NON-LINEAR SELF-ACCELERATION OF ELECTRONS EMITTED BY 
PLASMA CATHODE
A.V. Pashchenko, I.A. Pashchenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: paschenko@kipt.kharkov.ua
A non-linear non-stationary analytical theory of diode gap overlap by electrons emitted from the plasma cathode 
is represented. The fact of the non-linear self-acceleration at the electron flow front is confirmed.
PACS: 52.25.Sw, 52.35.Py, 52.60+h
1. INTRODUCTION
Electron self-acceleration takes place in a cathode-
anode  interval  during electron  emission  from a  metal 
cathode [1]. This effect provides the appearance of the 
corresponding  additional  source  of  current  in  the 
equivalent electrical circuit of the diode. It is interesting 
to ascertain as this effect is modified in case of electron 
emission  from  a  plasma  cathode.  This  problem  is 
investigated in the present report.
2. THEORY
Let us consider a plasma layer with a thickness of d 
in  the  Cartesian  coordinate  system  0XYZ  placed 
between  two  conducting  plates  x=0  and  x=l in  the 
interval (0,d), i.e. d<1.
Since a  moment  t=0 the plates  x=0 and  x=l were 
found  under  potentials  ϕ(0)=0  and  ϕ(l)=ϕ0(t)>0, 
respectively. So at t>0 an external electric field acts on 
plasma.  Then  to  a  moment  t not  so  greater  than  t=0 
electrons are displaced from the initial position so that 
the  whole  region  (0,l) is  divided  on  four  physical 
regions:  a) (0,k) - ions;  b) (k,d) - plasma;  c) (d,m) -  
electrons;  d) (m,l) - vacuum.  Let  us  consider  an 
electrodynamical problem for each of these regions with 
taking into account the non-linear  electron movement, 
make a joining of fields and potentials and satisfy their 
boundary conditions.
Let us start to proceed from the electron movement 
equation,  continuity  equation  and  Poisson  equation 
which in the Lagrange variables a,t, where a is the initial 
coordinate, t is the time, have a form:
Emet −=∂υ∂ , (1)
( )aCaxn 1=∂∂ , (2)
( ) axnneaE i ∂∂pi∂∂ −= 4 , (3)
( )ann ii = , (4)
where  e and  m are  the charge value and the  mass of 
electron.
Let us consider that at  t=0 and x=a:  υ=0, ni(a)=n0. 
Then C1(a)=n0 and from equation (3) we have( ) ( ),4 0 taxenE ξpi +−= (5)
where ξ(t) is the integral constant.
We have from equation (1):
( ),tax θ+= (6)
where function θ(t) satisfies the equation
( ) ,022 =++′′ tmept ξθωθ (7)
and besides θ|t=0=0, θ'|t=0=0.
Then we found that in this region
( ) ( )[ ],1 tHxem +−′′= θθϕ (8)
where H1(t) is the integral constant.
In this region the electron movement is described by 
the set of equations
( )



−=
=
−=
enxE
Cxn
Emet
pi∂
∂
ττ∂
∂
∂
υ∂
4
2 , (9)
where  τ is  the  time of  passing  by an  electron  of  the 
coordinate x=d.
The integration of the Poisson equation from (9) has 
shown that the electric field represents a sum of function 
t and function τ in this region:
( ) ( )tCdCeE 30 24 +−= ∫ τ ξξpi . (10)
It  permits  to  execute  the  integration  of  movement 
equation in quadratures:
( ) ( )  +−−= ∫ tCdCemetx i 30 222 4 τ ξξpi∂∂ , (11)
( ) ( )
( ) ( )∫
∫
+−
−−=
′
t
p
CdCm
e
tdCnx
τ
τ
τξξ
τξξω∂∂
,
t
43
0 20
2
(12)
( ) ( ) ( )
( ) ( ) ( ).
2
431
2
0 20
2
5
1
ττξξξ
τξξωτ
τ
ξ
τ
τ
−+−
−
−


+=
∫ ∫
∫
tCdCdme
tdCnCx
t
p
(13)
We find the integration constants from conditions at 
t=τ:
( ) ( ) ,,5 dtxC t =τ=τ τ= (14)
( ) ( ) ( ) ( )
τ
τθ
∂
∂
υ
∂
τ∂
τ ττ d
d
t
tax
t
txC dxtt ==== === ,,4 .  (15)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2002, № 2.
Series: Nuclear Physics Investigations (40), p. 95-96. 95
Now we can find the function  C2(τ) incoming into 
the continuity equation in (9):
.)(2 τττ τ∂∂τ∂∂τ∂∂τ ==== === tdxtt xnxnxnC
Since n|x=d=n0 and ∂x/∂τ|t=τ=−C4(τ)+dC5(τ)/dτ=−θ'τ(τ),( ) ( ).02 τθτ τ′−= nC (15')
Joining of electric fields at the point x=d leads to the 
formula
( ) ( ) ( )tentemtC θpiθ 03 4−′′−= . (16)
Then taking into account that
( ) ( ) 


−+′−= 21
22 τωτθ∂
∂ ttx p , (17)
we can find the potential in the region (d,m)
( ) ( ) ( )( ) .22, 220 τωτ −+= tntn p (18)
It can be seen from formula (18)  that  the electron 
density decreases  with time as we move off from the 
plasma layer boundary x=d.
The second boundary coordinate of the plasma layer 
is determined by the formula, getting from (13):
( ) ( ) ( )∫ ∫++== tp ddtdtxm 0 012 1,0 ξ ξξθξωθ . (19)
It can be seen that the electron coordinate changing 
at layer boundary differs from the corresponding change 
in the plasma region at the value ( )∫ ∫tp dd0 012 1ξ ξξθξω  that 
confirms the electron layer propagation.
Consideration  of  the  Poisson  equation  ∂E3/∂x=0, 
joining of  fields  and  potentials  in  the  point  x=m,  the 
connection  to  boundary  condition  on  the  border  x=l 
leads to the formulas:
( ) ( ) ( ),/, 23 θωθ pemtxE +′′−= (20)
( ) ( ) ( )( ) ( ),, 023 txlemtx p ϕθωθϕ +−+′′−= (21)
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ]∫ −+′+
+−−−′′−=
t
p
p
dt
dlltmetH
0
22
2
01
21 ξξξθξθω
θωθθϕ
, (22)
where ϕ0(t) is the potential of anode x=l.
A  positively  charged  layer  completed  by  heavy 
stationary  ions  is  appeared  near  the  cathode  after 
beginning of action of the external potential  ϕ0(t). The 
ion density is n0. Solving of the Poisson equation ∂E1/∂
x=4pien0 and joining of electrical fields and potentials in 
the point x=k give us:
( )txenE ξpi += 01 4 , (23)
( )xtxen ξpiϕ −−= 201 2 , (24)
( ) .2221 θωθθ ptH +′′= (25)
3. BASIC PROBLEM EQUATION AND 
RESULTS
Equating expressions (22) and (25), we receive the 
equation  for  determination  of  non-linear  electron 
displacement in the plasma layer θ:
( ) ( ) ( ) ( ) ( )∫ =−′−−−+ T dTTdTd 0
2
02
2
021 ξ
ξξρξρφρηρ , (26)
where ρ≡θ/l, T≡ωpt, η≡d/l, ( ) ( )( ) ( ) .2200 lmteT pωϕφ ≡
The  equation  (26)  is  the  head  result  of  present 
research.  The  numerical  solution of  this  equation was 
performed using the function
( ) ( ).10 TeT αγφ −−= (27)
The concrete values of parameters  η and γ are very 
important for result. Estimations give us the value  10-3 
for γ under l≈1 sm, n0=1012 sm-3 and potential amplitude |
ϕ0(t)|≈103 V.  The  level  of  effect  on  plasma  can  be 
characterized by the ratio ρ/η: the effect is strong for ρ/
η≤1, and the effect is low for ρ/η<<1.
Fig. 1-3  demonstrate  the  numerical  results  for  γ
=0,025,  η=0,5. In particular, Fig. 2 shows the velocity 
of one electron corresponding. This fact takes place due 
to the collective acceleration of electrons at the electron 
flow front.
0.00
0.04
0.08
0.12
0 1 2 3 4 5 6T
Fig. 1. Electron displacement in a plasma layer, ρ.
0.0
0.1
0.2
0.3
0 1 2 3 4 5 6T
I
II
Fig. 2.  Curve  I  -  velocity  of  electron  flow  front,  
curve II - velocity of one electron under the influence of  
Ф0(T) for the case a=d
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4 5 6T
m
Fig. 3. The coordinate of electron flow front
4. CONCLUSION
In  this  report  we  create  a  strong  non-linear  non-
stationary  analytical  theory  of  diode  gap  overlap  by 
electrons emitted from the plasma cathode. The fact of 
the non-linear self-acceleration at the electron flow front 
is confirmed.
REFERENCES
1. A.V. Pashchenko, V.E. Novikov, Y.V. Tkach. 
The  phenomenon  of  non-linear  cathode-emitted 
particle self acceleration. 10th IEEE International 
Pulsed  Power  Conference,  Albuquerque,  New 
Mexico, 1995, p. 852-856.
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