Dispersion and radiating ability of relativistic electronic gas
In the paper the dispersion and conditions of formation of nonequilibrium radiation in the relativistic electronic gas are considered. For a case of a high-density electron bunch in the cw-approach and a wave-zone unharmonic oscillator the general kind of transfer plane-parallel front of nonequilib...
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| Cite this: | Dispersion and radiating ability of relativistic electronic gas / V.V. Porkhaev, N.V. Zavyalov, V.T. Punin, A.V. Telnov, Ju.A. Khokhlov // Вопросы атомной науки и техники. — 2004. — № 1. — С. 150-152. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859686950176292864 |
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| author | Porkhaev, V.V. Zavyalov, N.V. Punin, V.T. Telnov, A.V. Khokhlov, Ju.A. |
| author_facet | Porkhaev, V.V. Zavyalov, N.V. Punin, V.T. Telnov, A.V. Khokhlov, Ju.A. |
| citation_txt | Dispersion and radiating ability of relativistic electronic gas / V.V. Porkhaev, N.V. Zavyalov, V.T. Punin, A.V. Telnov, Ju.A. Khokhlov // Вопросы атомной науки и техники. — 2004. — № 1. — С. 150-152. — Бібліогр.: 8 назв. — англ. |
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| description | In the paper the dispersion and conditions of formation of nonequilibrium radiation in the relativistic electronic
gas are considered. For a case of a high-density electron bunch in the cw-approach and a wave-zone unharmonic oscillator the general kind of transfer plane-parallel front of nonequilibrium radiation equation is obtained.
Розглянуті дисперсія й умови формування нерівновагого випромінювання в релятивістському
електронному газі. Для випадку великої густини електронного згустку в квазістаціонарному наближенні в
хвильовій зоні ангармонічного осцилятора отримано загальний вид рівняння переносу плоскопараллельного
фронту нерівновагого випромінювання.
Рассмотрены дисперсия и условия формирования неравновесного излучения в релятивистском электронном газе. Для случая большой плотности электронного сгустка в квазистационарном приближении в волновой зоне ангармонического осциллятора получен общий вид уравнения переноса плоскопараллельного
фронта неравновесного излучения.
|
| first_indexed | 2025-11-30T22:58:51Z |
| format | Article |
| fulltext |
DISPERSION AND RADIATING ABILITY
OF RELATIVISTIC ELECTRONIC GAS
V.V. Porkhaev, N.V. Zavyalov, V.T. Punin, A.V. Telnov, Ju.A. Khokhlov
The federal state unitary enterprise “The Russian Federal Nuclear Center –
All-Russia scientific research institute of experimental physics”
Russia, 607190, Sarov, the Nizhniy Novgorod region, 37, Mira st.;
E-mail: otd4@expd.vniief.ru
In the paper the dispersion and conditions of formation of nonequilibrium radiation in the relativistic electronic
gas are considered. For a case of a high-density electron bunch in the cw-approach and a wave-zone unharmonic os-
cillator the general kind of transfer plane-parallel front of nonequilibrium radiation equation is obtained.
PACS 539.12
In FEL the question about superradiation of electron
bunch in a working range of frequencies (the stimulated
radiation of oscillators) is basic. Formation of a stimu-
lated radiation field probably takes place only in the
nonequilibrium system. Therefore it is necessary to de-
fine the mechanism of nonequilibrium radiation forma-
tion in the initially equilibrium system. As a rule, occur-
rence of stimulated (prime) radiation is considered, as a
casual process. In paper [1] the mechanism of resonant
amplification of slow-wave plasma fluctuations in the
relativistic electronic bunch due to the interaction with
the electromagnetic wave of the external source is theo-
retically investigated. It has been shown, that under con-
ditions of exact resonance in the magnetized electronic
bunch with a small density of volumetric charge there
can be resonant amplification of slow plasma waves. In
earlier work [2] the theoretical analysis of the response
of excited oscillators with randomly distributed phases
on casual electromagnetic perturbation has been carried
out. As a result the conclusion has been made, that the
electron bunch is unstable to the external electromagnet-
ic perturbation when the essential nonlinearity of the
equations of electron movement in a field of external
wave takes place. Thus, field fluctuations that give a
phase shift in the equation of electron movement can be
sources prime radiations.
In [3] the general view of the transfer equation of
wave package in boundless electronic gas in
monovigour approximation is obtained. The equations
received in the given paper, are fair for a case of a small
density of the medium consisting from statistically inde-
pendent diffusers. In work [4] the initial excitations of
active medium, which are stochastic fluctuations of den-
sity charge in the bunch, forming the fields of nonequi-
librium radiation are considered.
RADIATING INSTABILITY
OF ELECTRONIC GAS
Let the bunch of relativistic electrons goes along an
axis 0Z in periodic magnetic field Н (z). We shall set
electron’s power distribution in a bunch of function F
(γ), submitted on fig., where −γ the relativistic factor.
Everyone electron is source of bremsstrahlung which in-
tensity of its field is equal ( ) 2/12
0
2 32 HgDrE ieiBs γ= ,
where D is the period of magnetic field structure, g -
average volumetric electron’s concentration in bunch,
0H -intensity of magnetic field on axis 0Z, 22 mcere =
-classical radius of electron.
0 γ
F(γ )
I II
1
γ c
nj
nk
Еj
Еk
nj
nk
j
k
k→ j j→ k
ckj EE ω=−
γ k γ j
Function of power distribution in the electron bunch F (γ)
Let us consider the radiating instability in the bunch
with a small electron density when Bsp ω< <ω , where
pω is the frequency of plasma fluctuations of electronic
gas, Bsω is the frequency of bremsstrahlung.
In approach of the anharmonic oscillator the parame-
ters of the system being considered are constant during
oT ( oT is the period of electron fluctuations in the mag-
netic field). We shall define the characteristic size of the
area of radiating interaction.
If in the electron bunch the energy of the interchange
by quanta cω takes place then from the principle of
uncertainty ( π=∆∆ kl where l∆ is the size of wave
package, k∆ is the interval of wave numbers) it follows
that π=′∆ω ′∆ t , i.e.
)( can
c
ω
π=ω ′∆ ,
where a is the average distance between electrons,
)( cn ω is the factor of refraction of electronic gas at a
frequency cω .
So as ( )λ ′∆λ ′= 22
min ca it is necessary that
cc λ ′λ ′∆ω ′=ω ′∆ . Hence, at 2λ ′<a it follows that the
spectral range of power interchange channel is broaden-
ing up to λ ′∆λ ′ ~c . Then we obtain the area of electron
interaction by bremsstrahlung is the spherical layer with
radius 20 λ ′=r and thickness λ ′∆=δ , i.e. shall take a
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 1.
Series: Nuclear Physics Investigations (42), p.150-152. 150
note that secondary sources of radiation field and the
centers of dispersion are concentrated in volume
( )( ) 34 3
0
3
0 rrVs −δ+π= .
In system K ′ the electrons is not relativistic. Then
we shall consider the radiation field in isotropic ap-
proach, cω< <ω∆ and with fixed degree of polariza-
tion. Within the framework of the made assumptions, in
spherical coordinates which center is connected to the
leading center of i- electron, the change intensity of
bremsstrahlung from radius is described by equation
drrI
r
rdI )(2)(
+ρ−= ,
where ρ is factor of proportionality which characterizes
properties of environment and generally is function of r.
Let's define ρ for basic process in medium is scatter-
ing by free electrons of plainly-polarized wave own
bremsstrahlung. In a general view, for one-photon pro-
cesses ρ it’s defined, how ( ))()()()( rNrNrr III −σ=ρ
[6], where )(r σ is the cross-section of dispersion plain-
ly-polarized wave by free charges, )(rN I and )(rN II -
the volumetric concentration of electrons in areas I and
II (see fig. 1). In dipole approach ( сBs ω≈ω )
γ
γ∆πλ ′
π
=σ
c
c 2sin
2
3 2 , (1)
where cλ ′ is the most probable for distribution )(γF
wave-length of bremsstrahlung in system K ′ .
From equation (1) follows, that in monovigorous
electron bunch the cross-section of resonant interaction
of electrons with field own bremsstrahlung is smallest.
Therefore formation of stimulated field radiations in
electron’s bunch is impossible for small density of
bunch. For width of power distribution of electrons in a
bunch γ∆ at which 1~2sin cγγ∆π spectral cleanliness
of radiation λ ′∆λ ′ does not allow to receive coherent
(stimulated) radiation (coherence length
alcoh < <λ ′∆λ ′= 2 ). Absence of interactions between
electrons in bunch with small density ( Bsp ω< <ω ) and
field of own bremsstrahlung for monovigorous bunch
give the equality 1)( =ω cn .
Let us define the general requirements to dispersive
properties of electronic gas at which process of forma-
tion of superradiation mode by excited oscillators in a
bunch of electrons is most effective. We shall show that
these properties can be received from task optimization
function of parameters of magnetic field and electron
bunch to intensity of radiation.
In approach of monovigorous bunch we shall con-
sider, that parametrical interaction of radiations field of
flat waves does not give collecting effect as the sum of
phases of oscillator’s fluctuations for the period oT is
not constant. Average time of synchronous fluctuations
in ensemble of oscillators we shall define from condi-
tion of synchronism: π=∆ω tc . Then the considered en-
semble of oscillators aN is made in the volume, the
limited spherical surface with radius equal to 2car λ ′=′ ,
and )6( 33
caa DggVN γπ== . Then, aBsBs NEE 22 = .
In laboratory system of readout (the system K ) for
the lowest fashion of generation bremsstrahlung in unit
volume of electron’s bunch we have
ω−ω
γ
π+γ= =γ tEgDH
cm
geI ciBs
i
ii )(sin
63
2)( 22
3
3
22
032
4
.
Function )( iI γ is not monotonous and has an ex-
tremum in point where takes place the equality
tcii
p
)sin(
6
2
2
ω−ωπ
=
ω
ω
.
Whence it is easy to obtain factor of refraction of en-
vironment for radiation field )(zH by frequency iω
which extend along an axis 0Z
t
n
ci
iz
)sin(
61)(
ω−ωπ
+=ω . (2)
From (4) it follows in the direction of an axis 0Z en-
vironment is not transparent for the most probable for
distribution )(γF of frequency cω bremsstrahlung. In
dense electronic gas, at ip ω>ω , the field BsE is weak-
ened as a result of resonant interaction with plasma fluc-
tuations of environment (polarizing losses) and on dis-
tance cλ ′~ falls down up to zero [7]. Absorbing energy
of the field BsE , electrons reemit it with displacement of
a phase on 2π . Thus such process provide creation of
nonequilibrium radiations field prE . As other channels
of loss energy field BsE are absent, in monovigorous
electron’s bunch is possible to accept Bscpr EE ≈ω )( ,
and the direction and degree of polarization of fields
BsE and prE are identical. In the equation (2) the vol-
ume tci )sin( ω−ω is the characteristic of linac. It has
been earlier shown, that in high-dense electron’s bunch
occurs of broadening spectral range radiating of power
interaction of oscillators. Therefore, at 1> >aN , in time
oT the volume oci T)sin( ω−ω is not determined also the
minimal value of parameter of refraction equally
5.1min ≈n . Then condition of existence of a field prE
and transition of ensemble of oscillators in a mode of
superradiation is ec rDg 226 πγ=
Let's consider the dynamics of separate electron in
system K ′ in case of the high-density electron’s bunch,
when cp ωω ~ . Then the cross-section of dispersion of
fields )( cprE ω by electrons of bunch is
( ) ( )πω−ωλ ′=σ 2cos3 2 Tcic .
At cγ< <γ∆ , we have ( ) ( )π−λ ′=ρ 23 2
IIIc NN .
For a case of the charge bunch high density having
place in the linac such as LU-50 [8] with volumetric
electron concentration in a bunch of about 1110 sm-3, it is
necessary to take into account the charge interactions.
___________________________________________________________
PROBLEMS OF ATOMIC SIENCE AND TECHNOLOGY. 2004. № 1.
Series: Nuclear Physics Investigations (42), p.150-152.151
Achievement of condition cp ωω ~ can be connected
both to a longitudinal grouping of electrons, and with
cross-section focusing a bunch. However phase drift of
electrons in a field of falling wave, resulting to longitu-
dinal grouping, is energetically more favourable, than
cross-section focusing of a bunch. Therefore we shall
take into account only effect of a longitudinal grouping,
and the cross-section size of bunch we shall count con-
stant.
For one-dimensional dependence of parameter )(rρ
from coordinate in electronic gas the change of this pa-
rameter is function of difference concentration of elec-
trons in areas I and II (see fig.1) )()()( rNrNrA III −= ,
which will defined by efficiency of process of phase
drift of electrons in field of prE . Then
drrA
c
nrdA crcdr )()()()(
λ ′∆
ωωυ−= ,
where рr
e
iri
idr E
m
e
c
n )()( ωλ ′
=ωυ is average speed of
phase drift of electron for oT .
In the cw-approach, at 1)( < <ωυ cn crdr , depen-
dence )(rρ has the following form
γ∆λ ′
ωγωυ−λ ′
π
=ρ r
с
nAr
c
crccdr
c
)()(1
2
3)( 0
2 .
In system K from the differential equation of trans-
fer of plane-parallel wave front of nonequilibrium radia-
tion we obtain
dzzIz
сD
nADzdI pr
cdr
cr
c
pr )()(1
2
3)( 02
2
γ∆
γυω−
γπ
−=
(transformation )(rρ to system K is carried out by re-
placement czr γ= and cc D γ=λ ′ ). Thus we receive
dependence of intensity of nonequilibrium radiation
from dimensionless coordinate Dz=ξ
ξ
γ∆
γυω−ξ
π γ
−=ξ 2
3
0
0 2
)(
2
3exp)(
с
nDAII cdr
cr
c
pr ,
where ( )323
0 6 cBsgEDI γπ= is the initial intensity of
nonequilibrium radiation.
From general kind of equations for transfer of plane-
parallel wave front of nonequilibrium radiation follows
action of nonlinear term in exhibitor's parameter is di-
rected to condition of balance, for which 0)( II pr =ξ .
REFERENCES
1. A.G.Bonch-Osmolovskij, S.N.Dolja, K.А. Reshet-
nikova // JTF. 1983, v. 53, p. 1055.
2. А.В.Gaponov // LETF. 1960, v. 39, p. 326.
3. J.N.Barabanenkov, V.D. Ozrin // Izvestia academee
nauk. Radiophysics. 1977, v. 20, p. 712.
4. A.M.Kondratenko, Е.L.Saldin // JTF. 1981, v. 51,
p. 1633.
5. L.D.Landau, E.M.Lifshits. Teoria Polia M.:”Sci-
ence”, 1973.
6. Ja.I.Khanin. Osnovi dinamiki laserov. M.: “Sci-
ence”, 1999.
7. L.Spitzer. Fizika polnostiu ionizovannogo gaza. M.:
“Mir”, 1965.
8. N.I.Zavjalov, I.A.Ivanin, J.A.Hohlov, etc. Pribory
and tehnika experimenta. 1990, v. 3, p. 56.
ДИСПЕРСИЯ И ИЗЛУЧАТЕЛЬНАЯ СПОСОБНОСТЬ
РЕЛЯТИВИСТСКОГО ЭЛЕКТРОННОГО ГАЗА
В.В. Порхаев, Н.В. Завьялов, В.Т. Пунин, А.В. Тельнов, Ю.А. Хохлов
Рассмотрены дисперсия и условия формирования неравновесного излучения в релятивистском электрон-
ном газе. Для случая большой плотности электронного сгустка в квазистационарном приближении в волно-
вой зоне ангармонического осциллятора получен общий вид уравнения переноса плоскопараллельного
фронта неравновесного излучения
ДИСПЕРСІЯ І ВИПРОМІНЮВАЛЬНА ЗДАТНІСТЬ
РЕЛЯТИВІСТСЬКОГО ЕЛЕКТРОННОГО ГАЗА
В.В. Порхаев, Н.В. Зав'ялов, В.Т. Пунин, А.В. Тельнов, Ю.А. Хохлов
Розглянуті дисперсія й умови формування нерівновагого випромінювання в релятивістському
електронному газі. Для випадку великої густини електронного згустку в квазістаціонарному наближенні в
хвильовій зоні ангармонічного осцилятора отримано загальний вид рівняння переносу плоскопараллельного
фронту нерівновагого випромінювання.
152
The federal state unitary enterprise “The Russian Federal Nuclear Center –
All-Russia scientific research institute of experimental physics”
Radiating instability
of electronic gas
references
Релятивистского электронного газа
В.В. Порхаев, Н.В. Завьялов, В.Т. Пунин, А.В. Тельнов, Ю.А. Хохлов
|
| id | nasplib_isofts_kiev_ua-123456789-78956 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T22:58:51Z |
| publishDate | 2004 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Porkhaev, V.V. Zavyalov, N.V. Punin, V.T. Telnov, A.V. Khokhlov, Ju.A. 2015-03-24T14:29:28Z 2015-03-24T14:29:28Z 2004 Dispersion and radiating ability of relativistic electronic gas / V.V. Porkhaev, N.V. Zavyalov, V.T. Punin, A.V. Telnov, Ju.A. Khokhlov // Вопросы атомной науки и техники. — 2004. — № 1. — С. 150-152. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS 539.12 https://nasplib.isofts.kiev.ua/handle/123456789/78956 In the paper the dispersion and conditions of formation of nonequilibrium radiation in the relativistic electronic gas are considered. For a case of a high-density electron bunch in the cw-approach and a wave-zone unharmonic oscillator the general kind of transfer plane-parallel front of nonequilibrium radiation equation is obtained. Розглянуті дисперсія й умови формування нерівновагого випромінювання в релятивістському електронному газі. Для випадку великої густини електронного згустку в квазістаціонарному наближенні в хвильовій зоні ангармонічного осцилятора отримано загальний вид рівняння переносу плоскопараллельного фронту нерівновагого випромінювання. Рассмотрены дисперсия и условия формирования неравновесного излучения в релятивистском электронном газе. Для случая большой плотности электронного сгустка в квазистационарном приближении в волновой зоне ангармонического осциллятора получен общий вид уравнения переноса плоскопараллельного фронта неравновесного излучения. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков Dispersion and radiating ability of relativistic electronic gas Дисперсія і випромінювальна здатність релятивістського електронного газа Дисперсия и излучательная способность релятивистского электронного газа Article published earlier |
| spellingShingle | Dispersion and radiating ability of relativistic electronic gas Porkhaev, V.V. Zavyalov, N.V. Punin, V.T. Telnov, A.V. Khokhlov, Ju.A. Динамика пучков |
| title | Dispersion and radiating ability of relativistic electronic gas |
| title_alt | Дисперсія і випромінювальна здатність релятивістського електронного газа Дисперсия и излучательная способность релятивистского электронного газа |
| title_full | Dispersion and radiating ability of relativistic electronic gas |
| title_fullStr | Dispersion and radiating ability of relativistic electronic gas |
| title_full_unstemmed | Dispersion and radiating ability of relativistic electronic gas |
| title_short | Dispersion and radiating ability of relativistic electronic gas |
| title_sort | dispersion and radiating ability of relativistic electronic gas |
| topic | Динамика пучков |
| topic_facet | Динамика пучков |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78956 |
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