A two dimensional nonstationary model of the propagation of a relativistic electron beam injected into a vacuum is considered. Collision effects are ignored and there are no external fields. Two types of the electron current propagation are shown from the computer simulation of the Maxwell-Vlasov equations

Dyachenko, V. F.

Ginzburg, S. L.

English

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NASA TECHNICAL MEMORANDUM NASA TM-75693 
TWO-DIMENSIONAL NONSTATIONARY MOCEL OF THE PROPAGA~ION 
OF AN ELECTRON BEAM IN A VACUUM 
S.L. Ginzburg and V.F. D'yacheG~o 
Translation of "Dvt..:mernaya nestatsionarnaya model' 
rasprostraneniya elektronnogo puchka v vakuume," Academy of 
Sciences USSR, Institute of Applied Mathematics imeni 
M.V. Keldysh, Moscow) Preprint No. 51, 1979, pp 1-22 
(NASA-TM-7569J) TWO-DIPIENSIONAL N80-10435 
RONSTATION1RY ~OD!L OF THE PROPlGATION OF AN 
ELECTRON BEAft IN A VACUU" (National 
Aeronautics and Space Administration) 14 p TJnclas 
HC A02/MF A01 CSCL 0ge G3/33 45946 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
WASHINGTON D.C. 20546 OCTOBER 1979 
,rW«r.Aet .... ~~.::"'C __ =C'"h~,CC~~"-~~ .. 
i 
. $-' _ ,eQ .444 a2 W -=----1!!Ziifi 
I ) 
f i 
I-~~_.:s~'~ T~~-;----'T Go~~~:~:'·A~c~ .. ·'o~-N~~·-··-·-
•. r,.I. ond Sub ... I. TWO-DHlENSIONAL NONSTA-
TIONARY MODEL OF THE PRCPAGATION CF AN 
ELECTRON BEAH IN A VACUUM 
q AND""'[') T'TI r t'g,r 
~:"k-~~;~;'e'~,.~co'OI~g t~·_ .• ~~~ 
S. Repo,. Do •• 
October 1979 
6. P."o,ming O'gon; lotiO" Code '. 
~----------.------------------------.---------~ 7. A .. ,ho,(.' i 8. P.,'o,ming O'gon;IO.;on R.II"" No. 
S.L. Ginzbu!'g and • ..·.F. D'yachenko, USSRL-.. _________ ~ 
Academy of SciencE's, Keldysh Ir.stitute 110. Wo,k Un;' No. 
of Applied Mathematics, Moscow I 
1-________________________ . __________ -J~ Con',oc' 0' G,on' No. 
9. P.,'o,m;nll O'gan;lo,;on No",. and Add,." 
i 1J 
Leo Kanner Associates 13. 1 JPe of R.po,' a"d Pe,illd Co"e,ed 
Redwood City, California 94063 
------------------------1Translation 112. Spon.o,;ng A;.ncy Nom. arid Add,." 
National Aercnautics and Space Adminis- 14. Spon.o,i"g Ag."cy Code J 
tration, Washington, D.C. 20546 
~----~-----------~------
'15. Supple/lle,,''''y No'n 
Translation of "Dvumernaya nestatsionarr.aya model' 
rasprostraneniya elektronnogo puchka v vakuume," Academy of 
Sciences USSR, Institute of Applied Mathematics imeni I M.V. Ke1dysh. Moscow, Preprint No. 51, 1979. pp 1-22 
i 16. Ab."oc' 
I
I A two-dimensional nonstationary model of the propagation of 
. ___ . __ ~:relativist 1 c -el..ec.tvon-l;}.e.am--4n,feet-ed-1-nt-o- Vd,\,;\.4WU .l.;,;) 
considered. Collision effects are ignored. There are no 
external fields. Results obtained by computer simulation of 
the Maxwell-Vlasov equations showed there are two types of 
the electron current propagation. If the injected current is 
less than some critical value, the beam remains regular, 
laminar. Otherwise, surface reflecting electrons arise. 
Part of them return; the rest leave along a cone. 
I 17. Key Wo,d. (sclecteCl by AuthOr!s») 18. O .. "i:u,io" S'otement 
Unclassified-Unlimited 
i ~ ____ . ________ ~r------· __ --·L-__ --------~-----------.----------,1 
19. ~.:ut.r. Clo •• ,I. :01 ,h,. t.~,.,) l~. 5.: .. ,oIy Cloud. (o,,~,. po •• ) 21. No. o' Poges 22. P,ice 
I I I Unclassified I Unclass'=.;i::.::.f.=.i.=.e~d~. __ .L-_____ -... _____ --,j 
~r __________ "' ____ ' __ --'--------
TWO-DIMENSIONAL NONSTATIONARY MODEL OF THE PROPAGATION 
OF AN ELECTRON BEAM IN A VACUUM 
S . L. Ginzburg and V. F. D'~' achenko 
USSR Academy of Sciences, Keldy h Institute 
of Applied Mathematics. . .OSCOW 
1. The dynamics of a rel~t1vi8tic electror. beam in its inherent /4* 
electromaguetlc field, injected into vacuum, are considered, without 
taking account of particle collisions, within the framework of the 
Maxwcll~Vlasov equations ~J 
~ 'tf -rotil". ~11' J~ 0, divE. - I,,,eh (1) 
~ ¥l .,Yotl·o di,ii~o 
M+irl{ -e(€+a/iflfii})/I"o (2) 
Here, E(t,x), H(t,i') are the eJt:ctric and magnetic fields, f(t,x,p) 1s 
the electron pulse distribution function, p, v=Wp is the velocity, 
W(p) = Im2c4+c 2p2 is the electron energy, m is its rest mass, e is the 
elemer.tary charge, c is the velocity of light. The electron density 
n(t,x) and currer.t density J(t,x) are expressed by the integrals over 
impulse space 
(3) 
The electrons are injected from a circular plane cathode, of radius 
L along the normal to it, with energy 
and producing a current 
the density of which 
1 me=' 0= () Ite 
. me ll 
Jo = 0 L;neL"2 
*Numbers in the margin indicate pag!natlon in the foreign text. 
--~ ---------------------~~----
(4 ) 
(5) 
(6) 
1 
rlll14A .• ~.:C"L!I'!!I''''''L!!'''. - .. -~.,~-~.-~-~~~-.-.•. ------~-.--~--.-... ...., ..... ~--=~ 
, 
! 
tl 
rl 
, 
i is constant (y and a are assigned constants). 
i· 
There are no external fields. The plane whi"!h contains the 
cathode is assumed to be equipotential and, cOusequently, here, the 
tangential componer.t of the electric field equals zel'o. 
In cylindrical coordinates r, z, ~, where z = 0 is the plane of the 15 
cathode, and r = 0 is the axis of symmetry, we obtain two dime~sional 
problem (1)-(3), with a/a<jl:o, relative to Er(t,r,z), Ez(t,r,z) ,H (t,r,z) 
and f= o(P<jl)F(t,r,z,Pr,Pz)' with initial data at t=O, 
£,.=[~aH.,=r",(), (7) 
with the boundary conditions at z=O 
at "'~L." p~ > 0 
at Y>l, p~ >0 
aud with i~trinsic conditions on the r=O axis 
E". = H.,.. 0 J r'( A..)::: ry-p,.) 
( 8) 
(9) 
(10) 
~i The following two dimensionless parameters in the problem thus 
[: formulated prove to be decisive 
(11) 
the injected electron energy and current. 
2. The problem formulated was solved by computer. To integrate 
Vlasov equation (2), a macroparticle mOQel was used. The latter con-
sists of presentation of the electron gas as a discrete set of macro-
part icle s, each of \.\'hi ch is a group of electrons with the same coor-
dinates and impulse. Cf course, the electron motion is described by the 
eque.tions 
(12) 
The electron density is calculated by the formula 
(13) 
2 
/ 
I 
i 
~ , 
I 
1 
t 
! 
! 
where summing is carried out over all macroparticles, and 0 is a deltoid 
function, with a carrier on the order of the dimensions of the calcu-
lation cell. The current density is calculated similarly. 
The calculation region is a cylinder 0<1'<1'1' O<z<zl' on the outer 
boundaries of which the following conditions are laid down 
1::' -H If :: 0 
Ei!.~H'f'::o 
at 2=2, 
at J'r- r", , 
(14) 
which sirr.ulates the absence of electromagnetic radiation from the outside. 
Electrons leaving the bou~daries z=O, z=zl' 1'=1'1' are excluded from 
the calculation, and those striking the 1'=0 axis are reflected from it, 
in accordance with (10). Quite large values of r 1 ,zl are chosen, so as 
not to affect the result. 
5-
1 
Fig. 1. Current at 
y=2, 0=0.2. 
3. We proceed to description of the 
calculation results. First and foremost, we 
note that, in all (with respect to y, 0) 
versic~s, practically steady state behavior 
is observed. We dWE;ll on transient details 
below, and we now give the general character-
istics of these behaviors. 
The two schemes of movement presented in 
Figs. 1 and 2 are typical. The current lines 
for two versions, which differ in the magnitude 
of the injected current (0=0.2 and 0=2, re-
spectively), with the same energy y=2, are 
presented here. 
Ir: the fir;:; t case, we have a uniform, ex-
panding beam, propagatir,g along the z axis. 
In the second case, the movement ~attern is 
com)letely different. There is an envelope 
in tte family of trajectories, the z=z*(r) 
sur:'ace, which ref] ect s elE;ctron:; ("pseudo-
cathode"), and a nonur.iform, r.'ultivelocity flow. 
3 
r 
I 
f 
I 
t 
I 
~. 
• r 
..... 
2 
Fig. 2. Current at y=2, 0=2 
The electrons flow past 
the r/z~l cone, and a sub-
stantial fraction of them 
returns to the cathode. 
4. A priori, it is 
clear that conditions are 
produced in the pseudo-
cathode behavior, for the 
development of the known two 
beam ir.stability [2]. There 
are two counterteams, for-
ward and reflected, in which 
t~e density of the fluxes 
is high in the reversal re-
gion and, consequently, 
high frequencies should be 
excited. 
On the other hand, there are stabilizing factors, steady state in-
jection frorr the nearby cathode, constant current drift and a decrease 
in density with distance from tte axis. 
As the calculation results show, the interaction of all these factors 
results in an os cillatol'y process, the amplitude of \,lhich 1s small, and 
the average state corresponds to the ~teady state behavior pointed out 
above. 
The transient effects intensify with increase in current. There-
fore, we turn tc the version with y~2, 0=4 to describe them. E (z) 
z 
curves on the r=O 8x1e at different moments of time are presented in 
Fig. 3. The points where E paeses through zero correspond to the 
z 
pseudocathode lccation z*. E vs. time is shc~n in Fig. 4, at a pojnt 
z 
located on the 1'=0 axi~ near the plO€udocathode (zrvO.25L), for the lOam€: 
ven:ion, y=2, 0=4. In this area, because of the laI'gc f~ cld grac ier.t 
in th:. direction of movem8nt cf its profile, the 2.mplltuG.c c.f the os-
cillations is comparable to the size of the field itself. 
4 
IL E 
Mil • 
t,. '.S'Llc i._ r Llc 
t,. 7.r Lie. 
f. - f() LIe. 
Fig.~. E
z 
profiles on axis at y=2, 0'-4. 
J 
2 
t ! It S' , 7 , I '0 
As a result of the 
pseudocathode pulsations, 
a small fraction of the elec-
trons periodically prove to be 
above it, and they are dumped 
in the space beyond the cathode. 
These pulsations give rise 
to transi~nt pher.omena in the 
entire flow. Electrons Khich 
reach tte pseudo cathode along 
Fig. 4. Ez(t) in pseudecathode area, at one current line acquire varlous 
Y=2, 0=4. 
VElocities upon reflection fro~ 
it and, then, movir.g pract~cally rectilinearly, they are located on 
curvcs, the shapes cf Which are similar to the jet from an oscillating 
hese. The movement of sucr. jets bas ically determines thc structure of 
the weak turbulent flow which arises. 
For this reason, in dc~criptlcn ef the steady state characteristics 
cf the behaviors by meens of the current lines, we have represented the 
5 
!3If!411 •.. -.aW _ 4 -WS 
latter provisionally in the figures. by a dashed line which shows the 
average movement. 
It can be said that the tran~j.ent nature of the process is reduced 
to the transient nature of the pseudocathode. 
1 i i 
s 
1 
1 t .. "" 
Fig. 5. Flow at yr.2, 0=0.5 
Fig. 6. E and P r,rofiJez on z z t-
axis, at Y=2, 0=0.5 
6 
5. We know dwell in greater 
d~tail on the dependence of the aver-
age steady state properties of the 
flow on current o. We consider ver-
sions with injected electron energies 
y-2. 
The results of calculation of the 
version with 0=0.5 are presented 1n 
Fig. 5. Aitho~gh, the trajectories 
intersect in a narrow boundary layer, 
on the \'lhole, the beam remains uni-
form, with pr~ctically constant current 
density over the cross section, which 
riees slightly at the boundary itself. 
Compared with the 0=0.2 version 
(Fig. 1), the rate of broadening of 
the beam has increased appreciably. 
Curves of the Ez(t) and pulse 
Pz(z) fields along the z axis are ~lver 
in Fig. 6, for the same version 0=0.5. 
The point where Ez=O, Pz=min corre-
sponds to the minimum potential. Upon 
passing it, the flow is accelerated. 
HOKever, electron energy W(P
z
) remains 
less tha~ the initial wo·ymc2 . 
On the same terms, the variant 
witt o~O.65 is shown in Figs. 7 and 8. 
C"nlformlty is preservea only in tLe 
.:;pg; - --- - - - _ £§RE&& 
;~-
,f: _ 
. &~ . ., ·I~··~.'·· 
~--~-" ~ 
4 , 
F1g. 7. Flow at y-2. a-0.65 
axial region. A pseudocathode 
torma in the outer layers. The 
outer ~oundary 01' the injected 
beam (current line V) provea to be 
inside the tlow. Eighty percent 
of the current meves outside it. 
The version a-O.75 is pre-
sented in Fig. 9. The pseudo-
cathode has formed completely and 
reached the ax's. The outer layer 
of the beam becomes the inner one 
(line V). The flow "was turned 
ins1de out". The main portion 01' 
the current. 8"%. f~ows in a cone 
between lines II and V. A negli-
g1ble fraction of the current (~%) 
leaks through the pseudo cathode in 
a transient manner. A reverse 
current develops. 
As we see, a qualitat1ve re-
arrangement of the structure and 
shape of the flow occurs over a 
small interval 01' change 01' a 
from 0.5 to 0.75. A critical "cur- -
rent value can be spoken of. in 
this case, acr~.6. 
F1g. 8. E and fz profiles on 
axis. at yl2, a-O.65. With further increase in 
current, the flow pattern does not change qualitat1vely. The pseudo-
cathode subsides 8r!d becomes flatter. The reverse current fract10n in-
creases. At a-2 (Fig. 2) 34% of the electrons return on the Z·O plane 
and, at a-4, 56%. 
In the outer zone, the flow retains the shape of a co~e witt the 
same angle '\145°. Of course, the fraction of the current traveling in 
7 
I 
i 
-.4-", __ 
" .. F~ , 
f J.. i 
.. ' 
f: i 
... 1 
" 
" ;-
.. I ;-;-
" , .",. 
.",. ;- ..,. 
" 
..,. .... ; 
'..,. ... 
..,. 
-- -, 
--~ ... 
--
_ ... 
--
-... 
4 
....... 1 
.. r . ty .. 
Fig. 9. Flow at y-2, a-0.7S. 
J 
z 
1 
i 
~ . 
l 
I ., s 
Fig. 10. 
8 
tt decreases. 
6. We have reported the 
calculation results of versions 
with injection energy y-2. At 
other y, the quant1tat1ve char-
acter1stics of the solution are , 
d1fferent, of course, but all 
the effecta descr1bed are pre-
served. Fig. 10 gives an idea 
of the presence of the pseudo-
cathode and 1ts height z, <at 
r-O), in versions with other y. 
Spec1fic 1nformation on z, 
and acr vs. y also can be obta1n-
ed, by means of analytic est1mates. 
7. In the limit, as a'-, the 
pseudocathode approaches the 
cathode without restriction, 
z,/L+O. But, with z,~<L, the 
fini1ie'·dlmensions of the cathode 
should not affect the processes 
, in the axial re.gion. Therefore, 
·it·· the increase of a occurs 
through an increase in cathode 
radius L, with the density of the 
injected current preserved, for 
determination of the asymptotes 
of the solution in the axial zo~e, 
a unidimensional versio~ of our 
problem, 3/ir=O, which corresponds 
to an infinite cathode, can be 
used. 
~ 
Th1s problem was'1nvestisated in 3. The steadf state verslon or 
1tbaa an anal)'t1oal solut1on, Wh1ch can be written in the tOni 
.r.J: IfR.llfll-/IIHl .... jE.J,) 
LeE, -,IIwlt:"', F,. -H.,-o 
(15) 
(16) 
1n wh1ch the relationsh1p between W(Pz ) and Z is determ1ned, w1th 
O<&<z" bf the 1ntegral 
(17' 
and z. itself r H"~. i J{xt,r~ 
f (18) 
Preciself these values for the con"uspc;r.d.1ns y ere represented 1n 
F1S. 10 by a dasbed l1ne. The asymptote 1s exprossed completely 
r cleerly • 
• ~, I e. Si~ple ~asonini permits an estim~te cf the critical current 
c 
~i acr(y) to lie obtained, at whieh a ct.enge 1n type of behavic;z, CCCU:!"D. 
I I ~ ! We find the ccnditione of the ex1stence ot a steady etate, single 
l velocity flow, unbounded along the z ax11. 
~ 
The motion ot aT. electron in a steady state tield with potential 
,. V.--!, occurs with retention of its total en,rgy, v-e •• const. Since, 
at the time of injection, all the electrons have the !~e energy 
wo•ymc2, and the cathode plane is equipotential, .Iz.o·.o' 
W-elf·Wo-t'~ ,I V~mea (19) 
for all electrons in the entire flo"" 
The potential aatie!'ies the Po1sson equation 
Li t.p ~ 411"(:17 
and 1t is finite at infin1ty. 
(20) 
9 
I 
, I 
= , 
"' I P 
~ 
f£ 
~or a !lncle veloolty flew, the current density 
By virtue ot the assumed bou~de4n.ss 
r, at each s, evidently, +++0 as ~ • 
.ent1nl the rad1al portion ot the Laplaoe 
(21) 
of the beam with ~e.p.ct to 
This &i ves a buls tor pre-
operator 1n the form 
(22) 
where 0 1s a dlmenslonless, lndfterm!~~te cottt1clent, and R 1, the 
etfectlve radius ot the beam, such that 
1s the total current. 
W1th (19), (21)-(23) taken into censideration, we rewrite equatlon 
(20) ln the torm 
(2" ) 
On the assumption that R/z <const, It ls easy to det~rmine that a 
necessary cond1t1on of the ex1stence of a bcunded solution of equat10n 
(2"), .. z .. , is the pOlsibl1lty of the reductlon ot lts rl&ht .1de 
to zero. otherwise, elther W+-. or the solution exlsts only ln a tinite 
2 Interval, 11nce W cannot be less thar. mc • 
~ By maklns the rl&ht alde of (2b) equal to zerc, with fixed IO and 
~ , 
I 
t 
l 
J 
0, we obtain an equatlon tor W. Simple analysls I~OWI t~at lt hae a 
re~l root, only ln the event 
(25) 
This formula give. sr. eltlmate ot troe value ot the critical cur-
rent CJcr(Y)' It 0'\4.3, it i8 ';'''1 lood agreement with tl:e calculation 
results (Fig. 10). In particu~ar, this Just1fies the "simplicity" of 
10 
P 2£& gws #¥,o X ux .. 
---- ._----
•• 
r 
I 
the reaaon1nc u •• d. 
9. The auth~rl thank V.S. l~h.n1t.k1y, X.V. Xralnobayev, O.V. 
Lokuts!yevsk1yand K.V. Khodatayev, the initiators of th1a work and 
con.tar.t ~art1c1pant8 1n fru1trul discuss1on., as well al N.M. Khokhlova 
and Ye.V. S~akhanQva for aaftiltanoe 1n the work. 
11 
~: __ _ !!"- __ c- -='=_~~-"="""'==-------"'=----
4.eaz:s .. 
\ 
~ 
1 
I 
f 
( 
• t 
REFERENCES 
1. Dav1dson. R •• Teoriya zary&zhennoy plazmy [Charged Plasma 'llheory]. 
Mlr Press. Moscow J 1978. 
2. Briggs. R., DvukhE~chkovaya neustoycr.1vost' ITwo Beam Instab111ty]. 
M1r Press, r.1oscow J 1974. 
3. Poukey, J.W., N. Rostoker J "One dirnensio~al model of relativistic 
e1~ctron beam propags.tion," Plasma Physics ll/10, (1971). 
12 


Two-dimensional nonstationary model of the propagation of an electron beam in a vacuum

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