Acceleration of plasmoids in waveguides by superhigh-frequency electromagnetic wav

Domrin, V. I.

Ramazashvili, R. R.

English

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National Aeronautics and Space Administration 
Goddard Space F l ight  Center 
Contract No.NAS-5-12487 
ST - PP - 10580 
ACCELERATION OF PLASMOIDS IN WAVEGUIDES BY 
A SUPERHIGH-FREQUENCY WAVE 
V. I .  Domrin 
R. R. Ramazashvili 
(USSR) 
I 
20 MARCH 1967 
. 
https://ntrs.nasa.gov/search.jsp?R=19670016341 2020-03-16T18:19:24+00:00Z
ACC1;LEMTION OF PIASMOIDS IN WAWXUI1)ES BY 
A SUPERHIGH-FREOUENCY WAVE" 
Yadernaya f izika 
.Journal of Nuclear Physics 
Tom 5, vyp.2, 312-317, 
Izd-vo '"AUKA", 1967. 
by V. I. Domrin C, 
R. R. Ramazashvili 
SUMMARY 
The acceleration is considered of an opaque plasmoid by an electromagnetic 
wave in a waveguide. 
when the group velocity of the wave in the waveguide is much smaller than the 
speed of light in vacuum. 
The acceleration process is studied in detail for the case 
* 
* * 
The acceleration process of a plasma layer of finite thickness by a plane 
electromagnetic wave has been investigated in the work by Gurevich and Silin [l]. 
Since in this work the relativistic effects are not taken into account, the 
formulas obtained are valid only for nonrelativistic velocities of plasmoid 
mot ion. 
In experiments plasmoids are usually accelerated in waveguides, in which 
the group velocity u of wave propagation may be significantly less than the pro- 
pagation of light c i n  vacuum. 
plasma acceleration in waveguides, the results of the work [l] will have to un- 
dergo certain changes. 
in a waveguide will naturally limit the maximum plasmoid velocity attainable in 
the waveguide. 
Therefore, when applying this to the case of 
In particular, the low group wave propagation velocity 
Let us consider a waveguide of infinite length, of which the dimensions 
We shall consider 
Assume that the 
and shape of cross-section are invariable along the axis. 
that the axis of the waveguide coincides with the axis z. 
plane z = 0 is the interface, to right of which is the Flasma, and to the left 
the vacuum. Assume further that plasma is uniform and cold. Let an electro- 
magnetic wave be incident upon the interface, with origin in the vacuum, and 
of which we shall consider the frequency w as being much lower than the Langmuir 
frequency of the plasma, thus enabling us to neglect the penetration of the 
field into the plasma. 
the field and the plasma remains plane in the course of the entire process. 
We shall consider, moreover, that the interface between 
* IJSKOFENIYA PLAZMENNYKH SGUSTKOV V VOLNOVODAKH SVCH VOLNOY 
3 
L . .  
where Tik is the system's energy-momentum tensor. Since we neglected the pe- 
netration of the field into the plasma, Tik is equal to the sum of energy-mo- 
mentum tensor of the electromagnet c field Ti/and of the energy-momentum tensor 
of particles T r k P  
3' f == - --- (--]i,b'p - ZZallp + (E2 + If?) 6@p ), 1 i 
4n I 
1 
T,,,'' : -- II'?, , f  = - - - p + l P ) ,  (u ,  p = 1 ,2 ,3 )  /In hn 
r<k= pL('2uiuh. 
Here vc2 is the energy density of plasma's 
scopic 4-velocity. 
particles. 
state of rest, and t1i is the riacro- 
When writing Ti<we neglected the mutual interaction of 
As is well known, two types of waves are possible in waveguides:, the E-yaves 
and the 14-waves. The relationship of fields' transverse components Eland I l l  
with the longitudinal components E, and H, in these waves is expressed by the 
following relations [3] 
~ 
- in the E-wave 
ik ,  dE,  ik ,  a E ,  E , = - -  E ,  = - --, 
n2 ax ' $ 1711 
= - -__._- 11, ---- - . 11, = 0. ito dE, io 3Ez 
r*?$ a!/ ' C X ?  as 
- in the 14-wave 
iw i3H, io d12, 
C X 2  ax 
= -- 17 -- - -__ , E, = 0, 9 '?.I  
cx2 a y  
0 2  
C2 
x 2 = _ - -  kz2, 
ik ,  dI1, , H , = - - -  i k ,  d H ,  I?,.= - -
n2 ax x2 dy ' 
(3) 
3 .  
I.'oi. kl-iiiitciwss wc shall consit1c.r in thc To1 lowing only I:.-w;ivcs, but a l l  
the results obtaincd will be also valid for 11-waves. 
Let us integrate the system (1) over the waveguide's cross-section. Making 
use of boundary conditions for fields in a waveguide, we obtain the following 
system of equations: 
The expressions for the components of the energy-momentum tensor depend 
on the velocity of the interface and on the velocity of particles reflected 
from it (because plasma is assumed to be cold, all the reflected particles will 
have an identical velocity). 
assumptions the system of equations (5) determined the motion velocity of the 
interface and of plasma particles. Averaging this system of equations in time, 
and taking advantage of the fict that the field does not penetrate into the 
plasma, we obtain 
Therefore, we may state that under the above 
expressions, in which the line above denotes the averaging in time. 
Let us denote by v the motion velocity of the interface and by v1 that 
of particles reflected-from the interface. In the system of counting, in which 
lies the interface, two groups of particles will be observed: particles incident 
upon the interface of which the velocity is -v - and particles outflying from it, 
of which the velocity will be denoted by vl'. Denoting the density of the num- 
ber of particles by N, and the cross-section area of the waveguide by S, we have 
Ut 5 T T d s  = - 
2 , 1 - v2/c2 
Here m is the mass of the ion (because of the smallness of me/m, the electronic 
part Sf the tensor rrkl'may be neglected) . 
In the above-made assumption about the total wave reflection from the inter- 
face, we shall obtain for the aggregate fields in the waveguide and in the inter- 
f x c  system the following expressions : 
E,  = %o(x, 9) cos ~ ' ~ ' G o s  k,'z', 
4. 
4 
Whcn wri t ing ( 8 ) ,  we took advantage of the cont inui ty  condition of E ,  a t  the 
in te r face .  Subst i tut ing (8) i n t o  ( Z ) ,  we obtain - 
[ J t i l i z ing  (7)  and (9 ) ,  we obtain from ( 2 )  
Ilence it is  easy t o  f ind  the h t e r f a c e ' s  motion veloci ty  
Here 
the  quant i ty  
u = c2kz / w is  the group veloci ty  of the wave i n  HP, and by a2we denoted 
When t h e  plasmas a r e  not too r a r e f i e d ,  we always have u - . f l .  
The motion ve loc i ty  of charged p a r t i c l e s  i n  HP, v l , i s  always easy t o  f ind  
from the  r e a l t i v i s t i c  law of veloci ty  addi t ion 
Tf t h e  plasma has a f i n i t e  thickness I ,  i t  is clear t h a t  i n  the time in te rva l  
1 / v  a l l  plasma p a r t i c l e s  will be imparted a veloci ty  equal t o  v l .  Passing t o  
the system moving with a ve loc i ty  v l ,  we again may u t i l i z e  formulas (11)-(13),  
provided we understand by u the group ve loc i ty  of the wave i n  the moving system. 
Pursuing f u r t h e r  our discuFsion, we may invest igate  i n  d e t a i l  the e n t i r e  process 
of plasmoid accelerat ion i n  the waveguide. 
The n o n r e l a t i v i s t i c  case n<r. i s  the one t h a t  lends i tself  t o  the most 
In t h e  l imit ing case, a f t e r  one cycle of accelerat ion complete invest igat ion.  
t h e  veloci ty  of the plasmoid is 2 y u ,  (y z a/(l + a) < 1). 
alongside with the plasmoid, the group veloci ty  u of waves i s  
In a system moving 
- 
(14) = u ( l  - 2y), 
(2) 
5. 
b 
uz = 2yu  + 2yu (1 -- 2y) .  (16) 
Generalizing t h i s  result  f o r  the  case of n acce lera t ion  cyc les ,  we obta in  - 
n-1 
un = 2yu  2 (1 - 2y)" = u { 1 - (1 - 2y)n) .  (17) 
m90 
I t  follows from t h i s  formula t h a t  i n  case of n o n r e l a t i v i s t i c  group velo- 
c i t y  of t he  waves, the maximum plasmoid ve loc i ty  i n  t h e  waveguide i s  exact ly  
equal t o  t i .  
may lead t o  the  form 
In case of very small values of y and f i n i t e  n (yn < 1) , (17) - 
showing t h a t  i n  t h i s  case the plasmoid ve loc i ty  iiicreases a f t e r  each cycle by 
one and the  same quant i ty  ZYU (cf. [l]). 
I t  i s  not d i f f i c u l t  t o  f ind  the  time necessary f g r  the plasmoid t o  a t t a i n  
a s p e c i f i c  ve loc i ty .  Since f o r  any s p e c i f i c  m-th cycle  the  t i m e  i n t e rva l  
is required,  t he  following time w i l l  be necessary €or the plasmoid t o  a t t a i n  the  
ve loc i ty  vn 
Zn = 
I 1-(1-2y)f3 2 Z(,) = - 2 9 u  ( 4  - 2y)n-f  * 
As may be seen from ( 2 )  , T n  rises i n f i n i t e l y  as  n +a. 
1 forniula (20)  takes the  form 
Therefore, t he  velo- 
c i t y  o r  the  plasmoid approaches the group wave ve loc i ty  asymptotically.  For 
Let us f ind  the d is tance  t h e  plnsmoid covers a f t e r  - n acce lera t ion  cycles:  
l l c ~ r ~ c c ~  i t  is easy to r ind  the accelcration 
‘Thercforc the acceleration rapidly decreases with the rise of Tn. In the 
case y t 1 - q  1 . ( 2 2 )  may be written in the form 
1 
which shows that in this limit case the motion of the plasmoid is unifmormly 
acce lerated. 
Ne may find the energy accumulated by a single ion as a result of n cycles - 
of :iccelcration 
‘I’his expression shows that the effective acceleration of the plasmoid takes 
place during a time of the order 
IJtilizing (21) we may find the law of En variation for VZ<! : 
‘I‘herefoi-e, at the beginning of the process the energy increases linearly 
‘I’hcn this dependence of En on 2, weakens gradually, and it with  thc distance. 
in;i>- be iicglcctcd at ;I distance of the order Tef u J (1 
!notion of plasnioids accelerated by a slow wave in the waveguide Com- 
p a r i s o n  \;it11 the results of the work [l] shows that fomiulas (18) ,  (21), (24) 
and (271, obtained in the assumption that y t ~ . .   1 ,  fully agree with the corres- 
ponding results of [l]. Naturally, a difference arises in the determination 
of the motion velocity of the boundary of the semi-infinite plasma in the wave- 
guide (fonnula (11)). This difference is linked with the fact that the group 
velocity of plane waves in free space is c, while in the case considered this 
velocity may be much less than c. 
o r  smallness of group wave velocity allowed us to obtain i-ormulas describing 
the motion of a plasrnoid through its attainment of a velocity equal to the 
group velocity of waves. 
2y) I / 2 ~ 2 .  
‘I’he fonnulns obtained by the above made assumptions fully describe the 
C u e c ) .  
Contra5 to the work [l] , the assumption 
7. 
W1ic.11 c l ( ~ r . r v  t r i i :  1 ti(> ;rl)ovc. y,c~t~c-r;iI c*xl)rc>ssioris, wc d i d  riot iiri1)osc ;iiiy 1 i r r i i -  
t ; i t  I O I I C ;  o i i  t l ic.  t i r r ; t r i t i ty  a. Ilowc‘vcr, i t  is not cliI‘ficu1t to ~ c ’ e  that ttic g ra -  
c l r i ; i l  accc1cr;ition o f  the plasmoid w i l l  take placc only for a < 1. As is secn 
I’rom (14), the plasrnoid, having attained the velocity 2yu after the lirst acce- 
lcration cycle, will move more rapidly than the wave, and no subsequent accele- 
ration will takc place. 
1,et us now pass to the consideration of the relativistic case u -c. First 
of rill we shall examine the case when the wave group velocity is so great that 
a -:(l - u2/c2) ‘ l 2 .  Then, as may be seen from (11), the velocity of the boundary 
will already be equal to the wave group velocity after one acceleration cycle. 
As to the plasmoid velocity, it will be 
2u 
- u - c .  v = ---- 
I + U 2 / G  
(It is clear that the sane pattern will take place in the case a,>! regardless 
of the magnitude of the group velocity. 
equal to 2u /(1 
the conditions a > 1 and (I)’< wpz are difficult to realize simultaneously). 
The velocity of the plasmoid will be 
+ u2/c2). - I t  is easy to see that for real fields and plasmas 
Most complex for the investigation is the inverse limiting case a<(l- u2/c2)’/? 
One may not succeed in obtaining for such a case a compact formula determining 
the velocity of the plasmoid in HP after an arbitrary number of plasmoid accele- 
ration cycles. 
with good precision for the nonrelativistic case by the formulas above. 
is that, because of the smallness of a/(l - u2/c2) 
process, the plasmoid velocity will remain nonrelativistic till the completion 
of a specific number of cycles, as this may be seen from (11 
tions arise because the denominator of (11) contains (1 - u /c ) 2and consequently 
changes in the process of plasmoid acceleration. However, for initial cycles we 
may apparently consider the group velocity as invariable in the denominator 
without committing great error, and thus utilize the above formulas in which Y is 
recognized for the expression u /(1 -u2/c2)’/2. Such a consideration will be valid 
until the plasmoid attains the relativistic velocity. 
However, the initial accelention cycle processes may be described 
The fact 
at the beginning of the 
Cy tain complica- 1. 2 J 
It should be borne in mind that the above assumptions may limit the appli- 
cability of the formulas obtained to a real experiment on plasmoid acceleration 
i n  waveguides. 
field-plasma is not in any way evident because of waveguide field dependence on 
transverse coordinates. However, cases may be indicated when this assumption 
may he fulfilled with good precision. For example, as is well known [ 4 ] ,  plas- 
moicls of specific configurations may fully close the waveguides in conditions 
when their dimensions are small by comparison with the wavelength. Under these 
conditions the field inhomogeneities at distance of the order of plasmoid dimen- 
sions in:i!- lie neglected and the acceleration process may be described by the 
above fonm1lns . 
It is clear that the assumption of invariability of the interface- 
8. 
I.ct 11s 1)oint ; ~ I s o  to onc 1 imitation that was discreetly assumed in the work. 
I ' o r w l i i  ( 1 I )  r l c ~ s c - r - i t w s  a stationary picture. 
stat  ioriary vc . loc i ty  the plasma interface wil 1 shift by a specific distance, which 
for the v a l  i t l i  ty of our reasonings should be much smaller than the dimensions of 
our  plasmoid. 
thickness and we neglected it completely. However, this question calls for tho- 
rough investigation. A more detailed research is also deserved by the questions 
of field penetration into plasma and its absorpt;Lonand scattering, for the field 
absorption may lead to plasma heating, which is going to complicate the pattern 
of plasmoid acceleration described here. 
Ilowcvcr, prior to reaching the 
We considered this distance to be of the order of skin-layer 
We convey our thanks to B. P. Silin for the theme proposed and for the use- 
f u l  discussions. 
Contract No.NAS-5-12487 Translated by ANDRE L. BR1C"T 
VOLT TECI fN I C U  CORPORATI ON 
1145 19th St. hW on 20 March 1967 
I). C. 20036; tel: 223-6700; 223-4930. 
REFERENCES 
1. 
2 .  L. ll. LANDAU, E. M. LIFSHITS. Teoriya Polya (Field Theory), FIZMATGIZ, 1960 
3 .  1,. D .  L-LMAU, E. M. LIFSHITS. Elektrodinamika sploshnykh sred (Continuum 
A. ~- 11. WREXCH, V. P. SILIN., Nuclear Physics (YA.F), 2, 250, 1965. 
Electrodynamics). Fizmatgiz, M. 1959. 
4. B. I. VEKSLER, I. R. GEKKER, E. YA. GOL'TS ET AL. - Trudy Mezhdunarodnoy 
konferentsii po uskoritelyam (Proc. of the 
1ntern.Conf. on Accelersttors). Dubna, 1963. 
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Acceleration of Plasmoids in Waveguides by a Superhigh-Frequency Wave

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