Account of the longitudinal temperature in cyclotron superradiance
The phenomenon of cyclotron Dicke superradiance (SR) in the inverted system of nonrelativistic electrons in low density magnetized plasma is considered. It is shown, that account of the longitudinal temperature increases the critical electron density which is needed for the nonequilibrium phase tran...
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irk-123456789-1118622017-01-16T03:02:49Z Account of the longitudinal temperature in cyclotron superradiance Novak, O.P. Fomina, A.P. Kholodov, R.I. Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина The phenomenon of cyclotron Dicke superradiance (SR) in the inverted system of nonrelativistic electrons in low density magnetized plasma is considered. It is shown, that account of the longitudinal temperature increases the critical electron density which is needed for the nonequilibrium phase transition to the SR regime. Розглянуто явище циклотронного надвипромiнювання Дiке (НВ) в iнвертованiй системi нерелятивiстських електронiв у розрiдженiй замагнiченiй плазмi. Показано, що врахування поздовжньої температури збiльшує критичну концентрацiю електронiв, необхiдну для нерiвноважного фазового переходу в НВ-режим. Рассмотрено явление циклотронного сверхизлучения Дике (СИ) в инвертированной системе нерелятивистских электронов в разреженной замагниченной плазме. Показано,что учет продольной температуры увеличивает критическую концентрацию электронов, необходимую для неравновесного фазового перехода в СИ-режим. 2013 Article Account of the longitudinal temperature in cyclotron superradiance / O.P. Novak, A.P. Fomina, R.I. Kholodov // Вопросы атомной науки и техники. — 2013. — № 3. — С. 69-73. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 52.30.-q, 52.25.Xz http://dspace.nbuv.gov.ua/handle/123456789/111862 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина Novak, O.P. Fomina, A.P. Kholodov, R.I. Account of the longitudinal temperature in cyclotron superradiance Вопросы атомной науки и техники |
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The phenomenon of cyclotron Dicke superradiance (SR) in the inverted system of nonrelativistic electrons in low density magnetized plasma is considered. It is shown, that account of the longitudinal temperature increases the critical electron density which is needed for the nonequilibrium phase transition to the SR regime. |
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Novak, O.P. Fomina, A.P. Kholodov, R.I. |
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Novak, O.P. Fomina, A.P. Kholodov, R.I. |
| author_sort |
Novak, O.P. |
| title |
Account of the longitudinal temperature in cyclotron superradiance |
| title_short |
Account of the longitudinal temperature in cyclotron superradiance |
| title_full |
Account of the longitudinal temperature in cyclotron superradiance |
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Account of the longitudinal temperature in cyclotron superradiance |
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Account of the longitudinal temperature in cyclotron superradiance |
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account of the longitudinal temperature in cyclotron superradiance |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Квантово-полевые и групповые подходы теоретической физики. Семинар памяти Петра Ивановича Фомина |
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| citation_txt |
Account of the longitudinal temperature in cyclotron superradiance / O.P. Novak, A.P. Fomina, R.I. Kholodov // Вопросы атомной науки и техники. — 2013. — № 3. — С. 69-73. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT novakop accountofthelongitudinaltemperatureincyclotronsuperradiance AT fominaap accountofthelongitudinaltemperatureincyclotronsuperradiance AT kholodovri accountofthelongitudinaltemperatureincyclotronsuperradiance |
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2025-07-08T02:49:05Z |
| last_indexed |
2025-07-08T02:49:05Z |
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1837045326781874176 |
| fulltext |
ACCOUNT OF THE LONGITUDINAL TEMPERATURE
IN CYCLOTRON SUPERRADIANCE
O.P.Novak1∗, A.P.Fomina2, R.I.Kholodov1
1Institute of Applied Physics NAS of Ukraine, 40000, Sumy, Ukraine;
2Bogolyubov Institute for Theoretical Physics NAS of Ukraine, 03680, Kiev, Ukraine
(Received January 1, 2013)
The phenomenon of cyclotron Dicke superradiance (SR) in the inverted system of nonrelativistic electrons in low
density magnetized plasma is considered. It is shown, that account of the longitudinal temperature increases the
critical electron density which is needed for the nonequilibrium phase transition to the SR regime.
PACS: 52.30.-q, 52.25.Xz
1. INTRODUCTION
The phenomenon of superradiance (SR) was con-
sidered first by Dicke [1] on the example of two-level
model. SR was studied in a number of works (see,
for example, reviews [2, 3, 4]), but a lot of interest-
ing and physically important questions remain not
investigated.
In the Ref. [5] (see, also [6, 7]) the theory of collec-
tive coherent SR in the system of inverted electrons
occupying high Landau levels [8] in low density mag-
netized plasma with
E⊥ = n~ωH , n À 1, (1)
ωH =
eH
mc
, (2)
has been developed.
It was shown in [5, 6], that under certain condi-
tions the polarization phase transition occurs to the
Dicke SR state [1] in such system due to the dipole-
dipole interaction between rotating electrons. The
phenomenon of SR arises in “coherence domains”
with the sizes R0 smaller than the radiation wave-
length λ when all N0 radiating dipoles gradually align
in the same direction due to the dipole-dipole inter-
action in a “near zone” R0 ¿ λ. As a result, the total
dipole moment of a domain becomes proportional to
the number of electrons N0 and the intensity of collec-
tive coherent dipole radiation of a domain increases
in N0 times in contrast to the intensity of uncorre-
lated dipole radiation and becomes proportional to
N2
0 . The transition to such correlated polarized state
is similar to the phase transition in magnetics and
the Weiss method of mean self-consistent field [9] was
used to find the criteria of self-polarization in a such
system.
The resulting nonlinear equation is similar to the
Weiss equation and determines the threshold of polar-
ization phase transition in a domain on the density of
inverted electrons. It is defined by the relation [5, 6]
ne > nec =
0.18H2kT
mc2E⊥
. (3)
In Ref. [7] the SR phenomenon theory is used to
explain the nature and the main features of the super
power decameter radiation (DCM) of the Jupiter-Io
system.
The sporadic DCM radiation of Jupiter was dis-
covered in 1955. Despite considerable progress in
studying the features of DCM radiation, there are no
generally accepted and consistent answers to many
important questions yet. The most important prob-
lem is the nature of the coherent collective mechanism
of radiation providing a gigantic peak power of the
DCM-pulses. It reaches ∼ 1017÷1018 erg/s that cor-
responds to the brightness temperature of the source
about ∼ 1017 K. Introducing the SR mechanism sim-
plifies the problem and allows us to explain the ob-
served power of DCM-pulses without involving of free
parameters.
However, in References [6, 7] the authors assumed
that electrons in a magnetic field rotate at circular
orbits with fixed centers and do not move along the
field. The purpose of the present paper is to investi-
gate how the longitudinal motion of electrons affects
the main features of the cyclotron SR.
2. ACCOUNT OF LONGITUDINAL
TEMPERATURE
It is convenient to consider first the situation
when the electrons in a coherence domain move along
and opposite to the magnetic field with the same con-
stant speed v. Thus, the electron trajectories are de-
∗Corresponding author E-mail address: novak-o-p@ukr.net
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2013, N3(85).
Series: Nuclear Physics Investigations (60), p.69-73.
69
fined by the expressions
~Rj(t) = ~rj(t) + ~rj⊥(t), (4)
~rj(t) = ~rj(0)± ~vt, ~v = (0, 0, v), (5)
~r⊥(t) = rL[cos(ωt + α), sin(ωt + α), 0]. (6)
Here, ~rj defines the position of the orbit center and
rL is the Larmor radius. Thus, it is assumed that
N0/2 electrons move along the field with the velocity
~v while the others N0/2 electrons move opposite to
the field with the same speed.
The potential energy of a trial dipole is
U = −~d0
~E, (7)
where ~d0 = e~r0⊥(t) and the total electric field ~E in a
near zone decomposes in the sum
~E =
N−1∑
j
3~nj(~nj
~dj)− ~dj
|~r0 − ~rj |3 , (8)
~nj =
~r0 − ~rj
|~r0 − ~rj | . (9)
Averaging (7) over the rotation period one should
note that radius vectors ~rj vary with time slowly in
comparison with ~rL. Substituting Eqs. (8), (9) into
(7) and averaging over the period we obtain
〈U(~r0)〉 = −d2
0
2
N/2∑
j=1
1− 3~n2
jz
|~r0 − ~rj |3 cos(α0 − αj)+
N∑
j=N/2
1− 3~n2
jz
|~r0 − ~rj |3 cos(α0 − αj)
, (10)
where d0 = erL and the first sum is taken over the
electrons which have the same longitudinal velosity
as the trial one while the second sum is taken over
the other group of electrons.
The position of the trial dipole should not be pre-
ferred, therefore it necessary to average (10) over ~r0.
After the replacing of variables according to the re-
lations {~r0, ~rj} → {~r = ~r0 − ~rj , ~R = 1
2 (~r0 + ~rj)},
the averaged potential energy can be expressed in the
form
〈U〉 = −d2
0ne
4
〈cos ∆α〉 × (11)
∫
Vc1
r2 − 3z2
r5
d~rj +
∫
Vc2
r2 − 3z2
r5
d~rj
,
where the sums are replaced with the integrals. Note
that cos(α0−αj) is replaced by its averaged over the
ensemble value 〈cos∆α〉 in accordance with the Weiss
method [9].
Aligning of dipoles is energetically favourable be-
cause it increases the negative contribution to the
potential energy 〈U〉. Therefore, the correlations will
occur only in the “coherence domain” defined by the
condition
r2 − 3z2 > 0. (12)
In the near-by domains the directions of the aver-
age polarization vectors should be close to opposite
to minimize the total potential energy of the system.
In the similar way magnetics and ferroelectrics are
broken into domains too.
As follows from Eq. (12), the ”coherence do-
mains” in relative coordinates look like flattened
cylinders with conical covers, as shown in Fig. 1,a.
Fig.1. Overal view of the coherence domains in
the cases of zero longitudinal temperature (a) and
presence of the motion along the magnetic field (b)
Both coherence domains Vc1 and Vc2 in Eq. (11)
have the same shape defined by Eq. (12). How-
ever, Vc2 moves along z axis since the second sum-
mand describes the contribution from dipoles that
move relative to each other. Consequently, align-
ing between such dipoles is possible only in the
area that belongs to the both volumes during the
phasing time τ It results in changing of the co-
herence domain shape, as shown in Fig. 1,b, Fig. 2.
Fig.2. Changing of the coherence domain due to
the electron motion along the magnetic field
It is convenient to carry out integration in Eq. (11)
in cylindrical coordinates.
In the integral over the variable r, the lower limit
should be chosen about Larmor radius rL since at
70
the smaller distance the electron-electron interaction
can not be described by dipole formulas. The upper
limit should be chosen ∼ R0, the characteristic size of
the “coherence domain” determined by the condition
rL ¿ R0 ¿ λ. After integration we obtain
〈U〉 = −1
2
d2
0ne〈cos∆α〉J(η) , (13)
where
J(η) =
4π
3
√
3
[
ln
√
2(x2 − χ1)(x2 − χ2)
η(x1 − χ1)(x1 − χ2)
+
√
6x1 − 3
√
3
1 + x2
1
+
√
6x2 − 3
√
3
1 + x2
2
]
, (14)
x1 =
√
3−1√
2
, η
√
2 < 1;
x1 =
√
1 + (10η − 1√
2
)2 − (10η − 1√
2
), η
√
2 > 1;
x2 =
√
1 + (η − 1√
2
)2 − (η − 1√
2
) (15)
and η is the dimensionless longitudinal speed,
η =
vτ
R0
. (16)
Let us proceed to the case of Maxwell distribution
f(v) for the dipole longitudinal speed,
f(v) =
√
m
2πkT‖
exp
(
−mv2
2kT‖
)
. (17)
In this case the total electric field of the electron sys-
tem in Eq. (7) can be written in the form
~E =
∑
i
∑
j
~Ei,j , (18)
where the sum over particle longitudinal speed is in-
troduced. Following the same procedure as before
and introducing the additional averaging over the
speed of the trial dipole, we obtain the average in-
teraction energy in the form
〈U〉 = −1
2
e2r2
Lne〈cos∆α〉
∞∫
−∞
dv′f(v′)
∞∫
−∞
dvJ(|v−v′|) ,
(19)
where the quantity J(|v−v′|) is the previously found
integral over the modified coherence volume,
J(|v − v′|) =
∫
V
d~r
r2 − 3z2
r5
. (20)
After changing of variables according to the rela-
tions {η, η′} → {ξ = η − η′, ξ′ = η′}, the averaged
potential energy can be expressed in the form
〈U〉 = −d2
0ne〈cos∆α〉
4η0
√
π
∞∫
−∞
e
− ξ2
4η2
0 J(|ξ|)dξ , (21)
where the function J(x) is defined by the Eq. (14) and
the quantity η0 is dimensionless longitudinal temper-
ature,
η2
0 =
kT‖τ2
mR2
0
. (22)
Thus, it is seen that the aligning of dipoles
with radiation of the released energy is energetically
favourable because of the negative contribution of
dipole-dipole interaction to the total potential energy
of the system. The thermal fluctuations resist to this
tendency. They are realized in low density magne-
tized plasma mainly in the form of plasma fluctua-
tions and Alfven waves. To determine the thresh-
old values of the parameters, when the transition to
the polarized state become possible, one can use the
Weiss method of mean self-consistent field [9]. Ac-
cording to the results of Ref. [6], the criteria of such
transition can be written as
〈U〉
2kT 〈cos∆α〉 > 1. (23)
It is convenient to introduce an auxiliary normal-
ized function g(η0),
g(η0) =
〈U〉
〈UT‖=0〉 =
3
√
3
8
√
π3
∞∫
−∞
e
− ξ2
4η2
0 J(|ξ|)dξ
η0 ln( mc2
5E⊥
)
, (24)
where 〈UT‖=0〉 is the potential energy of the dipole-
dipole interaction at zero longitudinal temperature,
obtained in Ref. [6],
〈UT‖=0〉 = − 2π
3
√
3
ln
(
mc2
5E⊥
)
ned
2
0〈cos∆α〉. (25)
Evidently, the condition g(0) = 1 is true. Func-
tion g(η0) has been calculated numerically and it is
shown in Fig. 3.
Finally, to determine the criteria of the transition
to the SR regime, it is necessary to substitute the
found potential energy 〈U〉 = 〈UT‖=0〉g(η0) to the
relation (23). With account of Eq. (25) the criteria
takes on the form
2π
3
√
3
ln
(
mc2
5E⊥
)
mc2
H2
neE⊥
kT⊥
· g(η0) > 1. (26)
71
Fig.3. The dimensionless function g(x) defining
the normalized potential energy of the dipole-dipole
interaction
3. PHASING DYNAMICS
As follows from the above consideration, longitu-
dinal motion in low density magnetized plasma re-
sults in electron outflow from the coherence domain.
Consequently, the total number of phased dipoles at
each moment of time t decreases with T‖ growing.
Let t′ < t be the moment of time when the number
of phased dipoles at zero longitudinal temperature N
equals to this number N ′(t) at T‖ 6= 0,
N ′(t) = N(t′). (27)
To find the number of phased dipoles in the moment
t + dt one should take into account the outflow dN :
N ′(t + dt) = N(t′ + dt)− dN(t). (28)
Assuming for estimation that the coherence do-
main is a cylinder with radius R0, one can find the
outflow of electrons having speed withing the interval
(v, v + dv) in the form
dNv = 2πR2dt · vdnv , (29)
where the fraction dnv is given by the Maxwell dis-
tribution,
dnv =
N ′(t)
V
√
m
2πkT‖
exp
[
−mv2
2kT‖
]
dv. (30)
Performing simple integration, we obtain the follow-
ing expression for the total outflow:
dN = N ′(t)
√
2
π
η0
τ
dt. (31)
Finally, expanding Eq. (28) into series in dt and
taking into account Eq. (27), one can find the follow-
ing equations determining the sought function N ′(t),
dN ′(t)
dt
=
dN(t)
dt
∣∣∣∣
t=t′
−N ′(t)
η0
τ
√
2
π
, (32)
N(t′) = N ′(t). (33)
To solve Eqs. (32), the function N(t) describ-
ing the phasing at zero longitudinal temperature is
needed, which is unknown. Nevertheless, some con-
clusions of how the longitudinal motion affects the
phasing can be inferred from treatment of a model
function N(t). Apparently, it should satisfy the fol-
lowing conditions:
1. N(t) ∼ exp( t
τ ) when t → 0.
2. N(t) → N0 when t →∞.
3. N(0) ∼ 1.
For example, let us choose the model function in
the form
N(t) =
N0
1 + N0 exp(− t
τ
)
. (34)
Then the solution of Eqs. (32), (33) can be expressed
as
N ′(t) =
γN0
1 + [γN0 − 1] exp(−γ t
τ )
, (35)
where
γ = 1− η0
√
2
π
(36)
and the initial condition is N ′(0) = 1. The plots of
Eq. (35) at different temperatures is shown in Fig. 4.
Fig.4. Phasing dynamics at different longitudinal
temperatures, N0 = 100
Thus, the total amount of phased electrons de-
creases by a factor γ while the phasing time increases
by the factor γ−1. In other words, account of the lon-
gitudinal temperature results in the following. The
coherence domain size decreasing is proportional to
the temperature, while phasing time is in inverse ra-
tio to the temperature.
Moreover, phasing is impossible when γ < 0.
Consequently, the following condition should be met:
η0 <
√
π
2
. (37)
72
Asknowledgement
The authors would like to thank V.Yu. Storizhko for
the support of this work and V.I. Myroshnichenko for
helpful discussions.
References
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3. A.V.Andreev // Usp. Fiz. Nauk, 1990, v. 160,
p.1.
4. L.I.Men‘shikov // Usp. Fiz Nauk, 1999, v. 169,
p.113.
5. P.I. Fomin, A.P. Fomina // Prob. At. Sci. Tech.
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Phys. 2002, v. 47, p.1001.
7. P.I. Fomin, A.P. Fomina, V.N. Mal’nev // Ukr. J.
Phys. 2004, v. 49, p.3.
8. L.D. Landau , E.M. Lifshitz // Quantum mechan-
ics, M.: ”Nauka”, 1974, p.752.
9. S. Smart. Effective field theoris of magnetism.,
W. Saunders Company, Philadelphia-London,
1966.
УЧЕТ ПРОДОЛЬНОЙ ТЕМПЕРАТУРЫ В ЦИКЛОТРОННОМ СВЕРХИЗЛУЧЕНИИ
A.П.Новак, А.П.Фомина, Р.И.Холодов
Рассмотрено явление циклотронного сверхизлучения Дике (СИ) в инвертированной системе нереляти-
вистских электронов в разреженной замагниченной плазме. Показано, что учет продольной темпера-
туры увеличивает критическую концентрацию электронов, необходимую для неравновесного фазового
перехода в СИ-режим.
ВРАХУВАННЯ ПОВЗДОВЖНЬОЇ ТЕМПЕРАТУРИ В ЦИКЛОТРОННОМУ
НАДВИПРОМIНЮВАННI
О.П.Новак, А.П.Фомiна, Р.I.Холодов
Розглянуто явище циклотронного надвипромiнювання Дiке (НВ) в iнвертованiй системi нерелятивiстсь-
ких електронiв у розрiдженiй замагнiченiй плазмi. Показано, що врахування поздовжньої температури
збiльшує критичну концентрацiю електронiв, необхiдну для нерiвноважного фазового переходу в НВ-
режим.
73
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