To the distribution of the dispersion equation roots for the parametric Cherenkov radiation
Distribution of the roots of dispersion equation of the parametric Cherenkov radiation (PCR) excited by uniformly moving charged particle in an ideally conducting metal waveguide filled with a spatially periodic layered dielectric is studied. The study both in long wavelengths range and in the range...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2020
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| Cite this: | To the distribution of the dispersion equation roots for the parametric cherenkov radiation / V.I. Tkachenko, I.V. Tkachenko, A.P. Tolstoluzhsky, S.N. Khizhnya // Problems of atomic science and tecnology. — 2020. — № 6. — С. 36-40. — Бібліогр.: 11 назв. — англ. |
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| author | Tkachenko, V.I. Tkachenko, I.V. Tolstoluzhsky, A.P. Khizhnyak, S.N. |
| author_facet | Tkachenko, V.I. Tkachenko, I.V. Tolstoluzhsky, A.P. Khizhnyak, S.N. |
| citation_txt | To the distribution of the dispersion equation roots for the parametric cherenkov radiation / V.I. Tkachenko, I.V. Tkachenko, A.P. Tolstoluzhsky, S.N. Khizhnya // Problems of atomic science and tecnology. — 2020. — № 6. — С. 36-40. — Бібліогр.: 11 назв. — англ. |
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| description | Distribution of the roots of dispersion equation of the parametric Cherenkov radiation (PCR) excited by uniformly moving charged particle in an ideally conducting metal waveguide filled with a spatially periodic layered dielectric is studied. The study both in long wavelengths range and in the range of wavelengths comparable and smaller than the period of non-uniformity is presented. It is shown that in the case of long wavelengths the dependence of the frequency of the excited PCR on the particle velocity is monotonic. For the case of wavelengths comparable and shorter than the non-uniformity period the spectrum of PCR is more complicated. Its characteristic features are determined.
Досліджується розподіл коренів дисперсійного рівняння параметричного черенковського випромінювання (ПЧВ), збуджуваного зарядженою частинкою, що рівномірно рухається в ідеально провідному металевому хвилеводі, заповненому просторово-періодичним шаруватим діелектриком як в області довгих хвиль, так і в діапазоні довжин хвиль, порівнянних і менших, ніж період неоднорідності. Показано, що у випадку довгих хвиль залежність частоти збуджуваного ПЧВ від швидкості частинки є монотонною. У випадку довжин хвиль, порівнянних і більш коротких, ніж період неоднорідності, спектр ПЧВ більш складний. Визначені його характерні особливості.
Исследуется распределение корней дисперсионного уравнения параметрического черенковского излучения (ПЧИ), возбуждаемого равномерно движущейся заряженной частицей в идеально проводящем металлическом волноводе, заполненном пространственно-периодическим слоистым диэлектриком как в области длинных волн, так и в диапазоне длин волн, сопоставимых и меньших, чем период неоднородности. Показано, что в случае длинных волн зависимость частоты возбуждаемого ПЧИ от скорости частицы является монотонной. В случае длин волн, сравнимых и более коротких, чем период неоднородности, спектр ПЧИ более сложный. Определены его характерные особенности.
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ISSN 1562-6016. ВАНТ. 2020. №6(130)
36 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2020, № 6. Series: Plasma Physics (26), p. 36-40.
https://doi.org/10.46813/2020-130-036
TO THE DISTRIBUTION OF THE DISPERSION EQUATION ROOTS
FOR THE PARAMETRIC CHERENKOV RADIATION
V.I. Tkachenko
1,2
, I.V. Tkachenko
1
, A.P. Tolstoluzhsky
1
, S.N. Khizhnyak
1
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: tkachenko@kipt.kharkov.ua
Distribution of the roots of dispersion equation of the parametric Cherenkov radiation (PCR) excited by
uniformly moving charged particle in an ideally conducting metal waveguide filled with a spatially periodic layered
dielectric is studied. The study both in long wavelengths range and in the range of wavelengths comparable and
smaller than the period of non-uniformity is presented. It is shown that in the case of long wavelengths the
dependence of the frequency of the excited PCR on the particle velocity is monotonic. For the case of wavelengths
comparable and shorter than the non-uniformity period the spectrum of PCR is more complicated. Its characteristic
features are determined.
PACS: 41.60.-m
INTRODUCTION
The effect of electromagnetic wave radiation during
the motion of a charged particle at a constant velocity in
the metal waveguide filled with a spatially periodic
layered dielectric was first studied in the paper [1]. The
authors called this radiation as parametric Cherenkov
radiation (PCR). The results of the study of this
radiation are presented in voluminous literature (see, for
example, [2–11] and the literature cited therein).
Studying different aspects of the PCR, various authors
gave it different names – resonant transition radiation,
X-ray transition radiation, parametric X-ray radiation
and others. Since the general condition for radiation
occurrence is the periodic variation of parameters, we
will follow the name PCR.
Since there is no analytical solution of the dispersion
equation for PCR in an interesting for practical
applications wavelengths (comparable or shorter than
the non-uniformity period), the analytical expressions to
calculate the generated fields and radiation power of the
assumed microwave generator can not be evaluated.
In this work the analytical solution of the dispersion
equation for the PCR in a waveguide loaded with
layered dielectric in the wavelength range of
comparable and smaller than the non-uniformity period
along with the case of long wavelength range is
obtained and compared with the results of numerical
calculations presented in [11].
ANALYTICAL STUDY OF SOLUTIONS OF
THE DISPERSION EQUATION OF PCR
Let us consider layered medium involved alternating
layers of two homogeneous and isotropic dielectrics. In
each longitudinal period L a b the first layer
0 z a has dielectric and magnetic permeabilities
1 1, , the second layer a z b has permeabilities
2 2, . Dielectrics are in a cylindrical perfectly
conductive metal waveguide.
The frequencies of the electromagnetic radiation of a
charged particle, moving with velocity v in such
medium, are determined by the roots of the dispersion
equation [1]:
cos( / ) ( ) 0kL g , (1)
where
1 2 2 1
1 2 1 2
2 1 1 2
( )
1
cos( )cos( ) sin( )sin( )
2
g
p p
p a p b p a p b
p p
and the following designations are introduced
2 2
1 1 1p k k is longitudinal wave number in the
first layer,
2 2
2 2 2p k k is longitudinal wave
number in the second layer, k=/c, k=n/R is
transverse wave number, R is waveguide radius, n is
n-root of the zero-order Bessel function, =/c is
dimensionless velocity of the charged particle.
Since the equation (1) is a dispersion equation of the
radiation arising in the layered dielectric (medium with
periodically varying parameters), the frequencies
determined by the roots of this equation correspond to
the waves propagating in such layered medium.
It can be seen that the solutions of the dispersion
equation should satisfy the condition: 1 ( ) 1g .
It is easy to show that at
1 2 1 2, the
value 1 2 2 1
2 1 1 2
1
1
2
p p
p p
and the equation (1) takes
the form
cos( / ) cos( ) 0L v pL . (2)
The solution of this equation gives the condition of
the Cherenkov radiation of the charged particle at a
uniform motion through the dielectric waveguide filled
with homogeneous dielectric
,zk v where
2 2
z p k kk . (3)
Since we consider waves propagating in the
waveguide with dielectric disks, the function ( )g is a
real function of frequency and can be represented as the
sum of two cosines
1 2 2 1( ) cos( ) (1 ) cos( )g p a p b p b p a , (4)
https://doi.org/10.46813/2020-130-036
ISSN 1562-6016. ВАНТ. 2020. №6(130) 37
where
2
1 2 2 1
1 2 2 1
( )
4
p p
p p
.
In the general case, the quantities p1 and p2 are
incommensurable quantities in the space of wave
numbers k. The function in the expression (4) depends
in a complicated way on the parameters of the dielectric
waveguide and frequency. Therefore, as a result, the
sum of terms in the right-hand side of (4) cannot have a
common period and to study analytically the solutions
of the dispersion equation (1) is not possible.
But only in some special cases it may be done. For
example, in the case of sufficiently thin disks, when the
conditions
1 1p a ,
2 1p b as well as / 1kL
(long-wave approximation) are satisfied, analytical
solutions of equation (1) can be found.
In this case, taking into account the quadratic terms
2 2 2
1 2( ) , ( ) , ( / )p a p b kL , the function ( )g is identical
to the expression
2 2 2 2
2 2 2 2 1 2 2 1
1 2
1 2
( )1
( ) 1
2
a b p p
g p a p b
and the dispersion equation (1) has a form [1]:
2
2 2
2
( )p
p
k
k k
, (5)
where 1 1 2 2
2 2 1 1
( )
( )
a b
a b
, 1 1 2 2( )
( )
p
a b
a b
.
The solution of equation (5) explicitly establishes
the correspondence between the wave number k and
particle velocity at fixed values of the parameters of
the medium and the diameter of the waveguide:
2 2 2
,
1
p
pp
k k
k
k k
. (6)
As it can be seen from (6), there is a monotonic
dependence of the wave number k of the excited PCR
on the particle velocity in the long-wave
approximation. In this case, PCR occurs when the
particle velocity is exceeded the threshold velocity
value min 1/ . Depending on the degree of
filling by the layers with a dielectric and the width of
the layers, the value lies within the range 1 2.
Fig. 1 shows the dependences of k()·L on for two
values of the spatial period of the disks
L – L=2.1∙10
-1
and 2.1·10
-2
cm.
It should be noted, that as the layer width decreases
the usability condition for dispersion equation (5)
expands to the high-frequency area.
On the other hand, for sufficiently large values of
2 2
i i k k ( 1,2i ), up to quadratic terms
2
2
i i
k
k
, the dispersion equation (1) can be represented
as:
1 1 2 2
2 2 1 1
cos( / ) cos (1 )
cos ,
a b kL
b a
k
k
(7)
where
2
1 1 2 2 2 1
1 1 2 2 2 1
( )1
4
.
In this approximation the function ( )g is a
periodic one with two periods in the wave number k :
1 1 2 2 2 2 1 1
2 2
,k k
a b b a
.
Fig. 1. The dependence of ( )k L on particle
velocity . For 1=2=1, 1=1.96, 2=4, k=1/R,
R=3 cm; a – L=2.1·10
-1
cm at a=1·10
-1
cm,
b=1.1·10
-1
cm; b – L=2.1·10
-2
cm at a=1·10
-2
cm,
b=1.1·10
-2
cm
In this case, the term with the wave number k
corresponds to the wavelength
1 1 2 2a b ,
and the term with the wave number k
corresponds to
the wavelength
2 2 1 1b a .
Thus, in the case under consideration the dispersion
equation of the PCR (1) is a periodic function on k with
three spatial periods +, ,L=L/.
To explain the phenomenon of superposition of three
waves, we will find a solution of equation (7).
From the equation (7) it follows that for sufficiently
large values k it is satisfied at some fixed values
ik ,
where
ik are the roots of equation (7). Since the
quantity does not depend on k , then the possible
roots of equation (7) are those for which the following
identities are valid:
1 1 2 2
2 2 1 1
cos( / ) cos
cos
ii
i
a b
a
L kk
b k
, (8)
Thus, the solutions of equation (7) are satisfied at
the following relations between the arguments for all
equalities (8):
1 1 2 2
2 2 1 1 2
/ 2 2
i
i ia b k
b a k
j m
n
k L
, (9)
where , ,j m n are arbitrary integers.
Therefore, it is not difficult to find a relation
determining the values of a certain sequence of
solutions of the dispersion equation satisfying the
conditions (8):
1 1( , , , , )N C a bk N , (10)
where
2
1
2
1( , , , , )
/ b
C a b
L
, ( )N n j .
38 ISSN 1562-6016. ВАНТ. 2020. №6(130)
To verify the validity of the obtained solutions, it is
necessary to substitute the values of the roots from (10)
into the original dispersion equation (1) and find by
numerical analysis the values of N at which
Nk is the
solutions of the original dispersion equation.
A numerical study of solutions of the PCR
dispersion equation and comparison of the analytical
solutions (9) of the dispersion equation (1) with the
results of its numerical analysis were carried out below.
NUMERICAL STUDY OF SOLUTIONS OF
THE DISPERSION EQUATION OF PCR
Graphs of the dependence of the function
( ) cos( / ) ( ) 0D k kL g on k and its spectrum
are presented by curves of Fig. 2, 7 for non-magnetic
media 1=2=1, 1=1.96, 2=4, a=1·10
-1
cm,
b=1.10·10
-1
cm, R=3 cm for two values of the particle
velocity =0.7 (see Fig. 2), =0.58333 (Fig. 7). They
show, that the dependence ( )D k is determined, as one
would expect, by the interference (beating) of three
cosine waves (see (1), (7)) at a wavelength /L L
equal to the characteristic length of the cos( / )kL
term variation and the wavelengths +, -,
corresponding to the characteristic lengths of g()
variation.
a
b
Fig. 2. The dependence ( ) cos( / ) ( )D k kL g
on k (a) and its spectrum SpD on =2/k (b) for
=0.7, k=1/R=2.404/3=0.80133 cm
-1
As can be seen from these diagrams, the
characteristic values of wavelengths L, +, for the
excited PCR found analytically coincide with sufficient
accuracy with their values obtained by numerical
analysis – L, = 0.3 cm, + = 0.36 cm, = 0.08 cm.
Further, we assume that all spatial variables are
measured in cm and the wave numbers k – cm
-1
.
To determine the spectral characteristics of the
radiation resulting from the interference of three waves
L, +, , the distribution of the roots of the dispersion
equation for the particle velocity =0.7 was numerically
studied.
Fig. 3 shows the dependence of the sequence of
wave number values
ik for the roots of the dispersion
equation D(ki)=0 as a function of the root number i .
Fig. 3. The dependence of the sequence of wave number
values ki/2 for the roots of the dispersion equation
D(ki)=0 of layered dielectric on the root number i for
=0.7
From Fig. 3 it is shown, that there is a violation of
the monotonicity of following of the wave numbers
values
ik , which consists in a periodic transition from
one straight line to another located below.
To clarify the nature of this nonmonotonicity the
dependence of the difference of neighboring roots of the
dispersion equation ( ) cos( / ) ( )D k kL g on the
values of these roots is presented in Fig. 4.
Fig. 4. The dependences of the difference
1( ) / 2i i ik k k of neighboring roots of the
dispersion equation D(ki)=0 of a layered dielectric on
the values of these roots 2ik for =0.7
As it can be seen from this figure, a strictly periodic
alternation of the difference of the roots of the equation
(1) is observed. Besides, as the numerical analysis
shows, there is a peculiarity in the values of the roots of
this equation – almost coincidence of some two
neighboring roots at a level of order 4~ 10ik with
values k near k36/2=50, k72/2=100 and k108/2=150.
With the same periodicity k/2=50 the remaining roots
of the dispersion equation alternate. At the same time,
as frequency increases, character of root following is
preserved.
Fig. 5 shows the dependence ( )ND k as a function of
the number N for the particle velocity 0.7
obtained from the formula (2).
As can be seen from this figure, the values Nk are
the roots of the dispersion equation (1) for the values
N= 8, 16, 24, …. At that, 2 2 0.0/ 8L b ,
ISSN 1562-6016. ВАНТ. 2020. №6(130) 39
C(a, b, , 1, 1)=12.5. In this case analytical roots
periodicity kanalit = 8C(a, b, , 1, 1)/2 = 50, root
periodicity obtained numerically: knum 49.99995
Fig. 5. The dependence D(kN) as a function of the
number N for =0.7
Thus, the values of the roots of the dispersion
equation (1) obtained by the formula (10)
kN= C(a, b, , 1, 1)·N correspond with a sufficient
degree of accuracy to the values of the roots obtained by
numerically solving this equation: kN=8 = k36,
kN=16 = k72., kN=24 = k108. Note, that with increasing the
number N the accuracy of calculating the expression
(10) increases and the relation between the roots follows
exactly the relations for periods L, +, :
1 2
2 1
p a p b
p b p a
,
2 1
/ LkL
p b p a
, 1 2
/L
p a p b
kL
.
This is confirmed by the data given in Fig. 6.
Fig. 6. The dependence of relations +/=4.5,
L/=3.75 and +/L=1.2 upon roots ki/2 for =0.7
From the relations for periods L, +, it follows
that, it is possible, for example, to superpose the periods
L and + by varying the particle velocities. So, at
kL>>1 and the particle velocity
1 2
=0.5833333
kL
p a p b
,
these periods are equal L=+.
It follows from Fig. 7 that wavelengths L, +
coincide ‒ L=+=0.36, =0.08.
As it follows from Fig. 8, there is a strictly periodic
alternation of the difference of roots of the equation (1).
It should be added that the numerical analysis shows
almost of some two neighboring roots with an accuracy
of the order ki 10
-3
.
The dependence D(kN) as a function of the number N
for a particle velocity =0.58333 obtained by the
formula (10) is presented in Fig. 9.
As it can be seen from this figure the values
Nk are
the roots of the dispersion equation (1) for the values
N=7, 14, 21,… At that, 2 2 0.1/ 4L b ,
C(a, b, , 1, 1)=50/7. In this case analytical roots
periodicity
1 114 ( , , , , 2 50)analit C a b of root
is equal to periodicity obtained numerically: knum
49.99995.
a
b
Fig. 7. The dependence ( ) cos( / ) cos( )D k kL
on k=/c (a) and its spectrum SpD (b) on =2/k for
=0.583333, k=1/R
Fig. 8. The dependence of difference
1( ) / 2i i ik k k of neighboring roots of the
dispersion equation D(ki)=0 on the values of these roots
2ik for ·=0.58333
Thus, the values of roots of the dispersion equation
(1) obtained by the formula (10) kN = C(a, b, , 1, 1)·N
correspond with sufficient degree of accuracy to the
values of the roots obtained by numerically solving of
the equation (1): kN=7 = k36, kN=14 = k72, kN=21 = k108. At
that, the relation between the roots exactly follows the
relations between the periods of the waves L, +, :
+/=4.5, +/L =1.
Calculations show that with increasing frequency, the
character of following of the roots is preserved.
Fig. 9. The dependence D(kN) as a function of number N
for·=/c=0.58333
40 ISSN 1562-6016. ВАНТ. 2020. №6(130)
CONCLUSIONS
Thus, as a result of the analytical and numerical study
of the dispersion equation of the PCR of a charged
particle moving along the axis of an ideally conducting
metal waveguide filled with a spatially periodic layered
dielectric, the following conclusions can be drawn.
1. In the range of wavelengths greater than non-
uniformity period (kL/<<1) the frequency of PCR
depends monotonically on the particle velocity. Under
condition kL/>>1, at fixed particle velocity , spectrum
of PCR is discrete. The dependence of values of the root
sequence of the dispersion equation on the root number is
non-monotonic. In this case, the difference of the roots of
dispersion equation alternate with a certain periods, the
values of these periods are determined numerically.
2. The analytical solution of the dispersion equation
of the PCR at 2 2
i i k k was obtained and the
periodic discrete nature of the spectrum caused by the
presence of three groups of wave numbers in the
dispersion equation (8) was revealed. The values of the
roots obtained analytically are in good coincidence with
the values of the roots determined numerically.
3. A peculiarity in the values of the neighboring roots
of the dispersion equation – the coincidence of some two
neighboring roots at the level of the order 3~ 10ik was
found.
ACKNOWLEDGMENTS
The authors express their gratitude to Professor
I.N. Onishchenko for fruitful discussions and useful
comments.
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(idei development) // Problems of Atomic Science and
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Article received 18.09.2020
О РАСПРЕДЕЛЕНИИ КОРНЕЙ ДИСПЕРСИОННОГО УРАВНЕНИЯ ПАРАМЕТРИЧЕСКОГО
ЧЕРЕНКОВСКОГО ИЗЛУЧЕНИЯ
В. И. Ткаченко, И.В. Ткаченко, А.П. Толстолужский, С.Н. Хижняк
Исследуется распределение корней дисперсионного уравнения параметрического черенковского
излучения (ПЧИ), возбуждаемого равномерно движущейся заряженной частицей в идеально проводящем
металлическом волноводе, заполненном пространственно-периодическим слоистым диэлектриком как в
области длинных волн, так и в диапазоне длин волн, сопоставимых и меньших, чем период неоднородности.
Показано, что в случае длинных волн зависимость частоты возбуждаемого ПЧИ от скорости частицы
является монотонной. В случае длин волн, сравнимых и более коротких, чем период неоднородности, спектр
ПЧИ более сложный. Определены его характерные особенности.
ПРО РОЗПОДІЛ КОРЕНІВ ДИСПЕРСІЙНОГО РІВНЯННЯ ПАРАМЕТРИЧНОГО
ЧЕРЕНКОВСЬКОГО ВИПРОМІНЮВАННЯ
В. І. Ткаченко, І.В. Ткаченко, О.П. Толстолужський, С.М. Хижняк
Досліджується розподіл коренів дисперсійного рівняння параметричного черенковського
випромінювання (ПЧВ), збуджуваного зарядженою частинкою, що рівномірно рухається в ідеально
провідному металевому хвилеводі, заповненому просторово-періодичним шаруватим діелектриком як в
області довгих хвиль, так і в діапазоні довжин хвиль, порівнянних і менших, ніж період неоднорідності.
Показано, що у випадку довгих хвиль залежність частоти збуджуваного ПЧВ від швидкості частинки є
монотонною. У випадку довжин хвиль, порівнянних і більш коротких, ніж період неоднорідності, спектр
ПЧВ більш складний. Визначені його характерні особливості.
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| id | nasplib_isofts_kiev_ua-123456789-194639 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:25:18Z |
| publishDate | 2020 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Tkachenko, V.I. Tkachenko, I.V. Tolstoluzhsky, A.P. Khizhnyak, S.N. 2023-11-28T10:34:36Z 2023-11-28T10:34:36Z 2020 To the distribution of the dispersion equation roots for the parametric cherenkov radiation / V.I. Tkachenko, I.V. Tkachenko, A.P. Tolstoluzhsky, S.N. Khizhnya // Problems of atomic science and tecnology. — 2020. — № 6. — С. 36-40. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/194639 Distribution of the roots of dispersion equation of the parametric Cherenkov radiation (PCR) excited by uniformly moving charged particle in an ideally conducting metal waveguide filled with a spatially periodic layered dielectric is studied. The study both in long wavelengths range and in the range of wavelengths comparable and smaller than the period of non-uniformity is presented. It is shown that in the case of long wavelengths the dependence of the frequency of the excited PCR on the particle velocity is monotonic. For the case of wavelengths comparable and shorter than the non-uniformity period the spectrum of PCR is more complicated. Its characteristic features are determined. Досліджується розподіл коренів дисперсійного рівняння параметричного черенковського випромінювання (ПЧВ), збуджуваного зарядженою частинкою, що рівномірно рухається в ідеально провідному металевому хвилеводі, заповненому просторово-періодичним шаруватим діелектриком як в області довгих хвиль, так і в діапазоні довжин хвиль, порівнянних і менших, ніж період неоднорідності. Показано, що у випадку довгих хвиль залежність частоти збуджуваного ПЧВ від швидкості частинки є монотонною. У випадку довжин хвиль, порівнянних і більш коротких, ніж період неоднорідності, спектр ПЧВ більш складний. Визначені його характерні особливості. Исследуется распределение корней дисперсионного уравнения параметрического черенковского излучения (ПЧИ), возбуждаемого равномерно движущейся заряженной частицей в идеально проводящем металлическом волноводе, заполненном пространственно-периодическим слоистым диэлектриком как в области длинных волн, так и в диапазоне длин волн, сопоставимых и меньших, чем период неоднородности. Показано, что в случае длинных волн зависимость частоты возбуждаемого ПЧИ от скорости частицы является монотонной. В случае длин волн, сравнимых и более коротких, чем период неоднородности, спектр ПЧИ более сложный. Определены его характерные особенности. The authors express their gratitude to Professor I.N. Onishchenko for fruitful discussions and useful comments. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics To the distribution of the dispersion equation roots for the parametric Cherenkov radiation Про розподіл коренів дисперсійного рівняння параметричного черенковського випромінювання О распределении корней дисперсионного уравнения параметрического черенковского излучения Article published earlier |
| spellingShingle | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation Tkachenko, V.I. Tkachenko, I.V. Tolstoluzhsky, A.P. Khizhnyak, S.N. Basic plasma physics |
| title | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation |
| title_alt | Про розподіл коренів дисперсійного рівняння параметричного черенковського випромінювання О распределении корней дисперсионного уравнения параметрического черенковского излучения |
| title_full | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation |
| title_fullStr | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation |
| title_full_unstemmed | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation |
| title_short | To the distribution of the dispersion equation roots for the parametric Cherenkov radiation |
| title_sort | to the distribution of the dispersion equation roots for the parametric cherenkov radiation |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/194639 |
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