Controlling Charge 2008

8/18/2019 Controlling Charge 2008

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Controlling charge and current neutralization of an ion beam pulsein a background plasma by application of a solenoidal magnetic eld:Weak magnetic eld limit

I. D. Kaganovich, E. A. Startsev, A. B. Sefkow, and R. C. DavidsonPrinceton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA

Received 22 February 2008; accepted 23 September 2008; published online 27 October 2008

Propagation of an intense charged particle beam pulse through a background plasma is a commonproblem in astrophysics and plasma applications. The plasma can effectively neutralize the chargeand current of the beam pulse, and thus provides a convenient medium for beam transport. Theapplication of a small solenoidal magnetic eld can drastically change the self-magnetic andself-electric elds of the beam pulse, thus allowing effective control of the beam transport throughthe background plasma. An analytic model is developed to describe the self-magnetic eld of anite-length ion beam pulse propagating in a cold background plasma in a solenoidal magnetic eld.The analytic studies show that the solenoidal magnetic eld starts to inuence the self-electric andself-magnetic elds when ce pe b, where ce = eB / mec is the electron gyrofrequency, pe is theelectron plasma frequency, and b = V b / c is the ion beam velocity relative to the speed of light. Thiscondition typically holds for relatively small magnetic elds about 100 G . Analytical formulas arederived for the effective radial force acting on the beam ions, which can be used to minimize beampinching. The results of analytic theory have been veried by comparison with the simulation resultsobtained from two particle-in-cell codes, which show good agreement. © 2008 American Instituteof Physics . DOI: 10.1063/1.3000131

I. INTRODUCTION

Background plasma can be used as an effective neutral-ization scheme to transport and compress intense chargedparticle beam pulses. To neutralize the large repulsive space-charge force of the beam particles, the beam pulses can betransported through a background plasma. The plasma elec-trons can effectively neutralize the beam charge, and thebackground plasma can provide an ideal medium for beamtransport and focusing. Neutralization of the beam chargeand current by a background plasma is an important issue formany applications involving the transport of fast particles inplasmas, including astrophysics, 1–4 accelerators, 4,5 and iner-tial fusion, in particular, fast ignition 6 and heavy ionfusion, 7,8 magnetic fusion based on eld reversed congura-tions fueled by energetic ion beams, 9 the physics of solarares, 10 as well as basic plasma physics phenomena. 11

Previous studies have explored the option of ion beampulse neutralization by passing t he beam pulse through alayer of plasma or a plasma plug. 12 The ion beam pulse ex-tracts electrons from the plasma plug and drags electrons

along during its motion outside the plasma plug region.There are several limitations of this scheme. When the in-tense ion beam pulse enters the plasma, the electrons streaminto the beam pulse in the strong self-electric and self-magnetic elds, attempting to drastically reduce the ionbeam space charge from a non-neutralized state to a com-pletely neutralized state. After the ion beam pulse exits theplasma, the beam carries along the electrons, with averageelectron density and velocity equal to the ion beam’s averagedensity and velocity. However, large-amplitude plasmawaves are excited in a nonstationary periodic pattern resem-

bling buttery wing motion. 13 Due to these transient effects,the beam may undergo transverse emittance growth, whichwould increase the focal spot size. 14 Smoother edges to theplasma plug density prole lead to a more gradual neutral-ization process and, in turn, results in a smaller emittancegrowth. 14

There are other limitations of this scheme in addition toa deterioration due to transient effects during the beam entry

into and exit from the plasma plug. As the beam transverselyfocuses after passing thorough the plasma plug, the trans-verse electron and ion beam temperature increases due tothe compression and can reach very high values. 16 As a re-sult, the electron Debye length can become comparable withthe beam radius, and the degree of charge neutralization isreduced considerably. This may result in poor beam focus-ing. Including gas ionization by the beam ions does not sig-nicantly improve the neutralization, mainly because theelectrons, which are produced by ionization, are concentratedin the beam path, whereas for effective neutralization of theion beam puls e, the supply of electrons should be from out-side the beam. 14

Therefore, neutralized ballistic focusing typically re-quires the presence of background plasma in and around thebeam pulse path for good charge neutralization. Reference16 showed that hot electrons cannot neutralize the beam wellenough; therefore, any electron heating due to beam-plasmainteractions has to be minimized. The presence of cold,“fresh” plasma in the beam path provides the minimumspace-charge potential and the best option for neutralizedballistic focusing. Experimental studies of ballistic transversefocusing have conrmed that the best results are achievedwhen both a plasma plug and a bulk plasma are used for

PHYSICS OF PLASMAS 15 , 103108 2008

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charge neutralization. 8,11 Hence, in the following we onlystudy the case when a large amount of cold backgroundplasma is available everywhere on the beam path.

The application of a solenoidal magnetic eld allowsadditional control and focusing of the beam pulse. 15 A strongmagnetic lens with a magnetic eld up to a few Tesla caneffectively focus beams in short distances order of a few tensof centimeters. However, due to the very strong magnetic

eld in the solenoid, the magnetic eld leaking outside thesolenoid can affect the degree of charge and current neutral-ization. In this paper, we show that even a small solenoidalmagnetic eld, typically less than 100 G, strongly changesthe self-magnetic and self-electric elds in the beam pulsepropagating in a background plasma. Such values of mag-netic eld can be present over distances of a few meters fromthe strong solenoid, and thereby affect the focusing of thebeam pulse. Moreover, a small solenoidal magnetic eld canbe applied to optimize propagation of a beam pulse througha background plasma over long distances.

In Refs. 18 and 19, the response of a magnetized plasmato intense ion beam injection was studied while neglectingelectron inertia effects, which corresponded to magneticelds of a few Tesla in ion ring devices. In the present paper,we analyze the opposite limit, corresponding to small valuesof magnetic eld. In the collisionless limit and without anapplied solenoidal magnetic eld, the return current is drivenby an inducti ve electric eld which is balanced by electroninertia effects. 20 Taking electron inertia effects into accountallows us to study the transition from the limit where thesolenoidal magnetic eld is small, i.e., where the presence of the applied solenoidal magnetic eld begins to affect the re-turn current in the plasma, and to determine the range of magnetic eld values which strongly affect the self-electric

and self-magnetic elds of a beam pulse propagating in abackground plasma. This allows us to study the beam pulseevolution over a wide range of solenoidal magnetic eldstrengths, from approximately zero to very large values, suchas when the beam pulse encounters an applied solenoidalmagnetic lens. Beam pulse propagation in a backgroundplasma immersed in an applied solenoidal magnetic eld hasbeen studied both analytically and numerically using twodifferent particle-in-cell codes to crosscheck the validity of the results.

This pa per is a considerably extended version of ourearlier letter 21 on this topic. In the present paper an analyticmodel is developed to describe the self-electromagnetic

elds of a nite-length beam pulse propagating in a coldbackground plasma in a solenoidal magnetic eld. Previ-ously, we developed an analytic model to describe the cur-rent neutraliza tion of a beam pulse propagating in a back-ground plasma 20,22 without an applied magnetic eld. Thesestudies provided important scali ng laws for the degrees of charge and current neutralization, 23,24 as well as served as acomputationally efcient tool for describing relativistic elec-tron beam transport in collisionless p lasma for modeling of the electromagnetic Weibel instability. 22

The electron response time to an external charge pertur-bation is determined by the electron plasma frequency, pe= 4 e2n p / m 1

/ 2, where n p is the background plasma density.

Therefore, as the beam pulse enters the background plasma,the plasma electrons tend to neutralize the beam pulse on atime scale of order pe

−1 . Typically, the beam pulse propaga-tion duration through the background plasma is long com-pared with pe

−1 . For electron beam pulses, some instabilitiescan develop very fast on a time scale comparable to theplasma period, 2 / pe . However, if the beam density issmall compared to the plasma density the instabilities’

growth r ate s are also small compared to the plasmafrequency. 22 As a result, after the beam pulse passes througha short transition region, the plasma disturbances are station-ary in the beam frame. In a previous study, we have devel-oped reduced nonlinear models, which describe the station-ary plasma disturbance in the beam frame excited by theintense ion beam pulse. 13,20 In these calculations, 20 we inves-tigated the nonlinear quasiequilibrium properties of an in-tense, long ion beam pulse propagating through a cold, back-ground plasma, assuming that the beam pulse duration ismuch longer than 2 / pe, i.e., b pe 2 , where b is thebeam pulse duration. In a subsequent study, 13 we extendedthe previous results to general values of the parameter b pe .Theoretical predictions agree well with the results of calcu-lations utilizing several particle-in-cell PIC codes. 13,20

The model predicts very good charge neutralization dur-ing quasisteady-state propagation, provided the beam is non-relativistic and the beam pulse duration and the beam currentrise time is much longer than the electron plasma period, i.e.,

b pe 2 . Thus, the degree of charge neutralization de-pends on the beam pulse duration and plasma density, and isindependent of the beam current if n p nb . However, thedegree of beam current neutralization depends on both thebackground plasma density and the beam current. The beamcurrent can be neutralized by the electron return current. The

beam charge is neutralized mostly by the action of the elec-trostatic electric eld. In contrast, the electron return currentis driven by the inductive electric eld generated by the in-homogeneous magnetic ux of the beam pulse in the refer-ence frame of the background plasma. Electrons are acceler-ated in the direction of beam propagation for ion beams andin the opposite direction for electron beams. From the chargedensity continuity equation, / t + · J =0 = e n p + Z bnb− ne , it follows that if the electrons neutralize the currentthey will neutralize the charge as well. The inductive electriceld penetrates into the plasma over distances of order theskin depth c / pe, where c is the speed of light. If the beamradius, r b, is small compared with the skin depth c / pe , the

electron return current is distributed over distances of orderc / pe. As a result, the electron return current is aboutr b pe / c times smaller than the beam current. Consequently,the beam current is neutralized by the electron current, pro-vided the beam radius is large compared with the electronskin depth, i.e., r b c / pe , and is not neutralized in the op-posite limit. This condition can be written as I b

4.25 bnb / n p kA, where b is the beam velocity normal-ized to the speed of light, and nb is the beam density.

This model has been extended to include the additionaleffects of gas ionization during beam propagation in a back-ground gas. Accounting for plasma production by gas ioniza-tion yields a larger self-magnetic eld of the ion beam com-

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pared to the case without ionization, and a wake of thecurrent density and self-magnetic eld are generated behindthe beam pulse. 25 In Ref. 25, beam propagation in a dipolemagnetic eld conguration and background plasma has alsobeen studied.

In the presence of an applied solenoidal magnetic eld,however, the system of equations describing the self-magnetic eld becomes much more complicated. A high so-

lenoidal magnetic eld inhibits radial electron transport, andthe electrons move primarily along the magnetic eld lines.For high-intensity beam pulses propagating through a back-ground plasma with pulse duration much longer than theelectron plasma period, one is tempted to assume that thequasineutrality condition holds, ne n p + Z bnb, where ne isthe electron density, n b is the density of the beam pulse, Z beis ion charge for the beam ions, whereas Z b =−1 for electronbeams, and n p is the density of the background ions as-sumed unperturbed by the beam . In the limit of a strongmagnetic eld, the plasma electrons are attached to the mag-netic eld lines and their motion is primarily along the mag-netic eld lines. For one-dimensional electron motion, thecharge density continuity equation, / t + · J =0, com-bined with the quasineutrality condition = e n p + Z bnb − ne

0 and absence of external current yields J 0. Therefore,in the limit of a strong solenoidal magnetic eld, the beamcurrent can be expected to be completely neutralized.

However, the above description fails to account for theelectron rotation that develops in the presence of a solenoidalmagnetic eld. Due to the small inward radial electron mo-tion, the electrons can enter into the region of smaller sole-noidal magnetic ux. Due to the conservation of canonicalangular momentum, the electrons start spinning with a veryhigh azimuthal velocity , which is much larger than the ion

beam rotation velocity. This spinning produces many unex-pected effects.The rst effect is the dynamo effect. 26 If the magnetic

eld is attached to the electron ow, the electron rotationbends the solenoidal magnetic eld lines and generates anazimuthal self-magnetic eld in the beam pulse. Note,though, that when electron inertia effects are taken into ac-count the generalized electron vorticity is frozen into theplasma electron ow, rather than simply the magnetic eldlines being frozen into the electron ow, as discussed in thenext section. Moreover, the electron rotation generates aself-magnetic eld that is much larger than in the limit withno applied eld. The second effect is the generation of a

large radial electric eld. Because the v B z force shouldbe balanced by a radial electric eld, the spinning results in aplasma polarization, and produces a much larger self-electriceld than in the limit with no applied eld. The total forceacting o n the beam particles now can change from always focusing 20 in the limit with no applied solenoidal magneticeld, to defocusing at higher values of the solenoidal mag-netic eld. In particular, an optimum value of magnetic eldfor long-distance transport of a beam pulse, needed, for ex-ample, in inertial fusion applications, 17 can be chosen wherethe forces nearly cancel. The third unexpected effect is thatthe joint system consisting of the ion beam pulse and thebackground plasma acts as a paramagnetic medium, i.e., the

solenoidal magnetic eld is enhanced inside of the ion beampulse.

With a further increase in the magnetic eld value, thebeam pulse can excite strong electromagnetic perturbations,including whist ler waves, corresponding to longerwavel ength s,18,27 and lower-hybrid-like or heliconwaves, 28,29 corresponding to shorter wavelengths. Both waveperturbations propagate nearly perpendicular to the beam

propagation direction. A similar excitation of helicon wavesduring fast penetration of the magnetic eld due to the Halleffect in high energy plasma devices, such as, plasma open-ing switches and Z pinches, has been observed in Refs. 30.Here, we consider relatively short ion pulses with pulse du-ration b 2 / pi, where pi is the background ion plasmafrequency, so that the background plasma ion response canbe neglected. For longer ion pulses, t he pla sma ion responsemay effect the plasma return current. 11,31,32

The organization of this paper is as follows: In Sec. II,the basic equations and model are discussed. Section III pro-vides a comparison between analytic theory and particle-in-cell simulations results. In Sec. IV, the dependence of theradial force acting on the beam particles on the strength of the solenoidal magnetic eld is discussed. Finally, Sec. Vdescribes the excitation of electromagnetic perturbations bythe beam pulse, including whistler waves and lower-hybrid-like helicon waves. In a follow-up pu blication the limit of strong magnetic eld will be discussed. 33

II. BASIC EQUATIONS

The electron uid equations together with Maxwell’sequations comprise a complete system of equations describ-ing the electron response to the propagating ion beam pulse.

The electron uid equations consist of the continuityequation,

ne t

+ · neV e = 0 , 1

and the force balance equation,

V e t

+ V e · V e = −e

mE +

1c

V e B , 2

where − e is the electron charge, m is the electron rest mass,and V e is the electron ow velocity. Maxwell’s equations forthe self-generated electric and magnetic elds, E and B, aregiven by

B =4 e

c Z bnbV b − neV e +

1c

E t

, 3

E = −1c

B t

, 4

where V b is the ion beam velocity, n e and nb are the numberdensities of the plasma electrons and beam ions, respectively,and Z b is the ion charge state for the beam ions, whereas Z b =−1 for electron beams.

We assume that the beam pulse moves with constantvelocity V b along the z-axis. We look for stationary solutions

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in the reference frame of the moving beam, i.e., where allquantities depend on t and z exclusively through the combi-nation V bt − z . We further consider cylindrically symmetric,long beam pulses with length, lb, and radius, r b, satisfying

lb V b/ pe , lb r b , 5

where pe = 4 e2ne / m 1/ 2 is the electron plasma frequency.

We also assume that the elds and electron ow velocity anddensity are in steady state in a reference frame moving withthe beam pulse. We introduce the vector potential,

B = A , 6

and make use of the transverse Coulomb gauge, · A = 0.For axisymmetric geometry, this gives Ar =0. The azimuthalmagnetic eld is

B = − A z r

, 7

and the perturbed by the plasma magnetic eld componentsare

B z =1

r

rA r

, Br = − A

z. 8

For long beams with lb V b / pe , the displacement cur-rent the nal term on the right-hand side of Eq. 3 is of order V b / pelb 2 1 compared to the electron current. Be-cause lb r b is assumed, the terms on the left-hand side of Eqs. 3 of order r b / lb 2 are neglected, as well. This gives

−1r

r r

A z r = 4 ec Z bnbV bz − neV ez 9

and

−

r

1r

rA r = 4 ec Z bnbV b − neV e . 10

The electron momentum equation, Eq. 2 , can be solvedto obtain the three components of electron velocityV ez , V er , V e . Howeve r, it is ea sier to use conservation of thegeneralized vorticity, 20,22,30,34 which states that the circula-tion C of the canonical momentum,

C p e − eA / c · r 11taken along a closed loop, which is “frozen-in” and movingtogether with the electron uid, remains constant. ApplyingThompson’s theorem, the circulation dened in Eq. 11 canbe rewritten as the surface integral of the generalized vortic-ity

C = p e − eA / c · r= p e − eA / c · S

· S , 12

where S is the uid surface element, and the generalizedvorticity is dened as

= p e − eA / c . 13

If electron inertia terms are neglected, the electron mechani-cal momentum can also be neglected in the expression forthe generalized vorticity, which gives −eB / c. The con-servation of generalized vorticity then becomes the well-known expression for the conservation of magnetic uxthrough a uid contour C = B · S =const. , e.g., see Ref.35.

Equation 12 can be rewritten in the differential form20

t + V e · = − · Ve + · V e . 14

Substituting · V e into Eq. 14 from the continuity equation1 ,

· Ve = −1ne

ne t

−V ene

· ne , 15

gives

t + Ve · ne

= ne · V e . 16This is a generalization of the “frozen-in” condition for themagnetic eld lines, when electron inertia terms areneglected. 35

As an example of application of the generalized vorticitylaw, we derive the magnetic dynamo effect using both inte-gral and differential forms of the conservation of generalizedvorticity, given by Eq. 12 and Eq. 16 , respectively. Con-sider a small element of the electron uid of size drdz , po-sitioned in a plane of constant ; then · S = drdz 0,as shown in Fig. 1. In the next time interval, t + dt , the uidelement moves and rotates. Due to the differential rotation

V e / z, the sides of the element rotate differently, and the

surface element opens in the z-direction. In the next timeinterval, · S =− zdrdz V e / zdt + drdz 1. Using thefact that the electron density is conserved in the uid ele-ment, drdzn e, the time derivative of the azimuthal compo-nent of vorticity, 1 − 0 / dt , can be written as

d

dt

ne=

1ne

z

V e z

. 17

This result can also be derived directly by taking the azi-muthal projection of Eq. 16 and neglecting the small radialcontribution on the right-hand side, because r z.

For simplicity, in the following we consider the mostpractically important case when the plasma density is large

dz

dr

FIG. 1. Schematic of the differential rotation of uid elements near thesymmetry axis.




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n p nb so that the changes in ne can be neglected in Eq. 16 .Also because n p nb, the effects of electron ows are smallcompared to the beam motion V ez V b , and we approxi-mate d / dt V b / z. Substituting into Eq. 17 , and integrat-ing with zero initial conditions in front of the beam pulsegives

= zV e V b

. 18

Here, we made use of the fact that z=− eB z / c is approxi-mately constant. From Eq. 13 , it follows that

− mV ez − eA z/ c / r , where only the radial derivatives aretaken into account, due to the approximation of long beampulses in Eq. 5 . Substituting the expressions for and zinto Eq. 18 , and integrating radially gives

V ez =e

mc A z +

eB z

mcV b r V e dr . 19

The rst term on the right-hand side of Eq. 19 describes theconservation of canonical momentum in the absence of mag-netic eld; the second term describes the magnetic dynamoeffect, i.e., the generation of azimuthal magnetic eld due tothe rotation of magnetic eld lines, 26 as shown in Fig. 2.Note that, if the inertia effects are neglected, Eq. 18 de-scribes the magnetic eld “frozen in” the electron ow, B = B zV e / V b.

Substituting V e from Ampere’s law in Eq. 10 , andassuming that the velocity of the beam rotation is small com-pared to the rotation velocity of the plasma electrons, gives

r V e dr = −

c

4 en p

1r

rA r

+r

Z bnbn p

V b dr . 20

Substituting Eq. 20 into Eq. 19 then gives

V ez =e

mc A z −

B z4 mV bne

1r

rA r

+eB z

mcV b r Z b

nbn p

V b dr .

21

Similarly, from the z projection of Eq. 16 , we obtain

r r r mV e − eA / c = −

eB zcV b

V bne − n p

n p− V ez , 22

and accounting for quasineutrality, ne − n p = Z bnb, and substi-tuting the expression for the current J z= Z ben bV b − en pV ezgives

mV e − eA /c =

B zcV bn pr r J zrdr . 23

Equation 23 describes the conservation of canonical angu-lar momentum

mV e =e

c A + rB z , 24

where r is the change in the radial position of the electronuid element inside of the beam pulse compared to the initialradial position in front of the beam pulse. Indeed, because of the conservation of current, · J =0, it follows that r J zrdr = er z neV er dz = eV brn p r , where r is the change in theradial position of a contour immersed in the electron uid.Equation 23 also describes the conservation of vorticityux in the z-direction through a circle in the azimu-thal direction, · S =2 0r rdr z=2 0r rdrd r mV e − eA / c / rdr =2 r mV e − eA / c − r 2eB z / c =const.

Making use of Ampere’s equation in the z-directiongives r J zrdr = cr / 4 A z / r , and

mV e −e

c A =

B z4 V bn p

A z r

. 25

Substituting Eqs. 19 and 25 into the corresponding com-ponents of Ampere’s equation then gives

−1r

r r

A z r = 4 ec Z bnbV bz − emc ne A z

+ B z

4 mV b

1

r

rA r

+eB z

mcV b r Z b

nbn p

V b dr , 26

and

−

r

1r

rA r =

4 ec Z bnbV b −

e

mc ne A

− B z

4 mV b

A z r . 27

As we shall see in the next section, under conditions of interest, the electron rotation is of order the electron cyclo-tron frequency times the ratio of the beam density to theplasma density, which is much larger than the ion rotation,which is given by the ion cyclotron frequency and the lastterm on the right-hand side of Eq. 26 can be neglected. Ingeneral, analysis shows that the electron inertia terms areimportant if

Vb

magnet c eld l neion beam pulse

magnetic flux surfaces

FIG. 2. Color online Schematic of magnetic eld generation due to thedynamo effect. The magnetic eld line is shown by the black solid line; acontour attached to the electron uid element is shown by the brown dashed

line in front of the beam pulse; and the dotted brown line indicates thiscontour inside of the ion beam pulse, the outline of which is shown by theorange, thin dotted line. The radial electron displacement generates a poloi-dal rotation; the poloidal rotation twists the solenoidal magnetic eld andgenerates the poloidal magnetic eld.




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ec r bV b

M m ,in the opposite limit of the electron motion can be describedin pure drift approximation.

III. COMPARISON OF ANALYTIC THEORYAND PARTICLE-IN-CELL SIMULATIONS

Figures 3 and 4 show the simulation results obtainedfrom the particle-in-cell PIC code EdPIC Ref. 20 for thedensity and magnetic eld of an ion beam pulse propagatingwith beam velocity V b =0.5 c in slab geometry, whereas Figs.5 and 6 show the sim ulation results obtained from the LSPcode, with V b =0.33 c.

36 In all simulations beams enter theplasma in the presence of a uniform solenoidal magneticeld. After some transitional period, plasma perturbationsreach a quasisteady-state in the beam frame. We have per-formed the PIC simulations in slab geometry, because thenumerical noise tends to be larger in cylindrical geometry

0

0.5

1

1.5

2

2.5

4.5 3 .6 2 .7 1 .8 0 .9 0 0 .9 1 .8 2 .7 3 .6 4 .5

12

9

6

3

0

3

6

9

12

15

xω pe /c

z

ω p e

/ c

15

FIG. 3. Color online The electron density perturbation caused by an ionbeam pulse moving with velocity V b =0.5 c along the z-axis. The beam den-sity is one-half of the background plasma density; the beam prole is attopwith smooth edges; the beam radius is r b =1.5 c / pe; and the beam half length is lb =7.5 c / pe.

φ

ω

ω

φ

φ

ω

δ

ω

ω

φ δ

φ

ω

δ

ω

ω

φ δ

φ

ω

δ

ω

ω

φ δ

(b)

(a) (c)

(d)

FIG. 4. Color online Comparison of analytic theory and EdPIC particle-in-cell simulation results for the self-magnetic eld and the perturbations in thesolenoidal magnetic eld in the center slice of the beam pulse. The beam parameters are the same as in Fig. 3. The beam velocity V b =0.5 c. The values of applied solenoidal magnetic eld correspond to the ratio of cyclotron to plasma frequency ce / pe: a 0; b 0.25; c 0.5; and d 1.




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due to the singularity on the axis r =0 . In Fig. 3, the beamdensity is one-half of the background plasma density; thebeam prole has a at top with smooth edges; the beamradius corresponds to r b =1.5 c / pe; and the beam half lengthis lb =7.5 c / pe , lead ions were assumed, however, and ionmotion was not important for short beam pulses. Figure 3shows that large-amplitude plasma waves are excited by thebeam head. The plasma waves are electrostatic, and, there-fore, the plasma waves do not have an effec t on the structureof the self-magnetic eld of the beam pulse, 20 except that thelocal value of the electron density is different from the pre-dictions of the quasineutrality condition ne = Z bnb + n p andaffects the value of the return current en eV ez. Such large den-sity perturbations are not accounted for in linear analytictheory nb n p , which is the reason for the difference be-tween the PIC simulations and the analytic predictions, aswill be shown below. Note that the presence of the solenoidalmagnetic eld results in an increase of the self-magneticeld. This is due to the magnetic dynamo effect caused bythe electron rotation, as discussed above see also Fig. 2 .

Another unusual effect is that the system consisting of the beam pulse together with the background plasma actsparamagnetically; the solenoidal magnetic eld is larger inthe center of the beam pulse than the initial value of theapplied magnetic eld. This effect can be found to originatefrom Eqs. 26 and 27 in the limit where the skin depth is

large compared with the beam radius c/

pe r b . In thislimit, the terms proportional to the return current ne A on theright-hand side of Eq. 27 can be neglected compared withthe terms on the left-hand side. Without taking into accountcontributions from the ions, and neglecting the term ne A ,Eq. 27 can then be integrated from r to , assuming that A =0 as r → . This gives for the perturbation in the sole-noidal magnetic eld

B z =1r

rA r

=4 e

c B z A z4 mV b . 28Note that B z is positive, i.e., the combination of the beamand plasma acts paramagnetically! In the follow-up

research, 37 we found that the beam plus plasma system re-sponse strongly depends on parameter ce / b pe . If

ce / b pe 1, the response is paramagnetic, if ce / b pe1, the response is diamagnetic.

Substituting Eq. 28 into Eq. 26 gives

−1r

r r

A z r = 4 ec Z bnbV bz − emc ne A z

+ B z2

4 m2V b2

e

c A z . 29

Note that the nal positive term on the right-hand side of Eq.29 proportional to B z

2 describes the dynamo effect, andleads to an increase in the self-magnetic eld. This increasebecomes signicant if

ne B z

2

4 mV b2 30

or

ce peV

bc , 31

where ce = eB z / mc is the electron cyclotron frequency. Thisis evident in Fig. 4 by comparing the value of the self-magnetic eld in Figs. 4 a –4 c with Fig. 4 d .

Figure 6 shows a comparison of analytic theory and LSPRef. 36 particle-in-cell simulation results for the self-

magnetic eld, the perturbation in the solenoidal magneticeld, and the radial electric eld in the ion beam pulse. Thebeam velocity is V b =0.33 c, and the beam density prole isGaussian, nb0 exp −r 2 / r b

2 − z2 / lb2 , where r b =1 cm, lb

=17 cm, nb0 = n p / 2=1.2 1011 cm −3 . The background

plasma density is n p =2.4 1011

cm−3

, except for case d ,where the beam density is nb0 =0.6 1011 cm −3 and theplasma density is n p =4.8 1011 cm −3 ; and case f , wherenb0 =0.3 1011 cm −3 and the background density is n p =2.4

1011 cm −3 . Figure 5 shows the electron density perturba-tion generated by the beam pulse. Because the beam head islong compared with the leng th V b / pe , the beam head doesnot excite any plasma waves, 20 and the quasineutrality con-dition ne = nb + n p is satised compare Fig. 3 and Fig. 5 .

For this choice of beam parameters, the skin depth isapproximately equal to the beam radius c / pe r b, so thatthe return current does not screen the beam self-magneticeld signicantly. Without the applied solenoidal magnetic

eld, the maximum value of the magnetic eld is 56 G seeFig. 6 a . The analytic theory agrees well with the PICsimulation results, because in thi s case the theory applieseven for the nonlinear case nb n p.

20 The radial electric eldis small and cannot be distinguished from numerical noise inthe PIC simulations. For the value of the applied solenoidalmagnetic eld B z0 =300 G, in Fig. 6 b , the parameter

ce / b pe =0.57, where b = V b / c is small. Therefore, thedynamo effect is insignicant according to Eq. 29 . Figures6 c and 6 e correspond to two and three times larger mag-netic elds B z0 =600 G and B z0 =900 G , respectively. Thevalue of the parameter ce / b pe =1.1,1.7, rises above unity,and the dynamo effect results in a considerable increase in

-10 -5 0 5 10

x(cm)

z ( c m )

140

180

2

4

3

FIG. 5. Color online The electron density perturbation caused by an ionbeam pulse moving with velocity V b =0.33 c along the z-axis. The beamdensity prole is Gaussian with r b =1 cm, lb =17 cm, and nb0 = n p / 2=1.2

1011 cm −3 .




8/16

the self-magnetic eld of the beam, also in agreement withEq. 29 . The 20% difference between the analytic and PICsimulation results is due to the fact that the theory of thedynamo effect is linear in the parameter nb / n p, whereas

nb / n p =0.5 in Figs. 6 b , 6 c , and 6 e . Figure 6 d showsresults for nb / n p =0.125, and the linear theory results arepractically indistinguishable from the PIC simulation results.Figure 6 f shows results for nb / n p =0.125, and the linear

φ

φ

φ

δ

φ δ

φ

δ

φ δ

φ

δ

φ δ

φ

δ

φ δ

φ

δ

φ δ

(b)

(a)

(c)

(d)

(f)

(e)

FIG. 6. Color online Comparison of analytic theory and LSP particle-in-cell simulation results for the self-magnetic eld, perturbation in the solenoidalmagnetic eld, and the radial electric eld in a perpendicular slice of the beam pulse. The beam parameters are the same as in Fig. 5 with n b0 = n p / 2=1.2

1011 cm −3 , except for d , where nb0 = n p / 8=0.6 1011 cm −3 , and f , where nb0 = n p / 8=0.3 1011 cm −3 . The values of the applied solenoidal magnetic eld, B z0, are: a B z0 =0 G; b B z0 =300 G; c and d B z0 =600 G; e and f B z0 =900 G.




9/16

theory results differs from the PIC simulation results by ap-proximately 30%. This is due to the assumption of quasineu-trality, which requires ce

2 / pe2 1 as shown below. For the

conditions in Fig. 6 f , ce2 / pe

2 =0.33, which accounts for the30% difference from the PIC simulation results.

The radial electric eld can be obtained from the radialcomponent of the momentum balance equation 2 . Neglect-ing the small radial electron velocity V er gives

E r =mV e

2

er +

1c

− V e B z + V ez B , 32

where V e is given by Eq. 25 . From Eq. 32 it follows thatthe radial electric eld increases strongly with increasing so-lenoidal magnetic eld, as is evident in Fig. 6. As shown inSec. V, the ion dynamics can reduce radial electric eld andhas to be taken into account for very long beam pulses lb

r b M / m 1/ 2.

As the electric eld increases with an increase in theapplied solenoidal magnetic eld, the assumption of quasineutrality may fail. To nd the criterion for validity of

the theory we estimate the electric eld value, consideringonly linear terms assuming nb n p. In this limit, the nonlin-ear terms in Eq. 32 can be neglected, which gives

E r = −1c

V e B z. 33

Equations 26 and 27 can be represented in dimensionlessform if the following normalization is applied:

r = p c

pe,

A z = mcV bze Z bnb0

n p,

A = B z p Z bnb0

n p,

V e =eB z p

mc

Z bnb0n p

,

where nb0 = nb 0 is the on-axis value of the beam density.Some straightforward algebra applied to Eqs. 26 and 27gives for the normalized components of vector potential, a z

= A z/

A z and a = A /

A ,

−1

a z

=nb r / p

nb0− a z +

ce2

pe2 b

2

1

a

, 34

1

a

= a + a z

. 35

Here, r / p. Note that the solutions of Eqs. 34 and 35depend only on two parameters: the ratio of the beam radiusto the skin depth through the beam density prole , and theparameter ce

2 / pe2 b

2, which characterizes the dynamo effectsee Eq. 31 .

The electron rotation velocity and azimuthal magneticeld are expressed through the normalized components of vector potential according to

V e = Z bnb0

n p

eB z pmc a + a z , 36

B = − Z bnb0

n p

mcV bz

e p

a z

. 37

Substituting Eqs. 36 and 37 into Eq. 33 then gives

E r = − Z bnb0

n p

mV bz2

e p

ce2

pe2 b

2

a z

+ a . 38

The quasineutrality condition requires

rE r r r

4 e Z b nb0 . 39

Substituting the estimate E r / r E r / p for E r into Eq. 38 ,and taking the normalized vector potentials to be of orderunity into Eq. 39 gives the condition

ce2

pe2

1. 40

The reason for the condition in Eq. 40 can be explainedas follows. The dielectric constant transverse to the magneticeld is given by

= 1 + pe2

ce2 − 2

. 41

In the analytic derivation, we accounted only for the plasmapart of the dielectric constant the last term on the right-handside of Eq. 41 , and neglected the displacement current.Apparently when ce , this is valid only if the conditionin Eq. 40 is satised. In order to account for a departurefrom the quasineutrality condition, we substitute into Eq.22 the perturbations in the electron density according to the

Poisson equation

Z bnb − ne + n p =1

4 er

rE r r

,

which gives

r mV e − eA / c

r r

= − B z

cn pV b−

V b rE r 4 r r

+ Z bnbeV b − eV ezn p . 42

Integrating Eq. 42 with respect to r gives

V e =e

mc A +

B zmcV bn pr r

J zrdr +rV b E r

4 . 43

Substituting Eq. 33 for E r ,

E r = −1c

V e B z, 44

and r J zrdr = cr / 4 A z / r into Eq. 43 then gives




10/16

V e 1 + ce2

pe2 = emc A + B z4 mV bn p

A z r

. 45

Equation 26 remains the same, but Eq. 27 is modied tobecome

− 1 + ce2 pe2

r

1r

rA r

=4 e

c Z bnbV b − emc ne A − B z4 mV b A z r . 46

The equations for the normalized vector potentials become

−1

a z

=nb r / p

nb0− a z +

ce2

pe2 b

2

1

a

, 47

1 + ce2

pe2 1 a = a + a z . 48

The electron rotation velocity, azimuthal magnetic eld, andradial electric eld are then expressed through the normal-ized components of vector potential according to

V e = Z bnb0

n p 1 + ce2

pe2

eB z pcm a + a z , 49

B = − Z bnb0

n p

mcV bze p

a z

, 50

φ

δ

φ

φ δ

φ

δ

φ δ

φ

δ

φ δ

φ

δ

(b)

(a) (c)

(d)

FIG. 7. Color online Comparison of analytic theory and LSP particle-in-cell simulation results for the self-magnetic eld, perturbation in the solenoidalmagnetic eld, and the radial electric eld in a perpendicular slice of the beam pulse. The beam velocity is V b =0.808 c . The plasma and beam parameters aren p =4.8 1011 cm −3 , nb0 =0.5 1011 cm −3 . The values of the applied solenoidal magnetic eld, B z0, are: a B z0 =0 G; b B z0 =900 G; c B z0 =1800 G; and d B z0 =3600 G.




11/16

E r = − Z bnb0

n p 1 + ce2

pe2

mV bz2

e p

ce2

pe2 b

2

a z

+ a . 51

Figure 6 f and Fig. 7 show the effects of the modica-tion of Eq. 27 to Eq. 46 . For the conditions in Fig. 6 f ,

ce2 / pe

2 =0.33, and this 30% correction brings the analyticresults much closer the PIC simulation results. Figure 7shows the self-magnetic and self-electric elds for a fasterbeam pulse than shown in Fig. 6, with V b =0.808 c . Figure7 a shows the case without any applied magnetic eld; thenotation “Nonlin. Anal.” denotes the results calculated fromEq. 26 where the perturbation in the electron density non-linear term in the return current ne A z, ne = n p + Z bnb is takeninto account; because nb / n p 0.1, this term accounts forabout 10% of the difference between the nonlinear and linear

theories. Figures 7 b –7 d show the results of linear theorywhen the solenoidal magnetic eld is applied. The notation“Full Anal.” denotes the results calculated from the system

Eqs. 26 and Eq. 46 , whereas the notation “Anal.” denotesthe system of equations corresponding to Eqs. 26 and 27 .The difference becomes noticeable for B=1.8 kG, where

ce2 / pe

2 =0.66. At the larger value of the magnetic eld B=3.6 kG, ce

2 / pe2 =2.6 and the solutions to Eqs. 26 and

27 show the excitation of waves, whereas the system of equations corresponding to Eqs. 26 and 46 does not, asdescribed in Sec. V.

Figure 8 shows the perturbation in the electron densityfor B=0 ,3.6,5.4 kG, which corresponds to ce / pe=0 ,1.6,2.4. It is evident that for cases b and c thequasineutrality condition breaks down, which corresponds to

ce pe. However, when the beam radius is increased, this

δ

δ

δ

δ

δ

δ

δ

(b)

(a) (c)

(d)

FIG. 8. Comparison of analytic theory and LSP particle-in-cell simulation results for the perturbation in the electron density. The beam velocity is V b=0.808 c. The plasma and beam parameters are n p =4.8 1011 cm −3 , n b0 =10 11 cm−3 . The beam density prole is Gaussian, n b0 exp −r 2 / r b

2 , where r b =1 cm,except for d . The values of the applied solenoidal magnetic eld, B z0, are: a B z0 =0 G; b B z0 =3600 G; c B z0 =5400 G. b shows the effect of the beamradius on the perturbation in the electron density for the parameters in case c , but with the beam radius equal to 2, 4, and 8 cm; only analytic calculationsare shown.




12/16

leads to a decrease in the radial electric eld according to Eq.51 , and consequently the quasineutrality condition is re-

stored for the perturbation in the electron density as shown inFig. 8 d .

IV. RADIAL FORCE ACTING ON THE BEAMPARTICLES

The radial force acting on the beam particles is

F r = eZ b −1c

V bz B + E r , 52

where the radial electric eld is given by Eq. 32 . Withoutthe solenoidal magnetic eld applied, substituting Eq. 32into Eq. 52 gives

F r = −eZ b

cV bz − V ez B , 53

and the radial force is always focusing, because the electronow velocity in the return current is always smaller than the

beam velocity, V ez V bz.20

However, in the presence of thesolenoidal magnetic eld, the radial force can change signfrom focusing to defocusing, because the radial electric eldgrows faster than the magnetic force − Z bV bz B , as the sole-noidal magnetic eld increases. To demonstrate this tendencyanalytically, let us consider only linear terms in the radialforce equation assuming nb n p. In this limit, the nonlinearterms in Eq. 32 can be neglected, which gives

F r = −eZ b

cV bz B + V e B z , 54

where V e is given by Eq. 25 .Substituting Eqs. 36 and 37 into Eq. 54 then gives

F r = Z b

2nb0n p

mV bz2

p

a z

− ce2

pe2 + ce

2 b2

a z

+ a . 55

From Eq. 55 , it is evident that, in the limit ce2 pe2+ ce

2 b2 or ce pe b b where b2 =1 / 1− b2 , the radial

force is focusing a z / r 0 , but if ce pe b b, the ra-dial force can become defocusing. Figure 9 shows the evo-lution of the radial prole of the normalized radial force fora nonrelativistic beam b 1 the term in the square bracketon the right-hand side of Eq. 55 acting on the beam par-ticles for various values of the parameter ce

2 / pe2 b

2. Theradial force is nearly zero when ce

2 / pe2 b

2 =1.5 for the main

part of the beam pulse. This value can be optimal for beamtransport over long distances to avoid the pinching effect.Note that the radial force is focusing at larger radius, whichcan help to minimize halo formation and produce a tighterbeam.

Figure 10 shows the optimum value of the parameter ce2 / pe

2 b2, ce2 / pe2 b2 op , plotted as a function of r b / p

corresponding to the minimum radial force for effectivebeam transport over long distances. Note that for smallr b / p, ce2 / pe2 b2 op is approximately equal to unity, and in-creases with r b / p to the limiting value 4; this value corre-sponds to the onset of excitation of whistler and lower-hybrid-like waves. For ce

2 / pe2 b

2 4 the structure of the

self-electromagnetic eld becomes rather complicated, 37 andthe transport of very intense beam pulses with r b / p 6 inthe presence of a solenoidal magnetic eld can be stronglyaffected by collective wave generation, as discussed in thenext section.

V. BEAM EXCITATION OF THE WHISTLERAND HELICON WAVES

In this section, we explicitly take into account that thebeam can be relativistic. As shown below, excitation of thewaves disappears in the limit of a relativistic beam with b

1. In the case of a dense background plasma, n p nb, theelectron velocity is much smaller than the speed of light; and

δ

FIG. 9. Color online The normalized radial force F r / Z b2nb0mV bz2 / n p pacting on the beam particles for different values of the parameter ce

2 / pe2 b

2.The gray line green online shows the Gaussian density prole multipliedby 0.2 in order to t the prole into the plot. The beam radius is equal to theskin depth, r b = p.

ω

ω

β

δ

FIG. 10. The parameter ce2 / pe2 b2 op plotted as a function of r b / p corre-sponding to the minimum radial force for effective beam transport over longdistances. The beams have Gaussian density proles with different values of r b / p.




13/16

relativistic correcti ons to the electron motion need not betaken into account. 20 Equations 26 and 46 support waveexcitations when

ce

pe2 b b

2 . 56

Indeed, looking for solutions of Eqs. 26 and 46 propor-tional to exp ikx for a uniform plasma in the absence of abeam pulse, some straightforward algebra gives

b2 1 +

12 k

4 p4 + b

2 1 +1

2 + b2

2 − 1 k 2

p2 +

b2

2

= 0 , 57

where = ce / pe . Equation 57 can be also derived fromthe general dispersion relation for electromagnetic wavessee for example, Refs. 38 and 39 ,

Akc

4

+ Bkc

2

+ C = 0 , 58

where A= sin 2 + cos 2 , B=− 1+cos 2 − 2

− g2 sin 2 , C = 2 − g2 . In the dispersion relation 58 ,, , g are components of the plasma dielectric tensor,

cos = k / k is the angle of wave propagation relative to themagnetic eld, k is the k -vector along the direction of thesolenoidal magnetic eld, and k = k . Here, we account forthe fact that for long beam pulses, only waves with k -vectorsnearly perpendicular to the beam velocity are excited, k

k k . The wave phase-velocity should coincide with thebeam velocity for a steady-state wave pattern in the beamframe, i.e.,

= V bk . 59

When small terms of order k 2 p2 and k 2 p

2 / 2 are neglectedin the general dispersion relation, Eq. 58 , the resultingequation becomes Eq. 57 . The solution to Eq. 57 is

k 2 p2 =

2 − 2 b2 b

2 2 2 − 4 b2 b42 b

2 b2 1 + 2

. 60

Therefore, when the condition in Eq. 56 is satised, wavesare excited. Note that the solutions to the approximate sys-tem, Eqs. 26 and 27 , without taking into account the termcorresponding to quasineutrality breaking down the termproportional to ce2 / pe2 on the left-hand side of the equationfor A , show the excitation of waves when ce / pe 2 b.The difference between this approximate condition and theexact condition given by Eq. 56 is sizable when b → 1. Forexample, for the conditions in Fig. 7, b =0.808 and for theconditions in Fig. 7 d , ce / pe =1.621 2 b =1.617, andwaves are not excited, whereas the approximate criterionpredicts excitation of waves. Particle-in-cell simulation re-sults show that waves are not excited even for twice largervalues of the magnetic eld because the critical value of

ce / pe is equal to 2 b b2 =4.7, which justies the criterion

given in Eq. 56 .

A. Excitation of helicon „lower-hybrid-like … waves

In the limit 2 b b2, the upper-root solution in Eq.

60 tends to k p = / b b 1+ 2 1/ 2, and substituting the

denition of gives

k → k + = k lh = ce pe

c b b ce2 + pe

2 1 / 2 . 61

This mode corresponds to the excitation of helicon lower-hybrid-like waves. Consider nonrelativistic beam pulseswith b 1, then the lower-hybrid frequency is

38,39

= ce pe

ce2 + pe

2 1 / 2 cos . 62

Figure 11 shows the excitation of helicon lower-hybrid-likewaves observed in simulations using the LSP particle-in-cellcode.

Substituting Eq. 59 into Eq. 62 and using cos = k / k , yields the limiting value k → k + for lower-hybridwaves given by Eq. 61 . As evident from Eq. 60 , for

2 b, k lh p 1 and the lower-hybrid waves have shortwavelengths, of order or smaller tha n the skin depth in agree-ment with PIC simulation result s.29 Lower-hybrid waveswere observed in PIC simulations. 27,29 Note that for relativ-istic beams there is an extra factor 1 / b in Eq. 61 comparedwith the derivation based on the lower hybrid frequency, Eq.62 . This is because the traditional analysis for the plasma

resonances including the lower hybrid frequency assumes A=0, whereas a more rigorous calculation shows that in thelimit cos → 0 the second term with the B factor has also tobe taken into account when solving Eq. 58 . Due to thissubtle difference we call these waves “lower-hybrid-like”waves not simply lower-hybrid waves. Similar to the low-hybrid waves if cos = k / k r b / lb m / M 1

/ 2 the ion dynam-ics has to be accounted for. 39 Therefore, this theory is validfor not very long beam pulses lb r b M / m 1

/ 2.In addition to a steady-state pattern of waves in the beam

frame, 27 nonstationary lower-hybrid waves were observedpropagating perpendicular to a strong solenoidal magneticeld when the beam parameters changes rapidly near thefocal plane. 29 A similar excitation of helicon waves duringfast penetration of the magnetic eld due to the Hall effect inhigh energy plasma devices, such as plasma openingswitches and Z pinches, was observed in Refs. 30 . Couplingof the helicon waves to the plasma or the beam ions can leadto develo pm ent of the electrostatic modied two-stream

instability.44

B. Excitation of whistler waves

The lower-root solution in Eq. 60 in the limit 2 b b

2 tends to k p = b b / and describes long wave-length perturbations. Substituting the denition of gives

k → k − = k wh = pe2 b bc ce

, 63

corresponding to whistler-wave excitation. Excitation of whistler waves in cylindrical geometry can be derived fromEq. 34 directly by assuming that the wavelength is large




14/16

compared with the skin depth k ws p 1. Then the terms onthe left-hand side of Eqs. 26 and 27 can be neglected, andneglecting the small ion beam rotation gives

e

mcne A = −

B z4 mV

b

A z r

. 64

Substituting Eq. 64 into Eq. 26 yields

cB z2

4 2emn eV b2

1r

r r

A z r + emc ne A z = Z bnbV bz . 65

Equation 65 describes oscillations with wavelength

wh =cB z

2en pV b, 66

which correspond to whistler waves. 18 Indeed, the dispersionrelation for whistler waves is 39

2 =

ce2 c2

pe4 k 2 + pi2c2 k 2 ,

where pi is the ion plasma frequency and k is the wave-number along the magnetic eld. Assuming that the beampulse length is not very long, i.e., k 1 / lb pi / c, the whis-tler wave dispersion relation becomes

= ce c

pe2 k k . 67

Because the perturbations correspond to a steady-state wavepattern in the beam frame, = V bk in the laboratory frame.Substituting Eq. 59 into Eq. 67 shows that the whistlerwaves are excited w ith the same wavenumber perpendicularto the beam velocity 18

-10 -5 0 5 10x(cm)

z ( c m )

230

240

0

2 10 cm

1

11 -3(a) (b)

(c) (d)

-10 -5 0 5 10

x(cm)

z ( c m )

230

240

-10 -5 0 5 10x(cm)

-10 -5 0 5 10x(cm)

-24

24 kV/cm

0

-16

16G

0

FIG. 11. Color online LSP particle-in-cell simulation results for the perturbations in electron density, self-magnetic eld, and self-electric radial eld. Thebeam velocity is V b =0.2 c. The plasma and beam parameters are n p =10 11 cm −3 , and nb0 =0.5 1011 cm −3 . The beam density prole is Gaussian,nb0 exp − z2 / lb

2 − r 2 / r b2 , where r b =2.8 cm, and lb =5.7 cm. The value of the applied solenoidal magnetic eld is B z0 =2839 G. a shows the beam density; b

the electron density; c the self-magnetic eld B y; and d the self-electric eld E x .




15/16

k wh = pe2 V b cec

,

which is equivalent to Eq. 63 or Eq. 66 .Particle-in-cell simulations show that structure of the

self-electric and self-magnetic elds excited by the beam inthe presence o f whistler and lower-hybrid waves becomesrather complex, 27,29 and will be discussed in future publica-

tions. Coupling of helicon waves to the beam ion oscillationscan lead to the development of the modied two-streaminstability. 44

VI. CONCLUSIONS

Application of a solenoidal magnetic eld strongly af-fects the degree of current and charge neutralization when

ce

pe b b , 68

b =1 / 1− b2

or equivalently,

B 320 b b n p cm −31010 G. 69The threshold value of B given in Eq. 69 corresponds torelatively small values of the magnetic eld for nonrelativis-tic beams. When the criterion in Eq. 69 is satised, appli-cation of the solenoidal magnetic eld leads to three unex-pected effects:

1 The rst effect is the dynamo effect, in which the elec-tron rotation generates a self-magnetic eld that is muchlarger than in the limit with no applied magnetic eld.

2 The second effect is the generation of a large radial elec-tric eld. Because the v B z force should be balancedby a radial electric eld, the spinning results in a plasmapolarization and produces a much larger self-electriceld than in the limit with no applied eld.

3 The third unexpected effect is that the joint system con-sisting of the ion beam pulse and the background plasmaact as a paramagnetic medium, i.e., the solenoidal mag-netic eld is enhanced inside of the ion beam pulse.

Application of the solenoidal magnetic eld can be usedfor active control of beam transport through backgroundplasma. Without the applied solenoidal magnetic eld, the

radial force is always focusing, because the magnetic attrac-tion of parallel currents in the beam always dominates theradial electric eld, which is screened by the plasma betterthan the self-magnetic eld. However, when a solenoidalmagnetic eld is applied, the radial electric force can becomelarger than the magnetic force, resulting in beam defocusing.Figure 10 shows the optimum value of the parameter

ce2 / pe

2 b2

op plotted as a function of the ratio of the beamradius to the skin depth, r b / p, corresponding to the mini-mum radial force for effective beam transport over longdistances.

For larger values of the solenoidal magnetic eld, corre-sponding to

ce

pe2 b

2 b , 70

or equivalently,

B 640 b2 b n p cm −31010 G, 71

the beam generates whistler and lower-hybrid waves. Fornonrelativistic beams b 1, the whistler waves have longwavelength compared with the skin depth

wh =cB z

2en pV b, 72

whereas helicon lower-hybrid-like waves have short wave-length compared with the skin depth

w =2 V b ce

2 + pe2 1 / 2

ce pe. 73

When collective waves are excited, the particle-in-cell simu-lations show that the structure of the self-electromagnetic

eld becomes rather complex, and the transport of veryintense beam p ulses can be strongly affected by thewave generation, 27,29 which will be discussed in futurepublications.

Beam propagation in a plasma is considered to be aneffective way to compress intense beam pulses both long itu-dinally and transversely by applying a small velocity tilt. 8,17

A number of possible instabilities during propagation of beam pulses t hrou gh a background plasma in a solenoidalmagnetic eld 40,41 can be effectively mitigate d by a smallvelocity tilt and plasma density inhomogeneity. 42,43

In a follow-up pu blication the limit of strong magneticeld will be discussed. 33

ACKNOWLEDGMENTS

We thank Mikhail Dorf, Bryan Oliver, Dale Welch, Jean-Luc Vay, and Alex Friedman for fruitful discussions.

This research was supported by the U.S. Department of Energy Ofce of Fusion Energy Sciences and the Ofce of High Energy Physics.

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