IEEE Transactions on Nuclear Science, Vol. NS-28, No. 6, December 1981
SOURCE-REGION ELECTROMAGNETIC PULSE ASSOCIATED WITH HIGH-ENERGY ELECTRON BEAMS*
W. E. Hobbs and K. S. SmithJ AYCOR
Santa Barbara, California
Abstract
Estimates are made of the electromagnetic (EM) fieldsgenerated by a highly relativistic beam in two configurations.In the first configuration, the beam impinges on a uraniumconverter and the subsequent bremsstrahlung radiationproduces electric currents in air. The absorbed radiation isderived from previous transport calculations. For the secondconfiguration the beam is directly injected into the ambient airand allowed to propagate in a self-pinched mode. Bothcalculations assume a beam with an electron energy of400 MeV, a current of 250 kA, a pulse rise of 2 ns, and durationof 100 ns.
The equations governing EM fields, background ionization, andsecondary electron currents are solved using finitedifferencing. The current of the primary electron distributionis found using a prescribed source or macroparticles. For thebeam/converter system, fields of 104 V/m and 20 A/m areexpected within a 50 cone about the beam axis at ranges up toa few hundred meters. When the beam is directly injected, theprimary currents are stronger throughout the simulationvolume and conductivity is typically larger by two orders ofmagnitude. The magnetic field is also increased to 200-300 A/m and the peak electric field is increased by an order ofmagnitude. In the highly conducting air t 4e electric fielddecays to an amplitude of the order of 10 V/m while themagnetic field remains large.
Introduction
In the last few years there has been a rapid development ofpulse-power sources. They have application to particle beamaccelerators for both energy and defense purposes. Fordefense these machines are attractive as potential sources forsimulating radiation effects in considerably larger volumesthan had heretofore been possible. In this article we presentresults of our numerical calculations of EM fields associatedwith powerful relativistic beams. We consider the beam'ssource region by analyzing the volume close to the beam axiswhere there are primary currents and resulting iknization.This work has the same emphasis as that of Cabayan but wasperformed independently. It il an extension of work performedby Vittitoe and Halbleib who considered both thebremsstrahlung conversion of high-energy beams and theresulting EM fields.
The primary motivation for this analysis is the possibleapplication of powerful beams as a nuclear source-region EM P(SR EM P) simulator. Machines which will yield short pulseswith megajoules of energy are on the design horizon. Theyhave several characteristics which are desirable for the driverof an SREMP simulator. Three of the more prominent ofthese features are as follows:
1. At energies greater than 10 MeV, electrons lose energy bybremsstrahlung, and a beam onto a high-Z converter will yieldan immense radiation field. Alternately, if the beam hassufficient current, it can propagate in a self-pinched modeanalogous to a very long exploding wire. As our calculationalresults show, the fields associated with the beam current alone(ignoring its bremsstrahlung) are also immense.
2. The current rise time at the beam tip will be relativelyfast since the pulse shape on a magnetically insulatedtransmission line tends to steepen.3 The rise time is shorterfor higher currents. Also, the beam itself may be conditionedby an intermediate pressure gas before ejection from theaccelerator. The result is that very fast (subnanosecond) risetimes are believed to be possible.This work supported by the Air Force Weapons Laboratory
3. Finally, and most important, the beam and its attendantshower will effectively irradiate very large volumes. Both thehigh-energy photons and ultra-high energy electrons havemean-free paths of the order of 100 meters. This makes theSREMP hardness verification of various military systems apossible application for the beams.
In a previous effort, Tumolillo, et al.,4 calculated the fieldsassociated with a model beam using the coupled Maxwell andair-chemistry equations. The variation of the near fields withthe various beam parameters was presented. The capability tocalculate the distant radiated fields from the beam was alsodemonstrated. The beam itself was represented by prescribedcurrents and the calculations used the same computer codeJRBEAM (see appendix) which was used for this report. Here,we present estimates of the radiation and resulting EM fieldswhich would result from an ultra-relativistic beam impingingon a uranium converter or directly injected into the air. Weconsider state variables at the same point for both calculationsfor comparison. The be m converter calculations are likethose of Tumolillo, et al., in that a prescribed source currentis employed, but they are different in that the prescribedcurrent results from detailed Monte Carlo calculations. Thebeam-alone calculations are similar to the Tumolillocalculations in objective, but they are much improved in thatthey utilize a more realistic macroparticle model for theprimary beam currents.
In this report we assume that we are given a highly relativisticbeam with an electron energy of 400 MeV and a current of250 kA. The beam pulse has a rise time of 2 ns and a durationof 100 ns. The electron accelerator might yield several ofthese pulses, each with an energy of 10 MJ. The beam has aGaussian radial profile with a characteristic length of1 meter. Even though this beam has a large cross section andrelatively poor quality, it represents a futuristic beam whichwill still require considerable research and development. Workat the joint Sandia National Laboratory/Air Force WeaponsLaboratory program, as well as in the Soviet Union is inprogress whNch might lead to such electron beam acceleratorcapabilities.
In our analysis, we do not address propagation issues such ashead erosion or beam stability. In these preliminarycalculations we are interested in the magnitudes of the fieldswhich can be expected and the volume over which they extendin space and time. We therefore assume cylindrical symmetryand thus ignore ground effects, the earth's magnetic field,etc. In some cases the cylindrical symmetry may be justifiedphysically since the plasma density in the beam's corona maybe large enough to isolate the internal dynamics. Ourcalculational model (JRBEAM) is essentially organized as astandard electromagnetic pulse (EM P) computer code and isbriefly described in the appendix.
A particle beam differs from a nuclear event in that itsradiation primarily flows in a specific direction, else it wouldnot be a beam. As we will see, this is reflected in the beam'sEMP since this asymmetry will drive a strong magnetic field.First, we will discuss the driving currents and then we willshow the results of the field calculations.
Source Currents
The results presented in the next section are from twocalculations using different driving currents. In the firstcalculation, the beam immediately impinges on a uraniumconverter and much of its energy is transformed intobremsstrahlung radiation. This radiation interacts with the
ambient air producing a Compton current which drives the EMfields in the usual EM P fashion. For the second calculation,the electron beam is directly injected into the ambient air.The fields generated tend to hold the beam together formingan intense source. The Compton electrons from the converterradiation are assumed to deposit their energy locally, and thus,we use a prescribed current for them and do not follow theirself-consistent orbits. For the direct beam calculation, thesituation is quite different. The beam electrons travel longdistances and we follow them in a self-consistent manner asthey move through the fields. The background ionization isdistributed along the electron paths. (An electron with energygreater than 10 MeV produces approximately 100 electron-ionpairs per cm of path; we use this constant for our beamelectrons.) For both calculations the secondary electrons(background ionization) are treated self-consistently in thatseveral of the coefficients (e.g., avalanche, attachment, andmobility) are functions of the local electric field magnitude.
For the beam/converter system, we use the radiation fieldwhich was calculated by Vittitoe and Halbleib.2 The electronand photon transport through the converter and the subsequentenergy depositien in the ambient air were calculated using theACCEPT code. The electrons are first transmitted through auranium converter equal in thickness to 0.4 times thebremsstrahlung mean-free path for the 400 MeV initialelectron energy. This thickness is chosen to maximize theforward 2r photon energy flux. The ACCEPT code is a three-dimensional coupled electron (positron)/photon Monte Carlotransport code. It provides a detailed description of theproduction and transport of the extensive shower resultingfrom a beam that is normally incident on a converter. Theresulting radiation is then transported through the air and thedose deposited per beam electron accumulated. A portion ofthe spatial distribution of energy deposition over a forwardhalf-space of nitrogen is shown in Figure 1. Note that our
0 4,o 10
°° 5° 1 ° 145 20° 250 30°
POLAR ANGLE MEASURED FROM BEAM DIRECTION
RE-04165 Figure 1.
Gamma-ray deposition for a 0.4-GeV electron Incident uponthe uranium converter, for a series of distances from theconverter
pulse has 0.025 coulombs and that the 10 rad point extends outto approximately 1 km on the beam axis. The y data areobtained by assuming that the power deposition follows thetime dependence of the source current pulse. The dose rateinformation is then used to determine a source current usingthe formula
2 2 x 10 y(rad/s)J(A/m ) = < E(e)>(1)
where y is the energy absorption rate and < E > is the averagephoto9 energy. This is the conversion formula for the Comptonevent with the average gamma energy added as a correctionfactor. This correction accounts for the reduction in currentas the pair-production event be2comes more probable at higherenergies. Vittitoe and Halbleib have given data verifying thisformula for the energy spectrum to which it is applied. We use< E > = 10 MeV in our calculations.
The current of Equation (1) is the radial component in sphericalcoordinates with the uranium converter at the origin. It iseasily converted to cylindrical coordinates for our numericalmodel. We also define 'zero' time as when the beam pulsefirst hits the converter, and we calculate a delay time for allthe points of the system. (Note that there will be no'retarded' time delay along the z-axis. Retarded time isexplained in the appendix.)
Our alternate calculation is of the beam directly injected intothe air with no converter. For comparison purposes we use thesame simulation grid and the beam is launched with arelatively large initial radius of 1 meter. This large radiuswould be adequate for the converter, but a smaller one wouldbe better to quickly develop the pinch magnetic field neededfor propagation. Nevertheless, in view of the very large fieldscalculated by Tumolillo, et al.,4 there was considerableinterest in how these EM fields would compare. In theseearlier calculations of Tumolillo, the beam parameters(current, radius, etc.) were chosen arbitrarily and there wasquestion about the viability of the resulting beams. The use ofself-consistent macroparticles will answer these to someextent. The beam launch condition is presented in Figure 2.
CURRENT:
ENERGY:
Ib = 250 kA
y = 783 (0.4 GeV)
RADIUS: a = 1.0 mn [f(r) = exp(-r2/a2)/(na2)]
TRANSVERSETEMPERATURE:
STP DRY AIR
250
i:Z-
w
C1
TI = 3.75 MeV
60 CENTIMETERS 3000 T61 0 T
2NANOSECONDS
CURRENT PROFILE
100
RE-04166
Figure 2.
Model beam pulse for macroparticle calculations
The primary currents in cylindrical coordinates are obtainedeither from the ACCEPT data and Equation (1) or directlyfrom the macroparticle analysis. These are then used to drivethe coupled Maxwell and air-ion equations which are written inthe retarded time appropriate to the beam.
Results and Discussion
The simulation volume for the calculations extends out to aradius of 40 m, down the axis to 1.2 km and back through thebeam pulse to 130 ns. Since both the driving photons andelectrons are moving with their signal, the forward movingsignal grows as it moves down the axis. The sources dissipatehowever, and eventually the signals decay. In our simulationswe see our signals maximize at about 100 m, but substantialfields are seen throughout the simulation volume.
For the beam propagation calculation, the beam throws offcharge, cools, and eventually evolves into a quasi-equilibrium
4452
state with its magnetic field, albeit at a greatly reducedcurrent. Subsequently, it diverges slowly and the fields remainapproximately constant.
To illustrate the EM environment, the fields are shown at thepoint with the axial coordinate z = 50 m and the radialcoordinate r = 8 m. These are representative of the timeprofiles for a point near the axis for the first few hundredmeters. The axial electric fields are shown in Figure 3, theradial electric fields in Figure 4, the azimuthal magnetic fieldsin Figure 5, and the air conductivities in Figure 6.
CONVERTER
15
12
9 _E~
- 6
3
9060
3060 -
:: 30t
0
RE-04 167
15
10
E
S.
5
50
50
RE-04168
Radial
t (ns)
BEAM ALONE
t (ns)Figure 3.
Axial electric fields at z 50 and r =8 m
CONVERTER
t (ns)BEAM ALONE
0
-7
E
-14
-28
90
-902e
-180
RE-04169
CONVERTER
0 2 5 78 1 0130
0 26 52 78 104 1 30
t (ns)BEAM ALONE
t (ns)
Figure 5.Azimuthal magnetic field at z = 50 m and r = 8 m
CONVERTER
t (ns)BEAM ALONE
30
24
18
- 12Is
RE-04170 t (ns)
Figure 6.Air conductivity at z = 50 m and r = 8 m
The electron current from the beam alone is considerablygreater (order of magnitude) than that resulting from thebeam/converter system throughout the simulation volume. Thecurrent resulting from the radiation from the converter is not
I_ t E 1so sharply peaked on axis as that of the beam alone. In this itis more similar to the nuclear SREM P case. There will be avery complex shower from the propagating beam but for theset (ns) calculations we have assumed that the beam energy lost isFigure4. absorbed instantly and locally. Due to high-energy secondary
electric field at z=50 m and r =8m radiation, the beam would actually have a corona surroundingit which would reduce the strong axial bias. As mentioned
4453
Z-ES
_
E-
-
LD
earlier, however, the beam would probably be launched with amuch smaller cross section and thus be more peaked along theaxis.
Because of the relatively large beam radius used, the magneticfield lags a bit behind the beam current and a displacementcurrent in the axial direction is driven. The limit of a very fatbeam reduces to a one-dimensional surface emission problemwhere a strong space-charge field is generated. The growth ofthe axial field is accentuated by the development of ionizationin the air and the flow of conduction currents to neutralize theradial electric field. This further retards the growth of themagnetic field and the axial field continues to grow. For athin beam this neutralizing of the radial field and simultaneousretarding of the magnetic field growth is the dominantmechanism for the generation of the axial field. Since themagnetic field is not able to maintain its vacuum profile in theincreasing conductivity, the radial field in effecA rotates tobecome the so-called inductive longitudinal field. This fieldis important since it drives the return current and this allowsthe beam to continue propagating. The beam does notcompletely current neutralize since there is still a net currentreflected by the magnetic field which is frozen in theconductivity. The same processes are involved for thebeam/converter EM fields in a less dramatic fashion.
The conductivity for the beam/converter system is limited byattachment. For the beam alone the ionization is somewhatmore dense and recombination is also a factor. For thepropagating beam the conductivity has a sharp peak at the tailof the beam. This results because the electric field is greatlyreduced in magnitude and the secondary electrons have a muchhigher mobility in the small fields. The axial field changes signat the end of the beam so that now the secondary current flowsto maintain the current of the beam and the frozen magneticfield. The higher mobility of the secondary electrons makesthis 'inductive kick' easier. The air conductivity is not highenough for this phenomenon to occur at the depicted radius forthe beam/converter system, however it does occur near theaxis.
CONVERTER
10
8
6
s 4
2
0
10
8
4
z (m)
BEAM ALONE
0 100 200 300 400 500
z (m)RE-041 71
Figure 7.
Contours. plots of maximum electric field magnitudes in kV/mn
In Figure 7 we have contour plots of the peak electric fieldmagnitudes for our two cases. The much larger values for thebeam alone are somewhat misleading since, as we have seen,the large peak field is transitory. The large propagating beamcurrent leads to a significantly larger conductivity(approximately an order of magnitude) for the self-consistentbeam. This has different effects on the electric and magneticfields. The electric field for the beam alone increases veryrapidly near the beam tip to a magnitude which is considerablylarger (again by an order of magnitude) than the field from thebeam/converter system. But this field decays quickly due tothe larger air conductivity to a value which is actuallycomparable to that when the converter is utilized. The largeconductivity effectively inhibits change in the magnetic field.This field freezes at a value an order of magnitude greaterthan the field which results using the converter.
Concluding Remarks
For the beam/converter system, fields of 104V/m and"30 A/m are expected within a 50 cone about the beam axis at
ranges up to a few hundred meters. The primary currents are
stronger out to 16 meters when the beam is propagated alone,and the air conductivity is typically larger by two orders ofmagnitude. (The beam is launched with a 1 m radius and doesnot strongly pinch.) The magnetic field is increased to 200-
300 A/m, and the peak electric field is also increased by an
order of magnitude but decays rapidly due to the higherconductivity. The final electric field is again in the 10 V/mregime.The two types of sources for the calculations in this report are
certainly not mutually exclusive. On the one hand, there willbe a large primary electron beam emerging from a converter
that is only a fraction of the bremsstrahlung mean-free path inthickness. The converter will give the beam a large transverse
temperature and reduce the energy of the individual electrons,but self-consistency will be important. On the other hand, thebeam in air will interact with the air nuclei via bremsstrahlungproducing a corona about the beam. The air nuclei are not
high-Z so the interaction is not as rapid but nevertheless an
intense radiation shower accompanies a propagating beam.
Thus, the complete situation in either case would involveelements of both the simulations here reported.The use and thickness of a converter depends on the
application desired in addition to the details of the radiationfield. The most apparent subject is the testing of coupling to
long lines. Here the geometries of the line and the beammatch and it is appropriate to shoot the beam parallel to theline.
An intriguing aspect of this study is that we were able to start
with a large area beam and still get significant propagation(granted we did begin with an enormous current). This has
implications for existing machines which should be furtherinvestigated. The variation of EM field mafnitudes with beam
parameters was found by Tumolillo, et al. For a variety ofreasons it would be very useful to perform a similar survey
using the self-consistent macroparticle model for the beamelectrons. The capability we have demonstrated in this papermakes this easy to accomplish, thus the beam EM P
characteristics with regard to the needs of a nuclear SREM P
simulator can be further evaluated.Appendix
introduction to the JRBEAM Code. The code JRBEAM treats
the self-consistent coupling of transverse electron dynamics,background air conductivity, and electromagnetic (EM) fields.The code is similar to the code BEM P (i.e., buried EM P) whichwas developed to study EMP phenomena in tunnels. In BEMP,the weapon gamma rays interact with the air to produce a
beam of Compton electrons which move forward at velocitiesnear the speed of light. These Compton electrons themselvesinteract with the air to produce conduction electrons which are
modeled using the standard three-species air-ion equations.The Compton current along with the return current of theionized air constitute the source for the EM fields in Maxwell's
4454
iS-
2
equations. With a few modifications, this algorithm wasapplied to beam problems per se. Our code solves Maxwell'sequations in essentially the same way, but we haveconsiderably improved the treatment of the primary andsecondary electrons.
Our code is conveniently divided into three modules:(1) FIELDS, which solves the Maxwell curl equations;(2) PARTCL, which calculates the self-consistent radialbehavior of the beam electrons; and (3) PLASMA, whichevaluates the currents generated in the ionized air. The majorassumption of our analysis is that beam electrons moveforward at the speed of light. The application of this code is,therefore, valid only for very energetic beams, (>50 MeV)where this is a reasonable approximation. With thisassumption, it is calculationally convenient to use a retardedtime coordinate T (instead of time) as an independentvariable. It is defined as follows:
T = ct - z (2)
where X = 0 is defined as the tip of the beam and thus X issimply the distance behind the tip of the beam along the axis.
We conceptualize the beam as being divided into a stack ofpoker-chip slices of thickness AT. The evolution of the beamis solved by analyzing the behavior of each of the slicessequentially. A typical slice is depicted in Figure 8. The beamis constrained to be cylindrically symetric. As a beam slicemoves out in z, it evolves radially according to the ambientfields from the previous slices. At the same time, it furtherionizes the air through which it passes. Its currents contributeto the evolution of the fields. We are, in effect, integratingback into the body of the beam in a frame moving with thebeam.
BEAM SLICE
Ionized Air
z = O
E
L. E
ZMAX
JzI Jr
RE-041 72
Figure 8.
Schematic of beam numerical model
Table 1 summarizes the various modules of JRBEAM and wewill next discuss a few aspects of these.
Fl ELDS. With the assumed cylindrical symmetry, only thetransverse magnetic set (Ez,E r8¢ ) of the field variables arenon-zero. Maxwell's equations are written in terms of the fieldvariables F and G, which are defined as follows:
Er hBI(3)
G -BErB
where Er is the radial electric field and B is the azimuthalmagnetic field. These variables are the sate as those in thetheoretical analysis of Longmire. The equations areimplicitly differenced and the resulting set is unconditionally
Table 1. The JRBEAM code
COMPONENT MODEL EQUATIONS NOTES
EM Fields Maxwell's aET + 4moEz a .4mJz + 2r a-[r(F-G)] Implicit, 2nd order
E , Er, BEquations
aF aEz Unconditionallyz r -z 2wjF -Finite +r stableFinite ~~~~~~~~~~~~~Transmitting BC in rFG Er + B Differencing 3G I a(F+G) aEZ Conducting BC in z
2T T2FZ -arCnutnBCizG= E - B, = ct - z
d(yvRadial Beam Macroparticles dt m t = Vi Particle noiselDynamics 1 imits accuracy
May use d e z Finite thickness ofPrescribed dtp Mc r = Xx + 2 particlesaz Jr Source reduces noise
Pt= z/c 1 d a (vxtve)--*(Var Yv) _ _ldt
Background
Ionization
J = aE
Air-Ion ane S-- Source from beam- ~~~~~~~AaR-- Avalanche & RecomnbinationEquations at
stable in the high conductivity diffusion limit. Thedifferencing increments Az are constant, but Ar is increasedwith radius. The algorithm is second order in all threeincrements (Ar,Az,A-r). The beam current densities ) z and Jand conductivity a are needed as sources.
PARTCL. The energetic beam electrons have long mean-freepaths relative to the computational volume, and it is thereforeappropriate to treat them as an ensemble of macroparticles.Each of these is, in effect, a sample electron which representsa larger number ('10 ) of beam electrons. The particles arecharacterized by four variables: q, which is their charge (i.e.number of electrons); y which is their relativistic mass factor;vr, and v8 which are their radial and azimuthal velocities; andr, which is their radial position. These evolve self-consistentlyaccording to the standard equations of motion (of course, theircharge remains constant). The current densities J z and Jr areevaluted at the appropriate grid points as needed for Maxwell'sequations. The beam charge density p is proportional to the zcurrent density, since vz is assumed equal to c. p is needed toevaluate the amount of air ionized.
We also include the option of modeling the primary beamelectrons using a prescribed source. This assumes that theself-consistent transverse beam dynamics are somehow knownor are not critical. For example, it is applicable for the low-energy Compton electrons which result from the collimatedgamma radiation.4 The code then solves the coupled Maxwelland air-ion equations and runs significantly faster than whenthe macroparticles are used.
PLASMA. Through ionization, the beam electrons depositenergy in the ambient air. The amount per unit path lengthvaries slowly with beam energy for relativistic electrons and isapproximately 3400 eV/cm for normal density air. Since theenergy needed to produce a secondary electron is 34 eV and thebeam electrons are assumed to move at the speed of light, a
beam electron produces approximately 3000 electron-ion pairsper nanosecond. At normal (sea level) air densities, these low-energy secondary electrons are collision dominated, and theircurrent Js is accurately given using Ohm's law Js= aE, where a
is the conductivity found using the charged-particle densitiesand an electric field dependent mobility. We use the fulllumped-parameter rate equations for the density which are
employed in state-of-the-art EM P analysis.
For lower ambient air densities at higher altitudes or
temperatures, the secondary electrons may exhibit space-charge effects and a more complex model for their behaviorwill be required. To account for this we follow the secondaryelectron current using a convectionless fluid equation.
dj e2n Et
= -c -vjdt m c s
References
1. H. S. Cabayan, 'lElectromagnetic Emission from ElectronBeams,' L awrence Livermore N ational L aboratory,presentation at NEM in Anaheim, California, August 1980;N E M 1980 Record, p. 76.
2. C. N. Vittitoe and J. A. Halbleib, Sr., Sandia NationalLaboratory, Albuquerque, NM, private communication.
3. J. Pouky, J. Applied Phys., No. 50, p. 4996, July 1979.
4. T. A. Tumolillo, J. P. Wondra, W. E. Hobbs, and K. S. Smith,IEEE Trans. on Nuc. Sci., Vol. NS-27, pp. 1891-6, December1980.
5. R. B. Miller and K. R. Prestwich, 'Radial AccelerationDevelopment Summary,' paper presented at US/UK JOWOG-6meeting, Sandia National Laboratories SAND 81-0337, April1981.
6. J. A. Halbleib, Sr., Nuc. Sci. Eng., Vol. 75, p. 200, February1980.
7. C. L. Longmire, IEEE Trans. Antenna and Propagation, Vol.AP-26, No. 1, pp. 3-13, January 1978.
8. C. L. Longmire, Mission Research Corporation, SantaBarbara, California, private communication.
9. C. L. Longmire, and H. J. Longley, Mission ResearchCorporation, Santa Barbara, California, privatecommunication.
(4)
where the symbols have their normal meaning. We have notincluded the pressure term since it has negligible influence atthe low electron densities. To find the drag coefficient we
simply evaluate the mobility and use its definition e/mv. Notethat exponential differencing of. the current equation recoversOhm's law for large values of v. Using Ampere's law we findthe criterion for Ohm's law; i.e., v > w where w is the plasmafrequency.
