Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action
The stability of electron fiow passage through the diode with a neutralized space charge has been investigated on the basis of the first-order approximation for hydrodynamic and Maxwell equations. Classification of solutions to the equations was developed in accordance with the magnitude and nature...
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| Zitieren: | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action / O.G. Melezhik, A.V. Pashchenko, S.S. Romanov, I.M. Shapoval // Вопросы атомной науки и техники. — 2014. — № 5. — С. 161-167. — Бібліогр.: 2 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860065227335270400 |
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| author | Melezhik, O.G. Pashchenko, A.V. Romanov, S.S. Shapoval, I.M. |
| author_facet | Melezhik, O.G. Pashchenko, A.V. Romanov, S.S. Shapoval, I.M. |
| citation_txt | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action / O.G. Melezhik, A.V. Pashchenko, S.S. Romanov, I.M. Shapoval // Вопросы атомной науки и техники. — 2014. — № 5. — С. 161-167. — Бібліогр.: 2 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The stability of electron fiow passage through the diode with a neutralized space charge has been investigated on the basis of the first-order approximation for hydrodynamic and Maxwell equations. Classification of solutions to the equations was developed in accordance with the magnitude and nature (potential or eddy) of the electric field arising at the diode cathode. The stability regions for these states have been found. It is shown that the presence of external electromagnetic action makes the stability region narrower than that predicted by Pierce.
На основе решения системы гидродинамических уравнений и уравнений Максвелла в первом приближении изучена устойчивость прохождения электронного потока через диод с компенсированным объемным зарядом. Проведена классификация решений уравнений в соответствии с величиной и характером (потенциальное или вихревое) электрического поля, возникающего на катоде диода. Найдены области устойчивости этих состояний. Показано, что наличие внешнего электромагнитного воздействия сужает область устойчивости по сравнению с предсказанной Пирсом.
На основi розв'язання системи перших наближень гiдродинамiчних рiвнянь та рiвнянь Максвела вивчена стiйкiсть проходження електронного потоку через дiод з компенсованим об'ємним зарядом. Проведена класифiкацiя рiшень рiвнянь у вiдповiдностi з величиною та характером (потенцiйне або вихрове) електричного поля, що виникає на катодi дiода. Найдено областi стiйкостi цих станiв. Показано, що наявнiсть зовнiшнього електромагнiтного впливу звужує область стiйкостi порiвняно з передбаченою Пiрсом.
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| first_indexed | 2025-12-07T17:06:44Z |
| format | Article |
| fulltext |
MODIFICATION OF THE PIERCE INSTABILITY OF THE
ELECTRON FLOW IN A DIODE WITH AN EXTERNAL
ELECTROMAGNETIC ACTION
O. G. Melezhik A. V. Pashchenko, S. S. Romanov, I. M. Shapoval∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received July 1, 2014)
The stability of electron flow passage through the diode with a neutralized space charge has been investigated on
the basis of the first-order approximation for hydrodynamic and Maxwell equations. Classification of solutions to
the equations was developed in accordance with the magnitude and nature (potential or eddy) of the electric field
arising at the diode cathode. The stability regions for these states have been found. It is shown that the presence of
external electromagnetic action makes the stability region narrower than that predicted by Pierce.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION
It has been demonstrated by V.Bursian and
V.Pavlov [1] that with injection of an electron flow
into a planar diode, the flow density can be increased
only to a certain value, called the Bursian-Pavlov
limit. An excess of this limit gives rise to instability,
which leads to the appearance of a virtual cathode
(VC) and to realization of the effects associated with
it. One of these effects is the reflection of a greater
part of the electron flow incident on the VC. This
means that the current that can be passed through
the planar diode is determined by the Bursian-Pavlov
limit. The limit has its quantification. The electron
flow state in the short-circuited diode is given by the
Bursian-Pavlov parameter
q =
ω2
0l
2
v20
, (1)
ω2
0 =
4πe2n0
m
, (2)
where l is the diode length, v0 and n0 are, respec-
tively, the velocity and density of electrons at the
input of the diode. The Bursian-Pavlov limit corre-
sponds to the value
qB =
16
9
. (3)
This limit occurs to be a natural power limiter of the
electronic devices being developed. In due time, vari-
ous methods were tried to increase the density of the
electron flow passing through the diode. The pro-
posal to use the phenomenon of potential well neu-
tralization of the electron beam by ions belongs to
Pierce [2]. Pierce has shown that at a complete space
charge compensation of the electrons in the absence
of external action, the limit determining the electron
flow passage increases up to
qP = π2 . (4)
This limit stems from the occurrence of the ”Pierce
instability” in the compensated electron flow. So, at
the same electron flow parameters at the anode input,
and at the same diode length in case of space charge
compensation, the ultimate current density increases
Λ times, where
Λ =
qP
qB
=
9
16
π2 ≈ 6 , (5)
The present paper deals with the electron flow pas-
sage through the diode with a compensated space
charge for both the case of absence of total diode
current disturbances, considered by Pierce, and the
case of presence of these disturbances. The solution
to the problem of electron flow passage through the
mentioned diode is of both the applied and method-
ological importance. The thing is that the state of
this electron flow is only one of the possible elec-
tron flow states in the diode, which is noted for the
compensated space charge and, therefore, is most
amenable for theoretical description and physical in-
terpretation. The two will be presented below.
2. THE DIODE WITH A COMPENSATED
SPACE CHARGE
The diode with a compensated space charge [2] is de-
fined as a diode with the electrodes being at the same
potential, in which the space charge of the electron
flow is compensated by stationary ions. The electron
∗Corresponding author E-mail address: shapoval@kipt.kharkov.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.161-167.
161
flow in the diode is described by the equation of elec-
tron motion and by the Maxwell equations. In the
description of the electron flow motion, for reasons of
v/c smallness, we shall neglect the Lorentz force com-
pared with the electric field strength. This makes it
possible to avoid finding the self-magnetic field and
to describe the electron flow in terms of the velocity,
density and the electric field. The equation of mo-
tion, the Maxwell equation for the electric field and
the equations of continuity form the set of equations
to describe the electron flow in the diode. As in ref
[2], we consider the problem in the one-dimensional
approximation. Then the set of equations takes the
form
∂v
∂t
+ v
∂v
∂z
= − e
m
E , (6)
∂n
∂t
+ v
∂nv
∂z
= 0 , (7)
∂E
∂z
= −4πe(ni − ne) , (8)
where ni and ne- are the densities of ions and elec-
trons, respectively. The dimensionless values for the
density n, velocity v, coordinate z and time t are
the characteristic flow parameters at the input to the
diode. These are n0, v0, the diode gap length l, and
the transit time l/v0. We shall investigate the stabil-
ity of ion-compensated electron flow passage through
the drift space. For this purpose we find the solu-
tion of the equations in the zeroth- order and first
approximations. Each of the parameters v, n, E will
be represented as a sum of the solutions of zero- and
first approximations:
f(ζ, t) = f0(ζ) + f̃(ζ, t) , (9)
where ζ = z/l. The time dependence of deviation
values from zero-order approximations will be sought
as
f̃(ζ, t) = ψ̃(ζ)e−iωt . (10)
The deviations from stationary values are considered
to be small, and from Eqs.(6)-(8) we obtain the lin-
earized set of equations:
−iωṽ + dṽ
dζ
= −Ẽ , (11)
−iωñ+
dñ
dζ
+
dṽ
dζ
= 0 , (12)
dẼ
dζ
= −qñ . (13)
Before finding the solutions to the equations of the
zero-order and first approximations, we shall dis-
cuss the potential distribution in the diode (Fig.1).
-l1 Ζ1 l Ζ2
Ζ
U
j
Fig.1. Diode with the inlet N1 (at z = 0) and outlet
N2 (at z = 1) grids, and the cathode at z = −l1 (l1
is the cathode gap length)
The potential φ is dimensionless due to mv20/e. In
the cathode-grid N1 gap, in the presence of the cath-
ode emission the potential increases by the Child-
Langmuir-Boguslavsky law and provides the poten-
tial U transferred by the external circuit from grid
N1 to grid N2. The potential sag caused by the elec-
tron flow in the drift space is eliminated by ions. As
a result, the stationary distribution of the potential
in the drift space of the diode takes on the uniform
shape. Then, the stationary solution of the set (6) -
(8) is written as
v0(ζ) = 1 ; n0(ζ) = 1 ; E0(ζ) = 0 ; φ0(ζ) = 1 .
(14)
Equations (11-13) form a system of ordinary linear
differential equations of first order with constant co-
efficients. Its solution is given by
ṽ(ζ) = D1e
i(
√
q+ω)ζ +D2e
−i(
√
q−ω)ζ +
+C
iω
q − ω2
, (15)
ñ(ζ) = −D1(1 +
ω
√
q
)ei(
√
q+ω)ζ −
−D2(1−
ω
√
q
)e−i(
√
q−ω)ζ , (16)
Ẽ(ζ) = −D1(i
√
q)ei(
√
q+ω)ζ +
+D2(i
√
q)e−i(
√
q−ω)ζ − C
ω2
q − ω2
,
(17)
where D1, D2, C are the integration constants. The
solutions (15) - (18) must satisfy the boundary con-
ditions:
ṽ(0) = 0 , ñ(0) = 0 . (18)
Conditions (18) are caused by the requirement that
the electron flow should not bring in any external dis-
turbance in the diode at frequency ω. The conditions
lead to the following equations for D1, D2, C:
D1 +D2 + C
iω
q − ω2
= 0 , (19)
D1(1 +
ω
√
q
) +D2(1−
ω
√
q
) = 0 . (20)
162
The electric field in the diode arises self-consistently
with the coordinate distributions of densities and ve-
locities; therefore, it can neither by itself nor through
Eq.(15) give the missing condition on D1, D2, C.
The equation corresponding to this condition can be
derived from the boundary conditions on the scalar
potential φ̃. In accordance with the determination
of the potential φ̃, the equations for its finding and
the type of boundary conditions, which the potential
must satisfy, we consider below some problems, which
describe the electron flow passage through the diode.
3. SOLUTIONS CLASSIFICATION
The constant C occurring in the solution of (15)- (17)
represents the electric field value on grid N1 (Fig.1)
from the diode side. This conclusion follows from
(18) and formula
Ẽ =
iq
ω
(ñ+ ṽ) + C . (21)
Formula (21) was derived from the set (11) - (13) as a
result of substitution and integration. Since it is im-
possible to assign the electric field at the beginning of
the diode gap as the electron flow propagates in the
diode, the investigation of the flow passage through
the diode must be continued until finding the electric
potential, which governs the electron flow motion. In
this case, one uses the definition of the electric field
by means of the potential
Ẽ(ζ) = −∂φ̃(ζ)
∂ζ
. (22)
The electric field in this formula is the potential field.
Below we consider several cases concerning the nature
of the constant C: a) C is of potential; b) C is of eddy
and c) constant C is of mixed nature.
3a. The Pierce problem (C = 0)
In ref. [2] Pierce has considered the electron flow
passage through the diode, where the space charge is
compensated by stationary ions, assuming that there
is no total current disturbance in the circuit. The
assumption that this disturbance is zero leads in-
evitably to the requirement that in formulas (15)-(18)
we should have
C = 0 . (23)
Indeed, from the definition of the total current it fol-
lows that the disturbance is proportional to
J̃ ∼ ∂Ẽ
∂t
− 4πe(ṽ + ñ) . (24)
Writing expression (24) in terms of dimensional vari-
ables and taking into account that by definition
Ẽ(ζ, t) = Ψ̃(ζ)e−iωt we draw the conclusion that the
right side of (24) and the left side of (21) are equal,
this just proving the conclusion about the fulfillment
of equality (27). Hence, at zero disturbance of the
total current the electric field is written as
Ẽ(ζ) = −D1(i
√
q)ei(
√
q+ω)ζ +D2(i
√
q)e−i(
√
q−ω)ζ .
(25)
Using the definition (17) and taking into account that
φ̃(0) = 0 , φ̃(1) = 0 (26)
we obtain the expression for φ̃(ζ):
φ̃(ζ) =
D1
√
q
ω +
√
q
(ei(ω+
√
q)ζ − 1)−
−
D2
√
q
ω −√
q
(ei(ω−√
q)ζ − 1) (27)
and the condition for D1 and D2:
D1(ω−√
q)(ei(ω+
√
q)−1)−D2(ω+
√
q)(ei(ω−√
q)−1) = 0 .
(28)
Besides, the coefficients D1 and D2 must meet the con-
ditions
D1 +D2 = 0 (29)
and
D1(ω +
√
q)−D2(ω +
√
q) = 0 , (30)
that follow from (15) at C = 0 and (20). The three-
equation system ((28)-(30)) for the two unknowns D1 and
D2 is the over determined system and can be fulfilled only
for certain values of ω and
√
q. It is just the equalities
follow: from Eq.(30) by condition (29)
ω = 0 (31)
and from Eq.(28) √
q = nπ . (32)
Thus, it appears that the mode with the zero disturbance
of the total current is possible only at some assigned val-
ues of the parameter
√
q (
√
q = nπ).
3b. The potential case, Pierce’s instability
In this case, the electric field in formula (21) is determined
by expression (26), so that after integration of Eq.(26) and
fulfillment of conditions (21) we obtain that the potential
disturbance is equal to
φ̃(ζ) = D1
√
q
ω +
√
q
(ei(ω+
√
q)ζ − 1)−
−D2
√
q
ω −√
q
(ei(ω−√
q)ζ − 1)− C
ω2
ω2 − q
ζ (33)
and the equation for the coefficients takes the form:
D1
√
q
ω +
√
q
(ei(ω+
√
q) − 1)−
−D2
√
q
ω −√
q
(ei(ω−√
q) − 1)− C
ω2
ω2 − q
= 0 . (34)
Equation (34), considered together with Eqs.(19) and
(20), leads to the dispersion equation (DE):
(ω −√
q)2(ei(ω+
√
q) − 1)− (ω +
√
q)2 ×
×(ei(ω−√
q) − 1) + 2i
ω2
√
q
ω2 − q = 0 . (35)
It is evident from Eq.(35) that in the case under consid-
eration, oscillations with the frequency
ω = ±√
q (36)
can be excited in the diode. ω is the complex frequency,
i.e.,
ω = P + iQ (37)
163
where P is the oscillation frequency, and Q is the incre-
ment at Q > 0 and is the decrement at Q < 0. From
(36) and (37) it follows that the solution (36) presents
steady-state sinusoidal oscillations with the frequency
P = ±√
q (38)
and the increment
Q = 0 . (39)
The dispersion equation (35)can be written as:
eβ [(K2 − β2) sin(K) + 2Kβ cos(K))]−
−2Kβ +
β2
K
(K2 + β2) = 0 , (40)
where
K =
√
q (41)
and
β = iω = −Q+ iP . (42)
The solution of Eq.(40) is presented in
Figs.2,a,b; 3a,b by two families of curves.
0 5 10 15 20 25
-20
-15
-10
-5
0
Q(k)
k
a
0 5 10 15 20 25
-20
-15
-10
-5
0
5
10
15
20 P(k)
k
b
Fig.2. The solution family A of equation (33) for the
case of potential C: top - a) decrement, bottom - b)
oscillation frequency
The first family of curves (Figs.2,a and 2,b) is com-
posed of nearly periodic structures having the points
K = 2nπ as their centers. Each structure consists of
two curves for Q tends to the −∞ and having the vertical
straight lines K1 = 2(n − 1)π and K2 = (2n − 1)π as
their asymptotes. These curves cross at point K = 2nπ
at Q = 0. To the left of the point, one of the curves
becomes positive, so the process, corresponding to it, be-
comes unstable (Q > 0). These curves are in accordance
with P = 0 (Fig.2); so they give the decrement or the in-
crement of the aperiodic process. To the right of the point
K = 2nπ, the solution for β becomes complex-conjugate,
the curves for Q merge and give the increment of the ex-
cited periodical process. It can be seen from Fig.2 that to
the right of point K = 2nπ, apart from the solution P = 0
that relates to the neighboring structure with the center
at K = 2(n+1)π, two solutions appear for the frequency
P , which characterize the arising periodical instability.
The solution of Eq.(40) at K = 2nπ+α, |α| << 1 has
the form :
β =
nπ
3
− α± (−α(6 + (1 +
2
nπ
)α))1/2 . (43)
Since by definition we have β = −Q + iP , it is evident
that to the left of the point K = 2nπ, i.e., at α < 0,
there are two solutions for Q: one solution is positive,
and the other is negative. This is in agreement with the
above given treatment and with Fig.3,a. For the right
neighborhood of point K = 2nπ, i.e., at α > 0, formula
(43) gives two complex-conjugate solutions. This means
that in this neighborhood two new solutions appear for
P : one is positive, and the other is negative (Fig.2,b).
Of special interest is the solution of Eq.(40) for the re-
gion 0 < K < 2π. Fig.2,a suggests the conclusion that
in this region only aperiodical process (P = 0) is possi-
ble; and Fig.2b shows that this process splits at K < π,
since Q < 0, and increases at K > 0, since Q > 0. This
is just the Pierce instability. Figs. 3,a and 3,b show
the straight lines for Q = 0 and P = ±K, which rep-
resent a stable current transport through the diode. In
this case the current amplitude changes with time har-
monically at the dimensionless frequency P = K. Be-
sides, Figs. 3,a and 3,b show the curves of one more fam-
ily. They characterize the damped oscillations (Q < 0,
P ̸= 0); so, these oscillations cannot be excited in the
diode. Figs. 3,a,b show the straight lines for Q = 0 and
P = ±K that correspond to the solutions (38) and (39).
0 5 10 15 20 25
-16
-14
-12
-10
-8
-6
-4
-2
0
k
Q(k)
a
0 5 10 15 20 25
-40
-30
-20
-10
0
10
20
30
40
P(k)
k
b
Fig.3. Solution family B of equations (40) for the
case of the potential C: top- a) decrement, bottom- b)
oscillation frequency
164
3c. The eddy case
In this case the constant C must be excluded from ex-
pressions (17) and (31) for the electric field. And yet,
the terms comprising this constant should be taken into
account in (15) and (16) for ñ and ṽ , and also, in (19)
and (20); so, as the final result, the expressions for Ẽ
contain the terms with C. Having done all the things, as
suggested above, we obtain:
Ẽ(ζ) = −D1(i
√
q)ei(ω+
√
q)ζ +D2(i
√
q)ei(ω−√
q)ζ +
+
q
ω2 − q
C ,
(44)
φ̃(ζ) = D1
√
q
ω +
√
q
(ei(ω+
√
q)ζ − 1)−
D2
√
q
ω −√
q
(ei(ω−√
q)ζ − 1)− q
ω2 − q
Cζ .
(45)
The condition φ̃(1) = 0 leads to the equation:
D1(ω −√
q)(ei(ω+
√
q − 1)−
−D2(ω +
√
q)(ei(ω−√
q) − 1)− C
√
q = 0 . (46)
This equation, together with Eqs.(19) and (20), permits
the derivation of the dispersion equation for the case un-
der consideration. The DE has the form:
(ω −√
q)2(ei(ω+
√
q) − 1)− (ω +
√
q)2 ×
×(ei(ω−√
q) − 1) + 2i
√
q(ω2 − q) = 0 . (47)
(40) Using the definitions (41) and (42) for β and K, Eq.
(47) can be written as:
eβ [(K2 − β2) sin(K) + 2Kβ cos(K))]−
−2Kβ +K(K2 + β2) = 0 . (48)
a
b
Fig.4. a – Decrement (increment) of oscillations for the
case of eddy C; b – Oscillation frequency for the case of
eddy C
The analysis and the plots in Figs. 4,a, 4,b show that
Eq.(40) has no unstable growing solutions. This means
that in the case, where C is the eddy constant, the diode
is in a steady state.
3d. The mixed case. Pierce’s instability
modification
In this case, the constant C is equal to
C = C1 + C2 , (49)
where C1 and C2 are, respectively, the potential and eddy
parts of the constant C. Using the parameter η that de-
scribes the contribution, which the potential constituent
gives to C, formula (49) can be written in the following
form:
C1 = ηC . (50)
Finding the corresponding potential and carrying out, as
before, the procedure of deriving the dispersion equation,
we obtain the equation, which differs from Eqs.(40) and
(48) by the last component. It has the form
1
K
[(β2 +K2η −K2](β2 +K2) . (51)
It is obvious that the DE derived at η=1 goes over into
Eq.(40), while at η=0 it changes to Eq.(48).
The decrease in the parameter η provides the transi-
tion from DE (40), which gives rise to Pierce’s instabil-
ity, to DE (48), at which a steady electron flow passage
through the diode is realized.
The evolution of solution of the DE with component
(51) is presented in Figs.5,a-f. At 0 < η < 1, a new so-
lution appears, which has a vertical asymptote K = 0
and crosses the abscissa axis in the interval 0 < K < π,
thereby reducing the region of electron flow stability in
the diode. At η → 1 , this region extends up to K = π.
From formulae (48),(51) at Q=0 we have the relation-
ship:
η = 1− Sin(G)/G , (52)
where G is the electron flow stability boundary.
The graph of the parameter G against η, obtained
from equality (52), is presented in Fig.6, which shows the
extension of the electron flow stability region at a rela-
tively decreased role of the external electromagnetic ac-
tion, i.e., the parameter η. The value of G for this region
is equal to the highest value of the parameter K, start-
ing from which Q becomes positive. It can be seen that
Pierce’s parameter K reaches the π value only at η=1,
i.e., in the absence of external electromagnetic action.
Physically, the decrease of the electron flow stability
region with an increase in the external eddy action is due
to the fact that the amplitude of the velocity disturbance
(formula (15)) grows, whereas the stabilizing effect due
to maintaining electrode potential disturbances at a zero
level (formula (26)) is attributed only to the potential
disturbance C1.
165
a) b) c)
d) e) f)
Fig.5. Increments (decrements) of aperiodical oscillations in the diode: a) η=0; b) η=0.5; c) η=0.881;
d) η=0.9085; e) η=0.95; f) η=1.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Η
G
HΗ
L
Fig.6. Pierce’s parameter G as a function of η
4. CONCLUSIONS
The paper has been concerned with the stability of
the electron flow passing through the diode with a
compensated space charge under an external electro-
magnetic action.
Classification of electron flow states in the diode
has been developed in accordance with the magni-
tude (zero or nonzero) and nature (potential, eddy
or mixed) of the electric field, which arises at the in-
put electrode from the diode gap-side, and is assigned
by the constant C. Consideration has been given to
the cases, where the electron flow passage takes place
without total current disturbances (Pierce’s prob-
lem). Corresponding dispersion equations have been
derived and solved for the case, where the constant
C is potential. It has been demonstrated that in this
case a stable electron flow passage through the diode
takes place until the Pierce’s parameter equal to π
is reached. As soon as the Pierce parameter exceeds
the π value, the Pierce’s instability takes place.
In the states with the eddy electric field at the
diode input, there is no instability at any values of
the Pierce parameter.
It has been shown that for the case, where the
constant C is of mixed type, there appears a new so-
lution, which modifies the Pierce instability and re-
duces the electron flow stability region in the diode.
With a decreasing contribution from the eddy
constituent of the constant C, arising in the diode due
to the external electromagnetic action, the stability
region extends up to the Pierce parameter equal to π.
This means, in particular, that in the experimental
conditions even an insignificant influence of the ex-
ternal electromagnetic signal can cause an essential
reduction in the limiting density of the electron flow
passing through the diode.
166
ACKNOWLEDGEMENTS
The authors are thankful to V. I.Karas’ and
V. I.Maslov for their interest in the work and useful
comments.
References
1. V.R.Bursian, V.I. Pavlov. About a particular
case of the space charge effect on the electron flow
passage in a vacuum. //Zhurnal Rus. Fiz.-Khim.
Obshchestva. 1923, v. 55, p. 71-80 (In Russian).
2. J. Pierce. Limiting stable current in electron
beams in the presence of ions // J.Appl. Phys.
1944, v. 15, p. 721.
ÌÎÄÈÔÈÊÀÖÈß ÍÅÓÑÒÎÉ×ÈÂÎÑÒÈ ÏÈÐÑÀ ÏÐÈ ÂÍÅØÍÅÌ
ÝËÅÊÒÐÎÌÀÃÍÈÒÍÎÌ ÂËÈßÍÈÈ ÍÀ ÝËÅÊÒÐÎÍÍÛÉ ÏÎÒÎÊ Â ÄÈÎÄÅ
À.Ã.Ìåëåæèê, À.Â.Ïàùåíêî, Ñ.Ñ.Ðîìàíîâ, È.Ì.Øàïîâàë
Íà îñíîâå ðåøåíèÿ ñèñòåìû ãèäðîäèíàìè÷åñêèõ óðàâíåíèé è óðàâíåíèé Ìàêñâåëëà â ïåðâîì ïðè-
áëèæåíèè èçó÷åíà óñòîé÷èâîñòü ïðîõîæäåíèÿ ýëåêòðîííîãî ïîòîêà ÷åðåç äèîä ñ êîìïåíñèðîâàííûì
îáú¼ìíûì çàðÿäîì. Ïðîâåäåíà êëàññèôèêàöèÿ ðåøåíèé óðàâíåíèé â ñîîòâåòñòâèè ñ âåëè÷èíîé è õàðàê-
òåðîì (ïîòåíöèàëüíîå èëè âèõðåâîå) ýëåêòðè÷åñêîãî ïîëÿ, âîçíèêàþùåãî íà êàòîäå äèîäà. Íàéäåíû
îáëàñòè óñòîé÷èâîñòè ýòèõ ñîñòîÿíèé. Ïîêàçàíî, ÷òî íàëè÷èå âíåøíåãî ýëåêòðîìàãíèòíîãî âîçäåé-
ñòâèÿ ñóæàåò îáëàñòü óñòîé÷èâîñòè ïî ñðàâíåíèþ ñ ïðåäñêàçàííîé Ïèðñîì.
ÌÎÄÈÔIÊÀÖIß ÍÅÑÒIÉÊÎÑÒI ÏIÐÑÀ ÏÐÈ ÇÎÂÍIØÍÜÎÌÓ
ÅËÅÊÒÐÎÌÀÃÍIÒÍÎÌÓ ÂÏËÈÂI ÍÀ ÅËÅÊÒÐÎÍÍÈÉ ÏÎÒIÊ Ó ÄIÎÄI
Î.Ã.Ìåëåæèê, À.Â.Ïàùåíêî, Ñ.Ñ.Ðîìàíîâ, I.Ì.Øàïîâàë
Íà îñíîâi ðîçâ'ÿçàííÿ ñèñòåìè ïåðøèõ íàáëèæåíü ãiäðîäèíàìi÷íèõ ðiâíÿíü òà ðiâíÿíü Ìàêñâåëà âèâ÷å-
íà ñòiéêiñòü ïðîõîäæåííÿ åëåêòðîííîãî ïîòîêó ÷åðåç äiîä ç êîìïåíñîâàíèì îá'¹ìíèì çàðÿäîì. Ïðîâå-
äåíà êëàñèôiêàöiÿ ðiøåíü ðiâíÿíü ó âiäïîâiäíîñòi ç âåëè÷èíîþ òà õàðàêòåðîì (ïîòåíöiéíå àáî âèõðîâå)
åëåêòðè÷íîãî ïîëÿ, ùî âèíèê๠íà êàòîäi äiîäà. Íàéäåíî îáëàñòi ñòiéêîñòi öèõ ñòàíiâ. Ïîêàçàíî, ùî
íàÿâíiñòü çîâíiøíüîãî åëåêòðîìàãíiòíîãî âïëèâó çâóæó¹ îáëàñòü ñòiéêîñòi ïîðiâíÿíî ç ïåðåäáà÷åíîþ
Ïiðñîì.
167
|
| id | nasplib_isofts_kiev_ua-123456789-80479 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:06:44Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Melezhik, O.G. Pashchenko, A.V. Romanov, S.S. Shapoval, I.M. 2015-04-18T13:24:48Z 2015-04-18T13:24:48Z 2014 Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action / O.G. Melezhik, A.V. Pashchenko, S.S. Romanov, I.M. Shapoval // Вопросы атомной науки и техники. — 2014. — № 5. — С. 161-167. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/80479 The stability of electron fiow passage through the diode with a neutralized space charge has been investigated on the basis of the first-order approximation for hydrodynamic and Maxwell equations. Classification of solutions to the equations was developed in accordance with the magnitude and nature (potential or eddy) of the electric field arising at the diode cathode. The stability regions for these states have been found. It is shown that the presence of external electromagnetic action makes the stability region narrower than that predicted by Pierce. На основе решения системы гидродинамических уравнений и уравнений Максвелла в первом приближении изучена устойчивость прохождения электронного потока через диод с компенсированным объемным зарядом. Проведена классификация решений уравнений в соответствии с величиной и характером (потенциальное или вихревое) электрического поля, возникающего на катоде диода. Найдены области устойчивости этих состояний. Показано, что наличие внешнего электромагнитного воздействия сужает область устойчивости по сравнению с предсказанной Пирсом. На основi розв'язання системи перших наближень гiдродинамiчних рiвнянь та рiвнянь Максвела вивчена стiйкiсть проходження електронного потоку через дiод з компенсованим об'ємним зарядом. Проведена класифiкацiя рiшень рiвнянь у вiдповiдностi з величиною та характером (потенцiйне або вихрове) електричного поля, що виникає на катодi дiода. Найдено областi стiйкостi цих станiв. Показано, що наявнiсть зовнiшнього електромагнiтного впливу звужує область стiйкостi порiвняно з передбаченою Пiрсом. The authors are thankful to V. I. Karas’ and
 V. I. Maslov for their interest in the work and useful
 comments. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Теория и техника ускорения частиц Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action Модификация неустойчивости пирса при внешнем электромагнитном влиянии на электронный поток в диоде Модифiкацiя нестiйкостi пiрса при зовнiшньому електромагнiтному впливi на електронний потiк у дiодi Article published earlier |
| spellingShingle | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action Melezhik, O.G. Pashchenko, A.V. Romanov, S.S. Shapoval, I.M. Теория и техника ускорения частиц |
| title | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| title_alt | Модификация неустойчивости пирса при внешнем электромагнитном влиянии на электронный поток в диоде Модифiкацiя нестiйкостi пiрса при зовнiшньому електромагнiтному впливi на електронний потiк у дiодi |
| title_full | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| title_fullStr | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| title_full_unstemmed | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| title_short | Modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| title_sort | modification of the pierce instability of the electron flow in a diode with an external electromagnetic action |
| topic | Теория и техника ускорения частиц |
| topic_facet | Теория и техника ускорения частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80479 |
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