius, with an energy of 3.6 MeV. The nominal ion massused these calculations is 1=10 that of helium (in order toreduce the overall evolution time required in these simu-lations). Specifically, two methods for determining thecentroid motion with fractional ionization will be em-ployed. In both methods, the ions’ self-fields will be in-cluded with the analytic form described above for theelectric field from an axisymmetric and uniform distribu-tion. To check the validity of this approximation, the forceon the electron slices’ centroids will be found by (1)assuming the same analytic form for the electric fieldfrom the ion distribution and comparing that to (2) theelectric field found by exact grid calculation. This com-parison is a good check on the analytic form, since bothmethods should give essentially the same result.

Without introducing the additional electric field from theion themselves, the ion distribution focuses over time. InFig. 8 we see this effect, over the first 10 nsec of the beam(each electron slice corresponds to 1=3 nsec). Note thatwithout the fractional ionization, the ion rms radius de-creases at exactly the same rate that the centroid moves, asit should. For this simulation, the grid spacing is 0.16 mmand the beam radius crosses two grid lines as it compresses.Although this simulation is nonphysical because it does notinclude the fractional ionization, it is useful since it dem-onstrates that the rms radius is being modeled very accu-

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FIG. 11. (Color) Ion-hose calculation in the IFR using method 2.The red line shows the electron slice centroid and the blue lineshows the ion centroid.

FIG. 9. (Color) Ion-hose calculation in the IFR using method 1.The red line shows the electron slice centroid and the blue lineshows the ion centroid.


rately with the fluid model, even as particles are clearlycrossing through grid intersection points.

With the fractional ionization included by method 1, theelectron slice and corresponding ion distribution centroidat 10 nsec into the pulse as a function of axial location isshown in Fig. 9. The electric field inside the ion distributionis assumed to be Er � �Ir=2"0�c�b2, where b is the iondistribution rms radius. The electron centroid is oscillatingabout the ion column centered at x � 2 cm, and the iondistribution is executing smaller oscillations. The electronoscillation betatron wavelength has increased due to theion oscillations as predicted, and there is a correspondingexponential growth in the electron centroid oscillationamplitude. The growth rate is very close to the analyticsolution [24].

In Fig. 10 we compare the horizontal electric fieldcalculated by the fluid model to that found from the ana-

FIG. 10. (Color) Comparison of the electron centroid grid andanalytic electric fields, using method 1. The solid red line showsthe electric field calculated on the simulation grid, the dashedblue line shows the electric field calculated by the analyticmodel, and the dashed green line shows the rms ion size.

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lytic solution for a uniform distribution, that the electronslice centroid sees. The small hash is noise from the grid,but the overall agreement is also very good. In addition, thegrowth rate is in good agreement with analytic estimates[22,24].

In the next two figures, Figs. 11 and 12, we redo thecalculation using method 2 to calculate the electron cen-troid motion. Here, the analytic form is used to describe theions’ self-fields, but an exact grid calculation is used for theelectron slice centroids. Essentially the same results occur,however there is some additional oscillations in both theions’ and electron slice’s centroids. This oscillation is dueto the long-term transverse plasma oscillations of the ions(leading to slight periodic nonuniformities in the distribu-tion), but does not appreciably change the results. Becausethese plasma oscillations are physical, method 2 is also

FIG. 12. (Color) Comparison of the electron centroid grid andanalytic electric fields, using method 2. The solid red line showsthe electric field calculated on the simulation grid, the dashedblue line shows the electric field calculated by the analyticmodel, and the dashed green line shows the rms ion size.

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FIG. 13. (Color) Ion-hose calculation in the IFR using method 2,without fractional ionization. The red line shows the electronslice centroid and the blue line shows the ion centroid.


considered to be an accurate way of representing fractionalionization for simulations in this regime.

The previous figures provide some important physicalinsight into the mechanism. The key feature leading to theinstability growth is the increase in the electron’s betatronperiod. However, without full fractional ionization, the rmsion distribution radius decreases (Fig. 8), and in fact theelectron’s betatron period will decrease. This phase rela-tion between the electron and ion oscillations will lead toattenuation of the electron centroid oscillation amplitude,as predicted in Eq. (21). In Figs. 13 and 14 we show the ionand electron centroids and the calculated electric fields, forthe case where the fractional ionization is not included. Wesee the attenuation of the oscillations, and also the samecentroid wandering as seen in Fig. 11, due to plasmaoscillations in the ion distribution. For this case, the in-creased electric field as the ion distribution pinches ac-

FIG. 14. (Color) Comparison of the electron centroid grid andanalytic electric fields, using method 2, without fractional ion-ization. The solid red line shows the electric field calculated onthe simulation grid and the dashed blue line shows the electricfield calculated by the analytic model.

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tually shortens the electron betatron period, and thebetatron amplitude then attenuates, also in agreementwith the simple model presented above.

IV. NOMINAL ACCELERATOR TRANSPORTCASES

In this section, we will use the hybrid-fluid model tosimulate some nominal long-pulse beam transport cases. Inparticular, we will model the ion-hose instability for vari-ous beam and transport parameters, for 5-meter long trans-port channels. The goal of these simulations is to identifykey features of the instability, especially growth rates andsaturation mechanisms. These insights will in turn be usedto help interpret long-pulse simulations of the DARHTsecond-axis beam line, discussed in the following section.

For these cases, we will use a 4 kA beam at 20 MeV.We will assume that the gas pressure is 5� �10�7� torrand the cross section for releasing hydrogen ions is 4��10�23� m2. Hydrogen is used for these simulations be-cause its light weight will lead to the shortest ion oscilla-tion period and greatest growth rate.

The ion-hose instability has a complicated functionaldependency on axial position and time. It is felt that plotsof both ion and electron centroid motion at various axiallocations as a function of time are the most illustrative ofthe physical mechanism, and will be the primary tool forthis discussion and the one in the following section.

We will consider four 20 MeV cases, with initial primarybeam sinusoidal offset amplitudes of 0.1 to 1 mm. First, weconsider the case where the beam is matched in a uniformmagnetic field with an edge radius of 3 mm. The ionoscillation period for this case is 4 nsec. We will drivethis instability with a sinusoidal electron-beam centroidoffset (with a maximum of 0.1 mm) also with a period of4 nsec. Following that, we will observe the ion-hose stabil-ity for the case where the beam is focused to about 3 mmedge radius with periodic thin solenoids. Next, we considerthe same beam, but with a 3 cm edge radius, and with aninitial centroid offset period of 40 nsec. The final simula-tion will be the same as the 3 mm case with solenoidalfocusing, but with an electron-beam centroid offset periodof 8 nsec to observe nonresonant excitation.

A. 20-MeV electron-beam, 3-mm edge radius

In this case, we will consider a 20-MeV, 4-kA beam,with a 3-mm radius, with a maximum offset amplitude of0.1 mm with a resonant period of 4 nsec. The beam ismatched in a solenoidal field of 0.125 T. With these pa-rameters, the scaling factor I=�a2B2

0� is about 28 (so theionization model used is very accurate). In Fig. 15 we seethe evolution of the ion centroid as a function of time atdifferent axial locations. The electron-beam centroid evo-lution is shown in Fig. 16. Each electron slice in thesesimulations corresponds to 0.4 nsec, for a total pulse lengthof 200 nsec. For these parameters, the ion oscillation

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FIG. 15. (Color) Ion centroid evolution for 20-MeV, 3-mm, resonant beam case at different axial positions. Each slice is 0.4 nsec long.The red line is for the horizontal offset and the blue line is for the vertical offset. (a) z � 0:625 m, (b) z � 1:25 m, and (c) z � 2:5 m,and (d) z � 5 m.

FIG. 16. (Color) Electron-beam centroid evolution for 20-MeV, 3-mm, resonant beam case at different axial positions. Each slice is0.4 nsec long. The red line is for the horizontal offset and the blue line is for the vertical offset. (a) z � 0:625 m, (b) z � 1:25 m, and(c) z � 2:5 m, and (d) z � 5 m.


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period is about 4 nsec. The ion centroid amplitude reachesa couple of millimeters, but the axial magnetic field re-stricts the electron-beam centroid offset to about 0.2 mm.The ion centroid oscillations saturate once the ions moveout of the potential well of the electron beam. The electroncentroid growth does not saturate for this pulse length at0.625 m, is close to saturating at 1.25 m, and saturates at2.5 m and beyond. This saturation is also reflected in theion centroid evolutions. At 2.5 and 5 m, the ion centroidevolution shows that the ions lose coherence with theelectrons, causing the saturation. The electron centroidamplitude is suppressed by the confining axial magneticfield to about one tenth that of the ions.

B. 20-MeV electron beam, 3-mm edge radius withperiodic focusing

In this case, we will consider a 20-MeV, 4-kA beam,with a 3-mm radius, also with a maximum centroid offsetof 0.1 mm. The beam is matched with a periodic lattice of10-cm long solenoids, spaced 1 m apart. Here, the scalingfactor I=�a2B2

0� is about 3 in the solenoids and infinitebetween them. Since the solenoids constitute a small por-tion of the transport, we anticipate the simulation modelwill be quite accurate. The rms beam radius evolution overthe transport distance is shown in Fig. 17. As in theprevious case, 0.4 nsec slices are used. However, the in-stability did not saturate in 200 nsec, so a total of 2500slices was used, for a total electron pulse length of 1 � sec .In Fig. 18 we see the evolution of the ion centroid as afunction of time at different axial locations. The electron-beam centroid evolution is shown in Fig. 19. The plots ofthe ion centroid and electron centroid evolutions are at thelocations of the periodic solenoids. We see that the ioncentroids have a similar amplitude to the previous casewith uniform focusing (about 2 mm), and now the electroncentroids have comparable amplitudes, a factor of 5 greaterthan the previous case. A major difference introduced in

FIG. 17. Beam edge radius versus transport distance, for theperiod focusing case. 10-cm long solenoids are located at meterintervals, with strengths of 0.41 T.

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this case relative to the uniform focusing cases is that herethe electron and ion centroids will hunt for a resonance,which is clearly seen in the electron centroid plots at 3, 4,and 5 m where a new period is established and in the ioncentroid plots where the ion centroids lose coherence. Wesee that at axial distances of 3 m and beyond, the period ofthe ion and electron centroid oscillations change dramati-cally. This shift is seen clearly in Fig. 19(f).

C. 20-MeV electron beam, 3-cm edge radius

In this case, we will consider a 20 MeV, 4 kA beam, witha 3 cm radius. A very low axial magnetic field is used tomatch the rms beam radius, leading to a very large scalingfactor I=�a2B2

0�. For this beam radius, the ion period is40 nsec, and in this simulation the electron centroid offsetperiod is also 40 nsec to resonantly excite the instability(with an initial maximum amplitude of 1 mm). In Fig. 20we see the evolution of the ion centroid as a function oftime at different axial locations. The electron-beam cen-troid evolution is shown in Fig. 21. Ion amplitude satura-tion again occurs for radii above 2 cm, and the electronamplitude does not show saturation either over the 5 m orover the 2 �sec pulse length. The total amplitude of theinstability is about the same as the case with a 3-mm radius,but the relative effect is smaller, because of the larger beamsize.

D. 20-MeV electron beam, 3-mm edge radius,nonresonant oscillations

In this case, we will consider a 20-MeV, 4-kA beam,with a 3-mm radius. The beam is matched in a solenoidalfield of 0.125 T. The difference between this case and thefirst case is that the initial beam centroid offset has a periodof 8 nsec, which is not resonant with the ion oscillationperiod of 4 nsec. Ths slices in this case are each 0.8 nseclong, with a total simulation pulse length of 400 nsec. InFig. 22 we see the evolution of the ion centroid as afunction of time at different axial locations. The electron-beam centroid evolution is shown in Fig. 23. The saturatedamplitudes are similar to those seen for the resonantlydriven case (about 0.2 mm), and by 1.25 m, the ion centroidoscillation has a dominant period of 4 nsec. In addition, theelectron centroids show a dominant oscillation of 4 nsec by5 m. It appears that resonant excitation is not needed,although nonresonant excitation will delay saturationboth in terms of axial location and time within the pulseitself.

E. Discussion

The results of the previous cases can be summarized bythe following statements. Application of a uniform externalmagnetic field will suppress the amplitude of the electroncentroid oscillations, as described in [3]. The ion oscilla-tions are limited by the size of the electron-beam potential

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FIG. 18. (Color) Ion centroid evolution for 20-MeV, 3-mm, periodic-focusing beam case at different axial positions. Each slice is0.4 nsec long. The red line is for the horizontal offset and the blue line is for the vertical offset. (a) z � 1 m, (b) z � 2 m, (c) z � 3 m,(d) z � 4 m, (e) z � 5 m, (f) blowup of first 500 slices at z � 5 m, and (g) blowup of last 500 slices at z � 5 m.


well. Nonresonant excitation shows roughly equivalentresults to resonant excitation. Periodic focusing reducesthe suppression of the electron centroids and lead to greatersaturated amplitudes, although the initial growth rate isabout the same. The ion and electron centroids are allowedto hunt for resonances with periodic focusing. Larger beamsizes will decrease the relative effect of the instability.

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V. DARHT BEAM LINE ION-HOSE SIMULATIONS

The DARHT second-axis accelerator is a long-pulseaccelerator, designed for a 2- � sec electron pulse.The accelerator has more-or-less continuous focusingalong it, with long sections with reduced or no fieldonly in the early part and in the downstream transport

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FIG. 19. (Color) Electron-beam centroid evolution for 20-MeV, 3-mm, periodic-focusing beam case at different axial positions. Eachslice is 0.4 nsec long. The red line is for the horizontal offset and the blue line is for the vertical offset. (a) z � 1 m, (b) z � 2 m, (c)z � 3 m, (d) z � 4 m, (e) z � 5 m, (f) blowup of first 500 slices at z � 5 m, and (g) blowup of last 500 slices at z � 5 m.


after the last accelerating cell. The downstream transportafter the accelerator is intended to match the electronbeam into a kicker section which transforms the beaminto a series of 50 to 100 nsec pulses. The acceleratoris intended to eventually produce a 18.4-MeV, 2-kA elec-tron beam for fast radiography (initially a 1.4-kA beam

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will be produced). The accelerator layout is shown inFig. 24. The ion-hose instability can be suppressed forthis accelerator with proper tuning. In this section, wewill study the instability in an upstream section with dis-crete magnetic focusing and for an alternative tuning in thedownstream transport where it can become significant

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FIG. 20. (Color) Ion centroid evolution for 20-MeV, 3-cm beam case at different axial positions. Each slice is 4 nsec long. The red lineis for the horizontal offset and the blue line is for the vertical offset. (a) z � 1:25 m, (b) z � 2 m, and (c) z � 5 m.


(basically transporting a smaller electron-beam radius). Wewill use a gas pressure in the beam pipe of 2� �10�7� torrand cross section for water ions of 2��10�22� m2. Waterions (H2O�) are considered the most likely candidate for

FIG. 21. (Color) Electron-beam centroid evolution for 20-MeV, 3-cmThe red line is for the horizontal offset and the blue line is for the

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ionization from impact ionization by the primary drivebeam. A full analysis of the instability for the DARHTaccelerator is beyond the scope of this paper. However,these examples can be used with the cases in the previous

beam case at different axial positions. Each slice is 4 nsec long.vertical offset. (a) z � 1:25 m, (b) z � 2:5 m, and (c) z � 5 m.

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FIG. 22. (Color) Ion centroid evolution for 20-MeV, 3-mm, nonresonant beam case at different axial positions. Each slice is 0.8 nseclong. The red line is for the horizontal offset and the blue line is for the vertical offset. (a) z � 0:2 m, (b) z � 0:625 m, and (c)z � 1:25 m, and (d) z � 5 m.


section to build insight into the instability mechanism. Inparticular, the widely spaced focusing will encourage theinstability, and the varying beam size will provide a verywide resonance.

FIG. 23. (Color) Electron-beam centroid evolution for 20-MeV, 3-mis 0.8 nsec long. The red line is for the horizontal offset and the b0:625 m, and (c) z � 1:25 m, and (d) z � 5 m.

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A. Commissioning geometry

For the early part of the DARHT accelerator, we use thenominal commissioning tune, with beam radius, appliedmagnetic field, and beam emittance evolution shown in

m, nonresonant beam case at different axial positions. Each slicelue dashed line is for the vertical offset. (a) z � 0:2 m, (b) z �

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FIG. 24. (Color) Schematic of the DARHT second-axis accel-erator, showing the layout of the injector, beam head clean-upzone, accelerator cells, and downstream transport.


Fig. 25. The commissioning geometry is slightly differentthan the upstream geometry for the final accelerator con-figuration, because additional solenoids are installed in theaxial region from 9 to 12.5 m for better matching into theinjector commissioning diagnostics. The beam energygrows from 2.6 MeV to 4.0 MeV in this section, with a

FIG. 25. Commissioning tune: (a) rms beam radius evolution, (bevolution.

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beam current of 1410 A. There are 8 accelerator cells, eachwith 175 kV, centered at axial positions of 2.1, 2.6, 3.1, 3.6,4.2, 4.7, 5.2, and 5.7 m. The beam radius, injected from adiode, starts at about 7 cm and is focused to about 2 cm. Aregion of the transport intended to scrape off the beamhead, called the beam cleanup zone (BCUZ), is from about6 to about 9 m, followed by special solenoids used only forthe injector commissioning. The BCUZ will also be part ofthe final accelerator configuration. For the commissioningtune, the solenoids in the BCUZ are detuned to providesmooth transport, instead of a series of waists. We see themagnetic field holes due to the discrete location of thesolenoids in this section, in Fig. 25(b). The emittanceevolution, shown in Fig. 25(c) is dominated by transverseplasma oscillations and other nonthermal effects, and iswell understood [25]. For these parameters, the normalizedscaling factor I=�a2B2

0� is typically about 50, and theassumption in numerical model neglecting the ionizationelectrons should be quite accurate.

The electron centroid evolution at the end of the com-missioning geometry (12.5 m) is shown in Fig. 26, forvarious initial offset periods, ranging from about 100 to600 nsec. For this analysis, an 8 � sec electron beam wasused in the simulations, because the instability will notsaturate in 2 � sec , for any of the initial offset periods. Forall cases, the maximum initial centroid offset was 0.1 mm,and 1000 electron slices of 8 nsec apiece were used. We seea relatively narrow resonance, peaked at a drive frequencyof about 2.3 MHz (440 nsec period), with a bandwidth of

) applied external magnetic field, and (c) rms beam emittance

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FIG. 26. (Color) Electron centroids at 12.5 m for an 8 � sec beam (2 � sec has elapsed by the 250th slice). The red line is for thehorizontal offset and the blue line is for the vertical offset. (a) period of 96 nsec, (b) period of 200 nsec, (c) period of 200 nsec ondifferent vertical scale, showing same envelope as the 96 nsec case, (d) period of 304 nsec, (e) period of 352 nsec, (f) period of400 nsec, (g) period of 440 nsec, (h) period of 480 nsec, (i) period of 520 nsec, and (j) period of 600 nsec.


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FIG. 27. (Color) Centroid of slice 800 (6:4 � sec into pulse)versus distance for the commissioning geometry. The red lineis for the horizontal centroid and the blue line is for the verticalcentroid.


about 0.8 MHz. At resonance [Fig. 26(g)], we see clean,single frequency, growth of the ion-hose instability, withno frequency hopping. Off resonance, we see beating andfrequency hopping. Note that the resonance is very one-sided, and the response quickly drops off as the drivingfrequency is decreased. In Fig. 26(a), we see the signatureof driving the ion-hose instability far off resonance. Noamplitude growth is seen. At a period of 200 nsec[Figs. 26(b) and 26(c)], the same low amplitude signatureis seen, until late in the pulse when frequency hopping ismade to approximately the second subharmonic and reso-

FIG. 28. (Color) Ion centroids at (a) 5 m, (b) 7.5 m, (c) 8.75 m, and (an initial offset period of 440 nsec. The red line is for the horizonta

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nance is found. At resonance [Fig. 26(g)], the growth of theelectron-beam centroid oscillations reaches 3 cm after8 � sec . The maximum growth after 2 � sec (theDARHT pulse length) is less than 5 mm.

In Fig. 27, we plot the axial evolution of the centroid ofthe 800th electron slice, to verify that the majority of theexponential growth does in fact happen in the region of theperiodic solenoids.

In Figs. 28 and 29, we plot the ion and electron evolu-tions for an excitation frequency of 2.3 MHz, at axialdistances of 5, 7.5, 8.75, and 10 m. The drive frequencyis off resonance at the local condition at 5 m, and very littleion oscillations are seen, and no growth in the electronamplitudes are seen. At 7.5 m, some limited growth is seen,but both the ion and electron centroids appear to saturate ata low level. Both centroids start seeing exponential growthat 8.75 m, and have large growth rates at 10 m. A keyinsight into the mechanism is that by 10 m, the amplitudeof the electron centroids is comparable to that of the ioncentroids, suggesting that the mechanism damping effectfrom the confinement of the electrons from the externalmagnetic field is failing for the periodic field structure. Thedrive frequency stays in resonance over the range from 5 to10 m because there is very little change in the beam radius.

B. Downstream transport

After the last DARHT accelerating cell is a region calledthe downstream transport. We will use a modified focusingtune where the 19-MeV, 1.41-kA electron beam is about2 mm at the end of the accelerator section just preceding

d) 10 m, for an 8 �sec beam in the commissioning geometry, forl offset and the blue line is for the vertical offset.

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FIG. 29. (Color) Electron centroids at (a) 5 m, (b) 7.5 m, (c) 8.75 m, and (d) 10 m, for an 8 msec beam in the commissioning geometry,for an initial offset period of 440 nsec. The red line is for the horizontal offset and the blue line is for the vertical offset.


this section, and evolves to about 5.3 mm at the end of thissection. Focusing solenoids are located at about 1.8 and7.6 m. The beam rms radius, external focusing magneticfield, and beam emittance are shown in Fig. 30. For this

FIG. 30. Downstream tune: (a) rms beam radius evolution, (b) appli

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simulation, hard-edged solenoidal fields were assumed.The normalized scaling factor I=�a2B2

0� is about 2 in thesolenoids but infinite between them. Since the solenoidlengths are short and we know the growth of the instability

ed external magnetic field, and (c) rms beam emittance evolution.

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FIG. 31. (Color) Electron centroids at 13 m (a) 26 nsec period, (b) 50 nsec period, (c) 100 nsec period, and (d) 200 nsec period, for a2 msec beam in the downstream transport. The red line is for the horizontal offset and the blue line is for the vertical offset.

FIG. 32. (Color) Centroid of slice 800 (1.6 msec into pulse)versus distance for the commissioning geometry. The red line isfor the horizontal centroid and the blue line is for the verticalcentroid.


is suppressed by high axial fields anyway, the error in thesolenoids is not significant.

For this case, the smaller beam radius leads to a muchshorter ion oscillation period, and the simulations are for atotal electron beam pulse length of 2 � sec , using 1000slices of 2 nsec apiece. In Fig. 31, we plot the electroncentroid offset at 13 m (the end of this transport section),for the case of an initial offset maximum of 0.1 mm, foroffset periods ranging from 26 to 200 nsec. In directcontrast to the previous case (the commissioning geome-try), we now see an extremely wide resonance, essentiallyequally driven at all frequencies. The resonance is domi-nated by an oscillation with a 80 nsec period, for all cases.Note that the large change in beam radius (from 2 to 15 mmat its maximum) allows a large range of drive frequenciesto couple into the instability. In Fig. 32, we plot the axialevolution of the electron centroid of the 800th slice, at1:6 � sec into the electron pulse. The centroid definitelyincreases after the second solenoid, at about 7.6 m. Thislarge magnification of the centroid offset is caused by thelarge zero-current phase advance of the solenoid. Themagnification of the centroid offset from periodic focusingis a form of the common thin-lens instability which occursif the focal length of the lenses is less than one quarter ofthe lens separation.

In Fig. 33, we plot the ion centroid evolution at 2.5, 5,7.5, and 10 m, for an initial offset period of 200 nsec. InFig. 34, we plot the electron centroid evolution at the samelocations. The ion centroid contains a strong 50 nsec com-

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ponent by 2.5 m. This oscillation period increases to about100 nsec by 7.5 m, indicating the that oscillations arestaying in phase with the local resonance of the ions dueto the local beam size. The electron centroid oscillationscontain all previous frequency components, and are rich invarious components after 2.5 m. As shown in the periodiccase in Sec. IV, the centroids are allowed to find, and drive,the local resonances. The second solenoid acts as alens magnifying the magnitude of these oscillations.

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FIG. 33. (Color) Ion centroids at (a) 2.5 m, (b) 5 m, (c) 7.5 m, and (d) 10 m for a 2 msec beam in the downstream transport, for aninitial offset period of 200 nsec. The red line is for the horizontal offset and the blue line is for the vertical offset.


Comparing this case with the one for the commissioninggeometry, we find that the instability at the higher energybeam has a larger growth rate. This is surprising, consid-ering the higher energy beam has a larger inertia and that

FIG. 34. (Color) Electron centroids at (a) 2.5 m, (b) 5 m, (c) 7.5 m, aninitial offset period of 200 nsec. The red line is for the horizontal o

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the large-scale changing of the beam radius must be detun-ing the instability somewhat. We can conclude that theclosely spaced periodic focusing in the commissioninggeometry still provides some suppression of the ion-hose

d (d) 10 m for a 2 msec beam in the downstream transport, for anffset and the blue line is for the vertical offset.

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instability that is absent in the downstream configuration. Itis important to point out that a focusing tune with largerbeam sizes will suppress the instability growth amplitudeto an acceptable level.

C. Discussion

With these simulations, we have demonstrated modelingof the ion-hose instability for actual accelerator configura-tions, over sizable distances (over 10 m) with relatively finestep sizes (on the order of millimeters), including details ofacceleration and external focusing. We have used thehybrid-fluid model to phenomologically describe the onsetand development of the instability in regions of interest inthe DARHT long-pulse accelerator.

Continuous, uniform focusing will tend to suppress thegrowth of the instability, by restricting the amplitude of theelectron motion. The ion centroid amplitude is tied to theelectron motion, because the ions fall out of resonanceonce the ions leave the potential well of the electrons andthe external field restricts the ability of the electron slicesto follow the collective ion motion. If the focusing is notcontinuous, but the gaps between solenoids are small,narrow resonances appear. These narrow resonancesprobably depends on both the rms focusing strength andits first derivative, and the resonances can be moved aroundby changing the focusing. Periodic focusing does not re-strict as much the electron tendency to follow, and thusdrive, the collective ion motion, and very large-amplitudemotion can result. If few solenoids are used with longdistances in between, wider resonance appear, with greatlyincreased growth rates. These wide resonances can befound by the electrons either by direct amplification orby harmonic amplification. The amplitude of the centroidoscillations may be augmented by overfocusing from thesolenoids, with excessive zero-current phase advance be-tween the solenoids. For this case, the electron centroidmotion will tend to follow the local ion resonances, and thegrowth rate and saturated amplitude will be relativelyinsensitive to both tuning and beam initial conditions.Use of multiple ion species may lead to some damping.

ACKNOWLEDGMENTS

This work was supported by funds from the DARHTproject at the Los Alamos National Laboratory, operatedby the University of California for the U.S. Department ofEnergy.

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[1] R. J. Briggs (private communication).[2] G. J. Caporaso and J. F. McCarrick, in Proceedings of the

2000 LINAC Conference, Monterey, CA, edited byAlexander Chao (SLAC Report No. SLAC-R-5671, 2000).

[3] T. P. Hughes and T. C. Genomi, in Proceedings of the 2001Particle Accelerator Conference, Chicago, IL, 2001, IEEECatalog No. 01CH37268, 2001.

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using a hybrid-ﬂuid model to simulate the ion-hose

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