Scattering of relativistic electron beams by magnetic field errors and beam‐induced waves

Relativistic electron beams propagating in long plasma columns must be well focused to cause efficient plasma heating. Expansion of the beam area due to scattering lowers efficiency. We calculate the beam spreading expected from errors in the ambient magnetic field. We then include scattering from both electrostatic and· magnetic waves generated by the beam itself. All these effects can be important in contemplated experimental regimes. However, it may prove possible to "tune" beam-plasma heating processes to avoid significant beam spreading.


I. INTRODUCTION
Recently, attention has focused on using intense r elativistic electron beams to heat long columns of plasma for fusion applications. 1 The beam and plasma would be confined by a solenoidal magnetic field and the beam energy deposited through a variety of beam-plasma instabilities.
The efficiency of such a device depends on focusing the beam to relatively small diameters (-1 cm). A natural question is whether the electron beam launched at one end of a long column (-100 m) will spread across the magnetic field lines, due to scattering by electric and magnetic fields in the plasma. If beam focusing erodes significantly as it propagates down the column, expensive countermeasures, such as increasing the ambient magnetic field, may be necessary. This paper calculates the beam scattering from (1) errors in the ambient magnetic field,' due to faulty positioning of magnets or fringing fields, (2) electrostatic beam-driven streaming instabilities, and (3) magnetic modes, such as the Alfven waves. Our scattering formalism is simpler than the early work of Hall and Sturrock, 2 from which a number of astrophysical applications have been derived . s-s However, s ince we deal'with spatial diffusion transverse to B, rather than multiple reflections of particles along B (a process leading to a density gradient), the extensively developed astrophysical results do not carry over di rectly. For the most part, we have derived approximate forms which are useful in light of the fact that we never. know in great detail the spectrum of either field errors or ·beam-excited waves.

II. SCATTERING FROM GUIDE-FI ELD ERRORS
We begin with the quasilinear expression for the perturbed beam distribution /u due to scattering by random elec~ric OE and magnetic OB fields, in terms of the equilibrium distribution f 0 (x, p), where q is the. particle charge, velocity is v, and momentum is p. The beam moves along a guide field B 0 , along the z axis.
=ct((oE+~xoB)· 0~J :/oE +~x 0B) · 0~/ 0 dt'). (1) Here the integration over t' follows the zero-order helical orbits of the beam particles. In Eq. (1), we have discarded the "ballistic" propagation forward of initial perturbations. Particles begin their orbits at time t 0 • Then, t -t 0 must be small compared to the time required to perturb the particles from their zero-orbit trajectories, and, in order to simplify Eq. (1) further, we must assume the field-error fluctuations, as seen by the particles, last a short time compared to tt 0 • This means particles diffuse in a stochastic "bath" of OB, passing by the field fluctuations quickly compared with a diffusive time scale. Obviously, if the field errors are systematic and lengthy, they will be highly correl ated along the particle orbit and this assumption will fail . If t » t 0 , we can set t 0 --oo in the integral and recover the standard (relativistic) quasilinear expression. In general, the fast electron gyromotion will contribute terms involving the gyroangle and, using cyl indrical coordinates, will yield a sum over Bessel functions . The complete form appear s in Eq. (12). However, to clarify matters, for the moment we anticipate that in practice we shall not know the fluctuation spectrum in enough detail to justify retaining such detail in the particl e dynamics. Thus, we assume that the zeroorder distribution / 0 is a slowly varying function of the guiding center orbit, Then we shall estimate how the change in particle pitch angles influences cross-field diffusion. We write the fluctuation power spectrum as a tensor (2) and take OE =0. Here (1, 2, 3) corresponds to (z, 6, r). Then, Eq. (1) becomes where 0 = eB 0 /mey.
spectrum S 0 can be adequately represented by the spectrum in k,.. This means the field errors perpendicular to B 0 are distributed in the same manner as those along B 0 • This assumption vastly simplifies the analysis. To see why it may be valid, consider that the most strongly affected particles are resonant with some portion of the power spectrum, k,.=n/vr. For pitch angle ip, this means k 11 r L =tanip, with r L the Larmor radius, vJ./n.
Diffusion is usually most important for kJ.rL -1. Thus, if tanl/J-1 for the bulk of the distribution, the power spectrum in k 11 can represent that in kJ. reasonably well. Thus, we take (5) If / 0 (x, p) is independent of x inside the beam to a good approximation, we can neglect the spatial gradients in Eq. (3). Writing µ = COSl/J and neglecting any beam density gradients along z, Eq. (3) becomes (6) The sine term in Eq. (6) will force the k,. integration to zero if the coherence length l* of S 11 is comparable to the distance a particle travels, vµt. We are studying expected field errors which are not correlated over distances exceeding a few rL, so vµt » l*. Then, the sine function becomes a o function and we find, after integrating over the beam cross section, (7) This is a diffusion equation for the density which must be integrated over µ and v for the beam; all particles presumably begin at the same axial position. An average scattering time T obtained from Eq. (7) describes a diffusion in pitch angle of tl.1/J -1, wherein particles steadily rearrange themselves with respect to B 0 • They step sidewise a distance rL whenever the diffusive scattering changes µ appreciably, so that the transverse spatial diffusion coefficient DJ. obeys Traversing a distance L along z in a time L/vµ, a beam particle diffuses. Averaging over the beam velocities and pitch angles, the increase in beam area is (9) We can visualize (S 11 (k 11 ==0./µv)) as the fluctuation strength ((oB) 2 ) times a characteristic dimension of the field error, le. Then denoting beam radius by a, and le represents an average over the field-error "spectrum".
For a 1-cm-radius beam traversing a 100-m system, with ( tan 2 ip) = 1, the beam area doubles when ((~ ))1:-4x 10-4 Designs for fusion systems use B 0 = 50 kG, so errors of the order of 1 kG can be tolerated if the average le is small, as seems probable.
A more general treatment, modifying the work of Hall and Sturrock, 2 gives If the field.error spectrum is well known in kJ. and k., one can assign average correlation lengths for each direction and carry out the sum. Note that Eq. (12) reduces to Eq. (9) for S 33 = 0 and n=-1, if ~(kJ.rL)="1 .
Equation (12) is a quite accurate representation of the essential physics. The approximations involved in the simpler form, Eq. (11), probably make it accurate to within a factor of 2.
We have treated resonant diffusion because, in the. context of quasilinear theory, nonresonant diffusion is "fake" diffusion, i.e. , memory of initial orbits is not lost (Ref. 7). It is possible to generalize quasilinear theory by including resonance broadening. 8 However, this demands knowledge of the statistical properties of the fluctuations, , which, in general, we do not have.
Also, nonresonant contributions are largest for very short fluctuations (k » n/ µv) (because short fluctuations are sensed as quick "collisions", whereas long fluctuations are adiabatic in the particle frame, and thus yield no diffusion. ) For systems with magnets spaced at intervals exceeding 10 cm, 9 the errors will probably be 10 cm or longer, whereas the resonant >...=211µv/U -1 cm for y =10, B == 50 kG, µ-0.5, and V"'C· Thus, there should be very little S,,(k»U/µv), and nonresonant scattering will be unimportant for this application.

A. Electrostatic modes
Resonant scattering of a beam occurs most easily when the beam itself produces waves in the background plasma which are very nearly resonant with the beam velocity; i.e. , those which corresi)ond to space-charge oscillations on the beam. An obvious candidate for such a wave is the familiar streaming instability. The electrostatic instability is usually dominant over the electromagnetic form, 10 and Eq. (12) can be modified easily to include electrostatic wave scattering by adding to S 11 a term (c/ µv) 2 Sf» where sf1 (k) = J dry (oE.(x)~E.(x +77)) exp(ik• 77).
and a similar form for S 33 • Then a nonlinear theory (for example, Ref. 10) for the saturated value of the electric fields and their spectral range can be µsed to calculate the diffusion, as in the previous discussion.

B. Magnetic modes
Since the beam-heated plasma is "high beta" in the sense of having plasma pressure comparable to magnetic field pressure, there ma.y be significant magnetic waves present to scatter the beam, In particular, waves transverse to B, are most effective because they exert a steady decelerating force in the particle rest frame. If the transverse wave magnetic field OB exerts a constant force parallel to B 0 through the Lorentz force v J. x OB, the pitch angle of the resonant electrons scatters during the correlation time 1 during which the field and particl e are in resonance.
Low-frequency modes transverse to B 0 are Alfven type when 11 where w,, is the ion plasma frequency. They are helicon type when 2w,, 1 « k 11 «y!i.
To excite Alfven waves requires 12 where vA is the Alfven velocity and Mis the ion mass. For gross confinement, the beam must be stable against filamentation, 13 which requires (Jy}112<10-sn, where J is current density in Al cm 2 and B is the field in kG. The 50-kG field satisfies this condition. Equation (16) may be written, taking µ ==0. 5 and the average atomic number of ions as 4, 5 n, y 10 3 • 3 ' 3 < 1()1 6 cm· 3 10 B ' so for our contemplated system of interest, this condition is · satisfied by a factor of 6. These waves can be excited either by the beam current itself or by background plasma electrons which are counterdrifting to carry a return current. However, the plasma electrons will generally not drift faster than v "' and we neglect instability due to them. Thus, we turn to solely beamgenerated turbulence.

C. Alfven-wave growth rate
We consider a relativistic electron beam with beam frequency and distribution function / 0 (p, µ}. A general form for the growth rate has been given by Lerche 5 for applications to cosmic rays; we adapt to our case and find the growth rate y-::: w~ere µ = p.f p. We expect the beam distribution / 0 will have a wide distribution in µ and will be peaked in momentum Pat P 0 =myvb. The integral in Y1r will then give a fa.ctor no smaller than p/ Ap, where Ap is a width of the distribution in momentum space. We can then estimate This is independent of nb, and y and yields for our parameters After -30 nsec, they should reach large amplitude and begin to resonantly scatter the beam. However, to make a useful calculation, we must estimate the saturated magnetic fields of the waves, for use in Eq. (12).

D. Nonlinear saturation of Alfven waves
Alfven waves will grow to an amplitude which is limited by their coupling to other lower-frequency modes such as the ion acoustic. The ion-acoustic spectrum may also be excited for a time by return-current instabilities. However, electron heating and nonlinear effects may stabilize the ion sound waves by the time the slower-growing Alfven modes rise to large amplitudes. We shall assume this in order to simplify the calculation. 4 Energy balance is expressed by Sagdeev and Gal eev as 7 where the initial Alfven wave k decays into another Alfven mode k' and an ion sound wave q is given by w = qc,, with c, the sound speed. w"N" is the energy density in the kth mode, both mechanical (kinetic) and electromagnetic, The transition probabilities v,,, 11 This implies that very monoenergetic beams (Diop« p 0 ) are more effectively scattered by their self-generated waves than "hot" beams, since they produce a stronger spectrum. This will be so as long as the growth rate, Eq. (19), is not so small that the Alfven waves never reach saturation in a beam pulse time. Equation (31) can be used in the general formalism of Eq. where l~ is the average length over which beam particles and the beam-generated fields are correlated. A crude estimate of this is (33) since the wave-particle phase relation depends on the "sharpness" of wk.v.-nn, and w «k.v •. Assuming Di-k, = 0, i.e., a single standing wave in the plasma column, From Eq. (32), estimating Di-p=p 0 Di-µ where aµ is the spread in µ , and using B 0 = 1 O' G again, so only if aµ< 0. 1µ is the wave scattering significant. However, only more detailed study of the nonlinear estimates made here will provide an estimate reliable to better than an order of magnitude. For example, if the ion-acoustic sp. ectrum is active because of return-current instabilities, ((oB.-I B 0 ) 2 ) may be larger. In this case, estimates of saturated field amplitudes have been given 12 and can be substituted in our formalism, Eq. (10) or (12), directly.

IV. CONCLUSIONS
We have found that expansion of a relativistic electron beam may come from several sources of field fluctuations. The errors in the ambient field B 0 may scatter quite effectively; Eq. (10) gives an approximate form, and Eq. (12) gives a more general result.
Scattering by beam-induced instabilities poses a larger number of unknowns, since we must first know the nonlinear saturated spectrum of waves. Modification of Eq. (12) for electrostatic modes is simple [see Eq. (13)], and some existing electrostatic wave theories 10 may give reliable results for the fields .
Magnetic modes, however, can be as effective as the electrostatic waves, particularly in the cases where plasma pressure is comparable to ambient magnetic field pressure . We have studied the scattering from beam-driven Alfven modes, attempting an approximate treatment of the nonlinear saturation of these waves by mode coupling to ion-acoustic waves. Our crude estimates, summarized in Eq. (32), suggest that scattering from beam-induced modes may not be as significant as scattering from errors in the ambient field. Calculations for particular regimes of int~rest may yield somewhat different estimates, however.
Fusion systems using intense electron beams cannot tolerate very much spreading of the beam, since thermonuclear ignition is the paramount problem. Thus, spreading of the heated cylinder reduces effectiveness so much tha~ a doubling of the beam radius (i.e., decrease of maximum attainable temperature by a factor of 4) is probably the tolerable upper limit on beam diffusion. (This is deduced from Refs. 1 and 9. ) Equation (35) is cause for optimism. It shows that a reasonably "hot" beam, with significant spread in pitch angles µ., does not produce a magnetic field spectrum large enough to yield significant beam diffusion.
Equations (11) and (12) suggest that, if field errors on a scale less than 1 cm can be avoided, no large diffusion occurs.
However, there remain two sources of fluctuations which depend on precisely how beam-plasma heating proceeds: (i) scattering by electrostatic waves [Eq. (13)] and (ii) scattering by ion waves caused by return current flow, as remarked below Eq. (35). Given a particular heating scheme, these sources of fluctuations can be tailored for maximum efficiency, including beam diffusion. Thus, the outlook for rapid heating by electron beams is rather favorable.