Non-linear self-acceleration of electrons emitted by plasma cathode
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.
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2002 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Zitieren: | Non-linear self-acceleration of electrons emitted by plasma cathode / A.V. Pashchenko, I.A. Pashchenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 95-96. — Бібліогр.: 1 назв. — англ. |
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Pashchenko, A.V. Pashchenko, I.A. 2015-04-12T06:52:10Z 2015-04-12T06:52:10Z 2002 Non-linear self-acceleration of electrons emitted by plasma cathode / A.V. Pashchenko, I.A. Pashchenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 95-96. — Бібліогр.: 1 назв. — англ. 1562-6016 PACS: 52.25.Sw, 52.35.Py, 52.60+h https://nasplib.isofts.kiev.ua/handle/123456789/80128 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. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration Non-linear self-acceleration of electrons emitted by plasma cathode Нелинейное самоускорение электронов, эмитированных плазменным катодом Article published earlier |
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Non-linear self-acceleration of electrons emitted by plasma cathode |
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Non-linear self-acceleration of electrons emitted by plasma cathode Pashchenko, A.V. Pashchenko, I.A. Theory and technics of particle acceleration |
| title_short |
Non-linear self-acceleration of electrons emitted by plasma cathode |
| title_full |
Non-linear self-acceleration of electrons emitted by plasma cathode |
| title_fullStr |
Non-linear self-acceleration of electrons emitted by plasma cathode |
| title_full_unstemmed |
Non-linear self-acceleration of electrons emitted by plasma cathode |
| title_sort |
non-linear self-acceleration of electrons emitted by plasma cathode |
| author |
Pashchenko, A.V. Pashchenko, I.A. |
| author_facet |
Pashchenko, A.V. Pashchenko, I.A. |
| topic |
Theory and technics of particle acceleration |
| topic_facet |
Theory and technics of particle acceleration |
| publishDate |
2002 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Нелинейное самоускорение электронов, эмитированных плазменным катодом |
| description |
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.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/80128 |
| citation_txt |
Non-linear self-acceleration of electrons emitted by plasma cathode / A.V. Pashchenko, I.A. Pashchenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 95-96. — Бібліогр.: 1 назв. — англ. |
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| first_indexed |
2025-11-26T00:17:40Z |
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2025-11-26T00:17:40Z |
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1850599212570378240 |
| fulltext |
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:
Em
e
t −=∂
υ∂ , (1)
( )aCa
xn 1=∂
∂ , (2)
( ) a
xnnea
E
i ∂
∂π∂
∂ −= 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 ξπ +−= (5)
where ξ(t) is the integral constant.
We have from equation (1):
( ),tax θ+= (6)
where function θ(t) satisfies the equation
( ) ,02
2 =++′′ tm
e
pt
ξθωθ (7)
and besides θ|t=0=0, θ'|t=0=0.
Then we found that in this region
( ) ( )[ ],1 tHxe
m +−′′= θθϕ (8)
where H1(t) is the integral constant.
In this region the electron movement is described by
the set of equations
( )
−=
=
−=
enx
E
Cxn
Em
e
t
π∂
∂
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 3
0
24 +−= ∫
τ
ξξπ . (10)
It permits to execute the integration of movement
equation in quadratures:
( ) ( )
+−−= ∫ tCdCem
e
t
x
i
30 22
2
4
τ
ξξπ
∂
∂ , (11)
( ) ( )
( ) ( )∫
∫
+−
−−
=
′
t
p
CdC
m
e
tdCn
x
τ
τ
τξξ
τξξω
∂
∂
,
t
43
0 2
0
2
(12)
( ) ( ) ( )
( ) ( ) ( ).
2
431
2
0
2
0
2
5
1
ττξξξ
τξξ
ω
τ
τ
ξ
τ
τ
−+−
−−
+=
∫ ∫
∫
tCdCdm
e
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
( ) ( ) ( )tente
mtC θπθ 03 4−′′−= . (16)
Then taking into account that
( ) ( )
−+′−= 21
2
2 τωτθ∂
∂ t
t
x
p , (17)
we can find the potential in the region (d,m)
( ) ( ) ( )( ) .22, 22
0 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):
( ) ( ) ( )∫ ∫++==
t
p ddtdtxm
0 01
2 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 ( )∫ ∫
t
p dd
0 01
2 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:
( ) ( ) ( ),/, 2
3 θωθ pemtxE +′′−= (20)
( ) ( ) ( )( ) ( ),, 0
2
3 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=4πen0 and joining of electrical fields and potentials in
the point x=k give us:
( )txenE ξπ += 01 4 , (23)
( ) xtxen ξπϕ −−= 2
01 2 , (24)
( ) .222
1 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
dTT
dT
d
0
2
02
2
0
2
1 ξξξρξρφρηρ
, (26)
where ρ≡θ/l, T≡ωpt, η≡d/l, ( ) ( )( ) ( ) .22
00 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.
96
REFERENCES
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