AD-ALG 726 AIR FORCE WEAPONS LAB KIRTLAND AFB NM FIG 20/8PROPAGATION OF A RELATIVISTIC CHARGED PARTICLE BEAM THROUGH AN --ETCIU)

MAY 80 R W LEMKE. K A DREYERUNCLASSIFIED AFWL-TR-79- 198 L

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AFWL-TR-79-1 98 E ,<AFWL-TR-

79-1 98LEVL$PROPAGATION OF A RELATIVISTIC CHARGEDPARTICLE BEAM THROUGH AN AIR MEDIUMWITH CONDUCTING BOUNDARIES

00) R. W. LemkeK. A. Dreyer

DTIC, ELECTE,

May 1980 DEC 18 19814d' 5 Final Report

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AIR FORCE WEAPONS LABORATORY

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AFWL-TR-79-198

The final report was prepared by the Air Force Weapons Laboratory, KirtlandAir Force Base, New Mexico, under Job Order 57972202. Mr R. W. Lemke (NTYP) wasthe Laboratory Project Officer-in-Charge.

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PROPAGATION OF A RELATIVISTIC CHARGED PARTICLE Final ReportBEAM THROUGH AN AIR MEDIUM WITH CONDUCTINGBOUNDARIES 6 PERPORMING OR*. REPORT NuMER

7 AUTOR(I) S. CONTRACT ON GRANT NUMUE0es)

R. W. LemkeK. A. Dreyer

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Electron Beam

Particle Beam

I20 'ABSTRACT (ContInue an POefefl stde If fle*0#9OP eng identify by block Mnnbe)

Many propagation experiments involve firing a relativistic charged particlebeam into a column of air bounded by the conducting walls of a drift tube.In this investigation, the extent to which the magnitude of the resultingreturn current and, hence, the stability of the propagating beam is affectedby the presence of conducting boundaries. Maxwell's equations are employedto derive the appropriate wave equation. This equation is transformed intoa diffusion equation which is numerically solved for various boundary param-eters.

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AFWL-TR-79-198

CONTENTS

INTRODUCTI ON 3

ASSUMPTIONS AND EQUATIONS 4

BEAM PROFILE AND PLASMA CONDUCTIVITY 8

NUMERICAL ALGORITHM 10

RESULTS AND CONCLUSIONS 12

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AFWL-TR- 79-198

INTRODUCTION

Some applications of relativistic electron beams (REB) necessitate that

they stably propagate some distance through air. Whether the beam will propagate

or not depends on the pressure of the air into which it is fired. The goal of

many REB experiments is to determine the pressure most conducive to beam

propagation.

Low pressure environments, into which an REB may be fired, can be created

by various means. The most practical means, experimentally, is to use a drift

tube whose interior pressure is adjustable. The beam is then fired into the

tube and monitored as it passes various points along the length of the chamber.

The drift tubes used in propagation experiments generally have cylindrical

symmetry, and are sometimes made of highly conductive metals. Hence, because of

the conducting boundary, firing an REB into a metallic drift tube may not be the

same as firing it into the unbounded atmosphere.

In this investigation, we endeavor to determine (by solving Maxwell's

equations with appropriate boundary conditions) a minimum drift tube diameter

such that conducting boundaries have a negligible affect on relativistic electron

beam-plasma interactions.

AFWL-TR-79-198

ASSUMPTIONS AND EQUATIONS

As is well known, Maxwell's equations, in Gaussian units, for material media

are:

V • E - 4wP (1)

'B 0 (2)

VxB= -E- (3)

S c C at (4)

To obtain the appropriate wave equation, apply the Vx operator to equation 3

and employ the identity

VxVxF V(V. -V2 "

to get

2 - V .EB (5)

Substituting into equation 5, the expressions for V • E and V x B yields

7 2 - 1 a~ 4Tal 6c2 at2 c at (6)

To investigate the problem, we need solve only the z component of equation 6.

To do so, we make the assumption of cylindrical symmetry. That is, we assume

that E., Br, and B are zero, and that the remaining components of E and B are

independent of the azimuthal coordinate 0. In addition, we assume that Ohm's law,

J = aE, is valid in the plasma. With the above assumptions, the z component of

equation 6 becomes

Ia a a2 41r a 1 a2E -z +T- z --2 a z "N t z T C

"4 + 2 + J zOo

where J' represents the REB current density.z

4

AFWL-TR-79-198

We simplify equation 7 via the transformation

= t -z (8a)c

Z- z (8b)

Now

- -- a (9a)

and

3 _ a 13(9b)az TZ - c (b

We now make the so-called frozen beam approximation; i.e., we assume that the

fields translate rigidly with the beam. For any given field component F, the

frozen beam approximation is equivalent to the mathematical statement

DF 3F FDt +v 0 (10)

or

aF _ F~-3t - (11)

where v is the velocity of the beam electrons. The energy of the beam electrons

is generally greater than 1 MeV. Hence. we can further assume that v - c. Thus,

aE aEa ---- (12)

Substitution of the operator identities 9a and 9b into equation 12 yields

Z Z Z (13)

Equation 13 is valid only if

aEz _ 0 (14)

That is, with respect to the coordinates Z and T, the frozen beam approximation

is equivalent to

aE

3z z " (15

. .. . . .... . . .. ... .( -' -

AFWL-TR-79-198

In view of equation 15, the identities 9a and 9b become

a .- a (16a)at aT

and

(16b)

In terms of the new coordinates, equation 7 can be written as

13 _ 4nr 3 4_r 3 4w (J1

r -r -g- Ez c 2 -T a Ez c T c 2 (17)

An REB propagating through air generates a plasma, resulting in space charge

neutralization (SCN). We account for SCN by letting

3'•P -= _C exp (- 4"n'oT) (18)

Substituting equation 18 into 17 yields

13 3 4T a 47T ar.E - E (19)r r r - z c cz c23

where

3'* - J' El - exp (-4vaT)J (20)z z

We assume that the conductivity of the drift tube is large enough to be con-

sidered infinite. Continuity of E across the plasma-wall interface requires2Ithat

Ez (R T) -0 (21)

where R represents the inner radius of the drift tube.w

Since the beam electrons are highly relativistic, Ez is approximately zero

at the head of the pulse. That is,

E (r,o) 0 (22)z

6

/ =

AFWL-TR-79-198

The 9 component of equation 3, in the new coordinates, is

-E (r,T) -B (r,T) (23)Dr z c at BEvaluating equation 23 at r - 0 yields

a E (r,T) I - 0 (24)r' O

Equations 19, 21, 22 and 24 uniquely determine E (r,T) inside the driftz

tube chamber.

7'

AFWL-TR-79-198

BEAM PROFILE AND PLASMA CONDUCTIVITY

The beam J' is assumed to have the following profile:z

J (T/T1 )/(l + r2/a2)2 T < 1 (25a)

z Jo /(1 + r2/a2)2 T< T < T (25b)

z b

0T > T (25c)

where and a represent the on-axis beam current density and Bennet radius,b

respectively. Also,

0 0Oia0 b I/Ta (26)

where 1 0 is the on axis beam current.b

We assume that the plasma conductivity (a) is generated with a radial profile

similar to that of the beam. That is, we let

a = T0/(l + r2/a2)2 (27)

where a0 is the on-axis conductivity. We employed existing air chemistry models

(Ref. l)* to numerically obtain the on-axis conductivity. Figure 1 is a graph of

on-axis conductivity versus distance (in seconds) back into the pulse. The beam

parameters used are indicated at the lower right-hand corner of the graph. The

rise length of the pulse is AL(= TIc) and its length is L(- Tzc).

1 Dreyer. ' A., EZectric and Magnetic Fielde of An Intense PuZse of ReZativistic:ect 6e Propagating Through Air, AFIT/DS/PH/79-1, AFIT, Wright-Patterson AFB-.,Lo, 1979.

• Johnston, R., private communication.

8

AFWL-TR-79-198

P 760. OT

lo1l

T, =1. 5MOV

a:- 2. ocm

L LSOVc(20ns)

108 L z4125cm(137.5ns)

Figure 1. On-Axis conductivity versus distance.

9

L1

AFWL-TR-79-198

NUMERICAL ALGORITHM

We solve equation 19 using an implicit finite difference scheme developed by

Crank and Nicholson (Ref. 2). Following this scheme, equation 19 becomes

47r 1 ri ) n+l n 11Ez n+lC a Ez )i 2 r i ri 6r zi

1 1 nrir~

+ 6 r 6 En2 ri r ir zi

47 1 ( jz*n+l - n' (28)c2 AT 1 2i i

The boundary conditions, equations 21, 22 and 24,become

E 0 0 (29)

zlE n al (30)IZEn+1 = n+l (31)zi+l zi

where

T - nAt (32)

r - iAr (33)

and

R - IAr (34)W

The step sizes, AT and Ar, were assigned the values 5.00 x 10 - 1 2 s and 0.01 cm,

respectively.

The z component of the plasma current density Jz is calculated using Ohm's

law. That is,

jn+l _n+l En+l (35)zi i zi

2. Carnahan, B., Luther, H. A., Wilkes, J. 0., Applied Numerical Methods,J. Whiley & Sons, Inc., New York, 1969.

10

AFWL-TR-79-198

We integrate equation 35, using Simpson's rule, to obtain the plasma current

(p) at each time step. The numerical expression for I is

SJn+l Jn+lIn~l 2 rJn1l+ 4r1 J + 2r2p 3 ozo Z1 z2

+ 2nr j i+l + 4r n+l n+131-3 JZ-3 r1 2 Jz- 2 + r -J- 1 )

where I w R /r.

i1

/,

AFWL-TR-79-198

RESULTS AND CONCLUSIONS

The results of our computer runs are summarized in Table I and Figure 2.

The beam parameters, with which equation 28 was solved, are given in the lower

right-hand corner of this figure. The plasma currents in this table and figure

were evaluated at the top of the current rise (T-T,) for three different pressures,

given in units of Torr.

TABLE 1. RESULTS OF COMPUTER RUNS

P-760T P-76T P-7.6T

3.0 3.3 3.8 8.5

5.0 6.5 7.1 14.0

8.0 9.5 10.0 17.9

10.0 10.7 11.2 19.3

12.0 11.7 12.1 20.3

15.0 12.7 13.0 21.3

20.0 13.9 14.1 22.3

35.0 15.7 15.9 23.8

50.0 16.7 16.8 24.5

The curves in Figure 2 indicate that, at each pressure, the plasma current

increases significantly out to Rw - 20.0 cm. Thereafter, out to Rw - 50.0 cm,

the plasma current increases only a few percentage points. That is, the conducting

boundaries significantly influence the plasma current when the drift tube radius

is less than 10 Bennet radii. Thus, for a beam with a 2 cm Bennet radius, the

minimum drift tube radius beyond which wall effects are insignificant is 10

Bennet radii.

On the basis of our results, it is suggested that regardless of beam size

wall affects are negligible when the ratio of drift tube radius to Bennet radius

is at least 10. That is, drift tube propagation experiments may simulate propaga-

tion of beams through the air, provided that the ratio of beam radius to drift

tube radius is greater than 10.

12

F I i

AFWL-TR-79-198

32.5

30.0

27.5

25.0 -P a7.6T

22.5 -

20.0

17.5

-15.0

12.5 0

10.0

7.5 Ibz i M30A

T* =1. 5MOV

5.0a =2.cm

& L 00cm(2Ons)

L v 4125cm(137.5ns)2.5

0.I I I I5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0

Rw(CN)

Figure 2. Results of computer runs.

13/14

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