Scattering in backward wave oscillators from turbulent electric fields

27 October 1997 PHYSICS LETTERS A

Physics Letters A 235 (1997) 159-163

ELSEVIER

Scattering in backward wave oscillators from turbulent electric fields Gregory Benford Department of Physics and Astronomy, University of California, Irvine 92697-4275, CA, USA

Received 13 August 1996; acceptedfor publication 23 July 1997 Communicatedby M. Porkolab

Abstract

Relativistic electrons resonating in microwave-emitting slow-wave structures can scatter from background turbulence. We calculate the expected scattering rate from strong fields characterized by a single mean correlation length. Comparison with recent data from a backward wave oscillator, using the measured magnitude of E-fields, shows agreement with the magnitude of outward scattering. The implied correlation lengths then explain why gas ionization in the same oscillator does not proceed exponentially, as earlier work showed was expected but not observed. Plasma electrons scattered over

the correlation length cannot gain the necessary energy to ionize many gas atoms. These results suggest that inadvertently produced plasmas may have a quick, decisive effect on beam geometry, ending microwave emission prematurely. @ 1997 Elsevier Science B.V.

1. Introduction

There appear to be at least four plausible explanations of pulse shortening in high power microwave devices [ l-71. Three involve production of plasma which destroys the effective corrugated geometry, either by ( 1) beam bombardment of walls, or (2) extraction of plasma by strong electric fields, or (3) in plasma-filled devices, further generation of plasma from ambient gas by strong oscillating Efields. A fourth possibility is that the strong microwave fields disrupt the beam particle orbits so that it cannot emit through the slow-wave coupling. This could arise from an instability, or simply by perturbing the beam electron orbits so much that they oscillate, reverse, etc. Beam geometry disruption is the most direct way to

cut off emission. Cross-B diffusion can do this from a variety of electric modes excited. Magnetic filamentation can destroy cylindrical symmetry, but requires a background plasma for instability. These seem the most probable instabilities which can cause emission cutoff. Zhai et al. [8] reported emerging wall plasma in BWOs, measured by spectral lines of H and C (earliest, probably from anode-cathode and wall) and Cu (later, from the wall). They used optical pipes and lensing to detect optical emission from the wide and narrow portions of the BWO structure, in the 8th and 9th ripple. All neutral atom radiation appeared about 100 ns after microwave cessation, with copper from the wall 200 ns afterward. Emission came first at the narrow portion. If copper then moved along the magnetic field to the wide portion, the implied temperature was several keV.

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G. Benford/Physics Letters A 235 (1997) 159-163

Later work on the same device yielded the first measurements of the oscillating electric fields during the beam pulse 191. They used Stark effect lines in a plasma-filled BWO. Strengths as high as 34 kV/cm accompanied 75 MW microwaves throughout the 60 ns. power pulse, with lower frequency fields of 10 kV/cm persisting until beam shutoff, presumably from Gould-Trivelpiece modes. Plainly many such effects are inter-related. This paper calculates the beam expansion which can arise from scattering of beam electrons off the strong microwave fields in an operating wave structure. Recent measurements of plasma density during operation of a relativistic backward wave oscillator (BWO) found the surprising result that ionization of the background gas did not increase exponentially with onset of strong emitted microwaves. Though ionization rates increased by factors of - 5, they remained linear in time. As well, measurements in the same BWO structure of beam expansion showed a profound broadening in the beam profile [ 8,101. Garate and Zhai measured the beam profile by placing a witness plate at the downstream end of the tube near the location of the rf input for the density measurement. A single shot sufficed to make a suitable damage pattern. Plasma was generated initially by background helium ionization from the electron beam. A Marx capacitor bank generated a 650 kV, 2 kA electron beam with 500 ns pulse duration. The beam was annular with 1.8 cm diameter and 2 mm thickness, injected into the BWO along a 15 kG guiding magnetic field. The BWO was a cylindrical waveguide with a periodically varying wall radius, R( z ) , sinusoidally rippled about the mean radius, Re, such that R( z > = Ro + h cos( koz >, ko = 27r/.r,0 where h = 0.45 cm is the ripple amplitude, ~0 = 1.67 cm the period and Ro = 1.45 cm, with two observation windows at the 9th ripple. Damage profiles in the vacuum BWO showed no broadening of the 2 mm annulus. Addition of 100 Torr helium produced plasma of density 1Or3crnw3 within N 200 ns. Damage profiles showed outward radial expansion by 2 mm and little inward motion (within the error of 0.3 mm). Note that 2 mm takes a beam electron beyond the innermost wall of theripple, suggesting scattering after the beam has left the BWO region.

These results leave us with three significant observed facts: (a) GHz electric fields permeate the BWO region of strength 535 kV/cm; (b) despite these, plasma ionization increases only linearly, not exponentially as suggested by conventional theory; (c) damage plates show outward beam expansion only when plasma is present. This paper aims to reconcile these observations. First we calculate the rate of beam expansion in high fields. 2. Quasilinear scattering We begin with the quasilinear expression for the perturbed beam distribution fi, due to scattering by random electric (SE) and magnetic (SB) fields, in terms of the equilibrium distribution fs ( X, p), where q is the particle charge, velocity is Y and momentum p. The beam moves along a guide field B, along the z-axis.

Jfo ,+u.vfo+~(uxBo).-$fo f

* $fodr’).

(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 to. Then t - to must be small compared to the time required to perturb the particles from their zero-orbit trajectories and, or .order to simplify Eq. ( 1) further, we must assume the field error fluctuations as seen by the particles last a short time compared to t - to. This means particles diffuse in a stochastic “bath” of SB and SE, passing by the field fluctuations quickly compared with a diffusive time scale. Obviously if the fields are systematic and lengthy, they will be highly correlated along the particle orbit and this assumption will fail. If t >> to we can set to to -4 -cc in the integral and recover the standard (relativistic)

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G. Benford/Physics L.etters A 235 (1997) 159-163

quasilinear expression. In general the fast electron gyromotion will contribute terms involving the gyroangle, and using cylindrical coordinates will yield a sum over Bessel functions. The complete form appears in Eq. ( 10). 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 particle dynamics. Thus we assume that the zero-order distribution fo 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. Hereafter we neglect SB, expecting that electrostatic turbulence dominates, since it grows much faster, and so we write the fluctuation power spectrum as a tensor C, (k) =

J

d3v (SEi (x) SEj (X + q)) e”‘”

(2)

and ignore SB for the moment. Here ( 1, 2, 3) corresponds to (I, 8, r) Then Eq. ( 1) becomes

If fo(x,p) is independent of x inside the beam to a good approximation we can neglect the spatial gradients in Eq. (3). Writing p = cos Y, and neglecting any beam density gradients along z, Eq. (3) becomes

00

X J

($

x &)

JmdkCij(k)exp(ik*r)

X ] (VX $.fo)jexp(-ik.r) -CO

dt’,

(3)

where L?= eB/mc y. We assume that the power spectrum Cij can be adequately represented by the spectrum in k, . This means the fields El perpendicular to B are distributed in the same manner as those along B. 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, = L?/uZ. For pitch angle Y this means k, r~ = tan P, with rL the Larmor radius, u~/0. Diffusion is usually most important for klrL N 1. Thus if tan N 1 for the bulk of the distribution, the power spectrum in k, can represent that of kl reasonably well, since we expect k, N kl in a BWO. Thus we take Cij (k) =Cll(k,)SlisjlS(k2)S(k3).

(4)

- fi/w)wtl

k, - fl/pv

afo ap

>.

-02

(5) The sine term in Eq. (6) will force the k, integration to zero if the coherence length I* of Cl1is comparable to the distance a particle travels, v,uut.We expect fields which are not correlated over distances exceeding a few q_, so v+ > e*. Then the sine function becomes a delta-function and we find, after integrating over the beam cross section,

- $Dp( =

vu* P2

sin[(k,

dkzCl\(kz1

1 - r2)z.

This is a diffusion equation for the density which must be integrated over p and u for the beam; all particles presumably begin at the same axial position. An average scattering time r obtained from Eq. (5) describes a diffusion in pitch angle of AP N 1, wherein particles steadily rearrange themselves with respect to B. They step sideways a distance N rL whenever the diffusive scattering changes p appreciably, so that the transverse spatial diffusion coefficient DI obeys DI M ;.r2

(7)

Traversing a distance L along z in a time L/u,,, a beam particle diffuses. Averaging over the beam velocities and pitch angles, the increase in beam radius is Ax, where ((Ax)~> 2 (tan2*)(Cll(kz

= fi/~u))-&.

(8)

We can visualize (Cl1(k, = f2/pu)) as the turbulent fluctuation strength (( SE)2) times a characteristic dimension of the fields, e,. Then denoting annular heam thickness by a,

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G. Benford/Physics Letters A 235 (1997) 159-163

hW2)

-

a2

11

(tan%)( ($q2)$

(9)

and .!JCrepresents an average over the field spectrum. A more general treatment gives

structure implies that beam-wall collisions are a major source of added plasma after scattering begins. Eq. (9) scaled to the relativistic backward wave oscillator is (E/35 kV/cm) 2 (B/15 kG)2

x [tm2PJ;+l

+5,

x (&)

(%) (c,,(kz=g>

(+) (c,,(kz=?))I*

(10)

If the spectrum is well known in kl and k,, one can assign average correlation lengths for each direction and carry out the sum. Note that Eq. ( 11) reduces to Eq. (10) for&=Oandn=-l,ifJi(klrL) M 1. Eq. (3) is a quite accurate representation of the essential physics. The approximations involved in the simpler form, Eq. (9>, probably make it accurate to within a factor of two. We have treated resonant diffusion because, in the context of quasilinear theory, non-resonant diffusion is “fake” diffusion, i.e. memory of initial orbits is not lost. Often it is possible to generalize quasilinear theory by including resonance broadening. However, this demands knowledge of the statistical properties of the fluctuations, which in general we do not have. Also, non-resonant contributions are largest for very short fluctuations (k > Lt/puo). Because short fluctuations are sensed as quick “collisions”, whereas long fluctuations are adiabatic in the particle frame, and thus yield no diffusion. 3. Comparison

with experiment

Measuring radial excursion of relativistic beams is notoriously difficult because a witness plate adds damage of all the beam pulse, whereas we seek knowledge of the microwave-emitting pulse time. However, while microwaves cease after tp, Langmuir turbulence persists for N ,US,so beam scattering over this time will dominate damage patterns. Assuming this, we now compare with observed beam broadening. Garate and Zhai [lo] found outward scattering with little inward beam broadening, arising perhaps from an unknown gradient. Observed broadening out to the inner BWO

(A)

bw

a-2. (11)

Here the wave-electron mean correlation length C is multiplied by the volume fraction of the turbulent fields, f. For an annulus the characteristic length a is the annular width. Garate and Zhai [ IO] measured beam expansion in an operating BWO, finding that an 18 mm diameter annulus of 2 mm width expanded mostly outward by 2 mm when 100 mT of helium was added to introduce a plasma. The measured plasma density was N 5 x lOI cmm3 at microwave emission onset and rose at An/At N 5 x 10” (cm3 ns)-’ thereafter. The observed expansion Ax/a x 1 requires, for our typical scaling values of E, B and L,

.fe=E.

(12)

Allowing for the simultaneous presence of both Trivelpiece-Gould (TG) modes and Langmuir (L) wave turbulence, we can write, with appropriate correlation lengths,

(fToeTo (E&) + fLeL

(E;))

= f.q, tan

(13)

where the field energy densities are normalized to (35 kV/cm)2. Previous measurements on this same BWO [9,11-131 imply (E) M 100 kV/cm at maximum emitted power. Earlier observations [ 141 found fL M 0.1, though in a helium plasma without the BWO oscillator. These results roughly imply that (fE2) N 1, so werequire (a) x 0.4cm/(tan2$), and cathode measurements show that (tan2 I,!?)- 1. For comparison, the beam electron gyration radius is re = 0.2 cm (B/15 kG)-’

(sin+)

(14)

and the typical scale of strong Langmuir turbulence [I419

G. Benford/Physics

D G 100A~

=7.4x IO-‘cm

Letters A 23.5 (1997)

(15)

where Tevis in electron volts and nr2 plasma density in units of lo’* cmm3. Thus to explain the observed scattering we need wave-electron correlations on the scale of r,, though large Langmuir structures N lo3 AD could suffice as well. The implied scattering time t, N &/c N 7 X lo-l3 s has further implications for plasma production by the E-fields, as observed by Zhai et al. [ 151. Straightforward ionization calculations imply rapid exponential growth, but only moderate linear growth appeared, though at a rate 5 times the direct beam rate. E-field turbulence implied by beam scattering explains this. Oscillating fields can drive plasma electrons to very high velocities in the N 35 kV/cm fields, but they are scattered by turbulence in a time t* N &,lu,, with vt, the electron thermal speed. The customary ionization rate for helium, v0 x ( v/GHz) x lo9 s-r must be modified by a factor << ( Y$*)-’ N 0.2, so the true ionization rate in the presence of turbulence is less by N 0.2. This explains why Zhai et al. found no exponential behavior; plasma electrons scattered by turbulent fields cannot gain the energy necessary to ionize many atoms. This leads to an ionization time comparable to that observed. A more detailed calculation and experiment is needed here.

4. Condusions Our principal results, Bqs. ( 10) and ( 11) , are compatible with the recent results communicated by Garate and Zhai [lo]. Langmuir turbulence measured extensively in Refs. [ 8,9,14] verify that strong scattering from intense E-field regions can dominate the turbulent characteristics of strongly driven stream instabilities. This links previous measurements of E x 35 kV/cm with the recently reported ionization in an operating BWO [ 151. Strong fields operating over reasonable correlation lengths L N 5 mm, can account both for scattering of relativistic electrons and for the interruption of the usual ionization by such strong fields in the background helium plasma and gas. Such a unified view then has implications for all strong microwave generators. These studies [ 12,151

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deliberately produced a controlled ambient plasma, but many generators probably suffer from plasma generation inadvertently. Beam electrons may strike walls and strong E-fields can draw out from walls dense, rapidly moving plasmas. These in turn can develop turbulence in fractions of a nanosecond, since streaming instabilities are very rapid. Scattering from nuggets of strong E-fields can then dominate gross beam dynamics - particularly crossB migration, which can disrupt the entire microwave emitting geometry. Further work with this in mind may clarify the ubiquitous puzzle of pulse shortening in microsecond microwave devices. The author thanks Eusevio Garate and X. Zhai for providing data, and James Benford for fruitful discussions. References [ I] GA. Mesyats,The Problem of Pulse Shorting in Relativistic Microwave Generators, in: ISPP-10 Piero Caldirola, High Power Microwave Generation and Applications, eds. E. Sindoni, C. Wharton, Sl’F, Bologna, 1992, p. 345. [ 21 J.M. Butler, Ph.D., Dissertation, Cornell University, 199 1, p. 61. 13 I Y. Cannel, J. lvers, R.E. Kriel, J. Nation, Phys. Rev. Lea. 33 (1974) 1278. 141 M. Friedman, Appl. Phys. L&t. 26 (1975) 376. 151 V.I. Belousov, V.V. Bunkin, A.V. Gaponov-Grekhov et al., Sov. Tech. Phys. Lett. 4 ( 1978) 584. 161 VS. Ivanov, N.F. Kovalev, S.I. Krementsov, M.D. Raizer, Sov. Tech. Phys. Lett. 4 (1978) 329. 171 Yu.V. Tkach, Ya.B. Fainberg. N.P. Gadetskii et al., Ukr. Fiz. Zh. 23 (1978) 1902. [ 81 X. Zhai,E. Garate, R. Prohaska, G. Benford, Appl. Phys. l&t. 60 (1992) 19. 191 X. Zhai, E. Garate, R. Prohaska, A. Fisher, G. Benford, Phys. lxtt. A 186 (1994) 330. [ 101 E. Garate, X. Zhai. private communication. [ 111 Y. Cannel, K. Minami, R.A. Kehs, W.W. Destler. V.L. Granatstein, D. Abe, W.L. Lou, Phys. Rev. L&t. 62 (1989) 2389. [I21 X. Zhai, E. Garate, R. Prohaska, G. Benford, Phys. Rev. A. 45 (1992) 12. 1131W.R. Lou, Y. Cannel, W.W. Destler, V.L. Granatstein, Phys. Rev. Lett. 67 (1991) 2481. [I41 D. Levron, G. Benford, A.B. Baranga, J. Means, Phys. Fluids 31 (1988) 2026. 1151 X. Zhai. E. Garate, R. Prohaska, G. Benford, submitted to Phys. Len (1996). 1161 S. Brown, Introduction to Electric Breakdown in Gases (Wiley, New York, 1966) p. 50.