OSCILLATIONS AND SUPERRADIANCE IN RADIATION SPECTRUM
OF ELECTRONS MOVING IN SPIRAL IN MEDIUM*
A.V. KONSTANTINOVICH, I.A. KONSTANTINOVICH
Chernivtsi National University, Kotsyubinsky St.,2, Chernivtsi, 58012, Ukraine
E-mail: aconst@hotbox.ru
Received September 14, 2009
The fine structure of the synchrotron-Cherenkov radiation of spectral
distribution of the radiation power of one and four electrons moving in a spiral in a
transparent medium with relativistic transversal velocity component (the component
perpendicular to the magnetic field) is researched. The oscillations and superradiance in
the synchrotron-Cherenkov radiation spectrum of four electrons at different time shifts
between electrons are investigated. The influence of the coherence factor on the
spectrum of synchrotron-Cherenkov radiation for four electrons is analyzed.
Key words: synchrotron-Cherenkov radiation, radiation oscillations, coherence factor,
superradiance in synchrotron-Cherenkov radiation.
1. INTRODUCTION
The properties of synchrotron radiation of charged particles moving in a
circle in vacuum were studied in papers [1–3]. The particularity of different
properties of synchrotron radiation of charged particles moving in magnetic field in
vacuum was examined by Ternov in report [4]. Investigations of the generation of
coherent electromagnetic radiation from a system of non-interacting electrons
moving in a spiral in constant magnetic field in vacuum were reported in papers
[5–12].
The electromagnetic radiation spectrum of one electron moving in a medium
in magnetic field was under investigation in papers [13–18]. The oscillations in
synchrotron-Cherenkov radiation spectrum of one electron were obtained at its
motion in a circle [13] and in a spiral [18].
The coherence effects in the radiation spectrum of a system of noninteracting electrons moving one by one along a spiral in a transparent medium
*
Paper presented at the 10th International Balkan Workshop on Applied Physics, July 6–8,
2009, Constanţa, Romania.
Rom. Journ. Phys., Vol. 56, Nos. 1–2, P. 45–52, Bucharest, 2011
A.V. Konstantinovich, I.A. Konstantinovich
46
2
were considered in papers [19–24]. If the dimension of a system of electrons is
smaller in comparison to the radiation wavelength, both for a quantum-mechanical
system [25] and for a classical system of electrons [6, 24, 26] the superradiant
regime is possible.
The results on the anomalous Cherenkov rings [27], the subluminal Cherenkov
radiation [28] as well as rigorous theoretical investigations [29] are of a great
theoretical and experimental interest in this domain.
The aim of this paper is to investigate the oscillations and superradiance of
the radiation spectrum of four electrons moving in a spiral in magnetic field in
transparent media by using the exact integral relationships for the spectral
distribution of radiation power. The influence of the coherence factor on the
spectrum of synchrotron-Cherenkov radiation is analyzed. The Doppler effect
influence on peculiarities of the radiation spectrum of one and four electrons
moving in a spiral in medium is investigated, too.
2. SPECTRAL DISTRIBUTION OF RADIATION POWER OF FOUR ELECTRONS
MOVING ALONG A SPIRAL IN TRANSPARENT MEDIUM
The time-averaged radiation power P rad of a system of electrons moving in
magnetic field is expressed in [3, 30] as
G
G
T 
Dir G

G
∂
A
r ,t )
(
1
1
G
G ∂ϕ Dir ( r , t )  G 

rad
(1)
− ρ (r ,t )
P = lim
dr  dt .
  j (r ,t )
T →∞ 2T ∫ ∫ 
c ∂t
∂t
−T 
τ 
 
G G
G
Here j ( r , t ) is the current density and ρ ( r , t ) is the charge density. The
integration is over some volume τ . According to the hypothesis of Dirac [31], the
G
G
G
scalar ϕ Dir ( r , t ) and vector A Dir ( r , t ) potentials are defined as a half-difference of
the retarded and advanced potentials:
ϕ Dir =
G
G
1 G
ADir = Aret − Aadv .
2
(
1 ret
(ϕ − ϕ adv ) ,
2
)
(2)
Then according to [6, 22], the source functions of the system of four electrons
are defined as
4 G
4
G G
G
G
G
G G
G
j ( r , t ) = ∑Vl ( t ) ρ l ( r , t ) , ρ ( r , t ) = ∑ ρl ( r , t ) , ρ l ( r , t ) = eδ ( r − rl ( t ) ) , (3)
l =1
l =1
G
G
where rl ( t ) and Vl (t ) are the motion law and the velocity of the l th electrons,
respectively.
3
Radiation spectrum of electrons moving in spiral in medium
47
Here we study a system of four electrons moving one by one in a spiral in
transparent media. The law of motion and the velocity of the l th electron are given
by the expressions
G
G G
G
G
dr ( t )
G
. (4)
rl ( t ) = r0 cos {ω0 ( t + ∆tl )} i + r0 sin {ω0 ( t + ∆tl )} j + V|| ( t + ∆tl ) k , Vl ( t ) = l
dt
Here r0 = V⊥ω0−1 , ω0 = ceB ext E −1 , E = c p 2 + m02 c 2 , the magnetic induction
H
G
vector B ext ||0Z, V⊥ and V|| are the components of the velocity, p and E are the
momentum and energy of the electron, e and m0 are its charge and rest mass.
The time-averaged radiation power of the system of four electrons we obtain
after substituting expressions (2)–(4) into (1). Then
∞
P rad = ∫ W (ω ) d ω ,
(5)
0
 n (ω )

ωη ( x ) 
sin 
2
c
2e
 cos ω x V 2 cos ω x + V 2 − c  , (6)
W (ω ) = 2 ∫ dxµ (ω )ω S4 (ω ) 
( 0 ) || 2 
 ⊥
n (ω ) 
πc 0
η ( x)

2 ∞
ω 
sin 2  0 x  , µ (ω ) is the magnetic permeability, n (ω )
ω
 2 
is the refraction index, ω is the cyclic frequency, and c is the velocity of light in
vacuum.
In the case of four electrons moving one by one along a spiral the coherence
factor S 4 (ω ) takes the form [6, 22]:
where η ( x ) = V||2 x 2 + 4
V⊥2
2
0
S 4 (ω ) = 4 + 2cos (ω∆t 12 ) + 2cos (ω∆t 23 ) + 2cos (ω∆t 34 ) + 2cos {ω ( ∆t 12 +∆t 23 )}
+2 cos {ω ( ∆t 23 +∆t 34 )} + 2 cos {ω ( ∆t 12 +∆t 23 +∆t34 )} ,
(7)
Starting from relationships (5) and (6) the contribution of separate harmonics
to the averaged radiation power can be written as
∞
π

e2 ∞
  n (ω )

P rad = 3 ∑ ∫ d ωµ (ω ) n (ω ) S 4 (ω ) ω 2 ∫ sin θ dθδ ω  1 −
V|| cos θ  − mω0  ×
c m =−∞ 0
c
 

0


  m 2
 
c2  2
× V⊥2  2 J m2 ( q ) + J m′2 ( q )  +  V||2 − 2
 J m ( q )  .
n (ω ) 
 

  q
(8)
A.V. Konstantinovich, I.A. Konstantinovich
48
where q =
4
n (ω ) ω
V⊥ sin θ , J m ( q ) and J m′ ( q ) are the Bessel function with integer
c ω0
index and its derivative, respectively.
From relationship (8) one can conclude that each harmonic is a set of
frequencies, which are determined from the solution of the equation

ω 1 −
n (ω )


V|| cosθ  − mω0 = 0 .
c

(9)
After integrating in (8) over the θ variable we have obtained the spectral
distribution of electron radiation power on harmonics
∞
e2 ∞
P rad = 2 ∑ ∫ d ωµ (ω ) ω S 4 (ω ) η ( u 2 ( m ) ) ×
c V|| m =−∞ 0
  m 2

 
c2  2
× V⊥2  2 2
J m2 ( q1u ( m ) ) + J m′2 ( q1u ( m ) )  +  V||2 − 2
 J m ( q1u ( m ) )  , (10)
n (ω ) 
 
  q1 u ( m )

where the function η ( u 2 ( m ) ) is:
2
1, u 2 ( m ) > 0
c 2 (ω − mω0 )
n (ω )V⊥ ω
2
=
−
u
m
1
. (11)
q
,
=
,
(
)
1
2
c
ω0
n 2 (ω )V||2ω 2
0, u ( m ) < 0
η (u2 ( m)) = 
This function determines the band boundaries in the radiation spectrum.
3. FINE STRUCTURE OF THE RADIATION SPECTRUM OF FOUR
ELECTRONS MOVING ALONG A SPIRAL IN MEDIUM
It is interesting to compare the radiation power spectral distribution for one
electron (curve 1 in Fig. 1 and curve 3 in Fig. 3) to that of four electrons (curve 2 in
Fig. 2 and curve 4 in Fig. 4, respectively). Our numerical calculations of the
radiation spectra were carried out on the basis of equations (5) and (6). The spectral
distribution of synchrotron radiation power was obtained for B ext = 1 Gs, µ = 1 ,
n = 1.7 ,
V⊥vac = 0.2 ⋅1011 cm/s,
V||vac = 0.15 ⋅ 1010 cm/s,
r0 j = 1530 cm,
ω0 j = 0.1307 ⋅10 rad/s, c = 0.2997925 ⋅ 10 cm/s (j=1,2,…,6).
8
11
int
−12
erg/s is determined
The radiation power for one electron Pmed
1 = 0.2163 ⋅ 10
after integration of relationships (5) and (6) when S 4 (ω ) is substituted by
S1 (ω ) = 1 (curve 1 in Fig. 1).
5
Radiation spectrum of electrons moving in spiral in medium
49
(2)
For the time shifts ∆t12(2) = ∆t23
= ∆t34(2) = 0.001 ⋅ π / ω02 the coherence factor
S4 (ω ) ≈ 16 and at low harmonics four electrons radiate as a charged particle with
the charge 4e and the rest mass 4m0 (curve 2 in Fig. 2), i.e. by a factor of sixteen
int
int
−11
higher than a single electron ( Pmed
erg/s). In the
2 = 15.71 ⋅ Pmed 1 = 0.3398 ⋅ 10
frequency range of 0 − 50ω02 we have obtained a superradiant regime (curve 2 in
Fig. 2) for such the electron system so far as the dimension of this system is smaller
in comparison to the radiation wavelength [25].
120
200
-16
W(ω)⋅ω0j (10 ⋅erg/s)
100
-15
W(ω)⋅ω0j (10 ⋅ erg/s)
160
80
120
1
60
2
80
40
20
40
0
0
0
10
20
ω/ω0j
30
40
50
1
0
10
20
ω/ω0j
30
40
50
Fig. 2. Superrradiance in synchrotron-Cherenkov
Fig. 1. Synchrotron-Cherenkov radiation
radiation spectrum. Curve 2. Four electrons
spectrum. Curve 1. One electron moving in a
moving one by one in a spiral in medium for
spiral in medium with a radiation power
int
int
(2)
int
−12
erg/s, Pmed
Pmed
∆t12(2) = ∆t23
= ∆t34(2) = 0.001 ⋅ π / ω02 with
2 / Pmed 1 = 15.71 .
1 = 0.2163 ⋅ 10
int
−11
erg/s.
Pmed
2 = 0.3398 ⋅ 10
We have found the oscillations in the radiation spectrum of one electron
(curve 3 in Fig. 3) as well as in that of four electrons moving one by one with a
small selected time shift of 0.001π/ω04 (curve 4 in Fig. 4). This result is in good
agreement with [18]. The coherence factor decreases with increasing number of
harmonics (curve 4 in Fig. 4).
-16
W(ω)⋅ω0j (10 ⋅erg/s)
400
-15
W(ω)⋅ω0j (10 ⋅erg/s)
400
300
300
3
200
200
100
100
0
0
0
40
80
ω/ω0j
4
120
160
200
3
0
40
80
ω/ω0j
120
160
200
Fig. 3. Oscillations in synchrotron-Cherenkov
radiation spectrum. Curve 3. One electron moving
in a spiral in medium with radiation power
Fig. 4. Oscillations in synchrotron-Cherenkov
radiation spectrum. Curve 4. Four electrons
moving in a spiral for
int
−11
Pmed
erg/s.
3 = 0.3985 ⋅ 10
( 4)
( 4)
( 4)
∆t12
= ∆t 23
= ∆t34
= 0.001 ⋅ π / ω04 with
int
int
Pmed
4 / Pmed 3 = 12.49 .
int
−10
radiation power Pmed
erg/s.
4 = 0.4977 ⋅ 10
A.V. Konstantinovich, I.A. Konstantinovich
50
200
6
800
-15
-15
W(ω)⋅ω0j (10 ⋅ erg/s)
W(ω)⋅ω0j (10 ⋅ erg/s)
150
6
600
2
100
400
50
3
200
5
0
0
10
20
ω/ω0j
30
40
50
0
0
40
80
120
160
ω/ω0j
200
Fig. 5. Synchrotron-Cherenkov radiation spectrum
at low harmonics. Curve 5. Four electrons moving
one by one in a spiral in transparent medium for
Fig. 6. Oscillations in synchrotron-Cherenkov
radiation spectrum. Curve 6. Four electrons
moving one by one in a spiral in transparent
(5)
( 5)
( 5)
∆t12
= ∆t 23
= ∆t34
= 0.1 ⋅ π / ω05 with
medium for ∆t12 = ∆t 23 = ∆t34 = 0.1 ⋅ π / ω06
int
−12
radiation power Pmed
erg/s,
5 = 0.8804 ⋅ 10
int
−10
with radiation power Pmed
6 = 0.1598 ⋅ 10
int
int
Pmed
5 / Pmed 1 = 4.070
int
int
erg/s, Pmed 6 / Pmed 3 = 4.010
(6)
( 6)
( 6)
We have found the oscillations in the radiation spectrum of one electron
(curve 3 in Fig. 3) as well as in that of four electrons moving one by one with a
small selected time shift of 0.001π/ω04 (curve 4 in Fig. 4). This result is in good
agreement with [18]. The coherence factor decreases with increasing number of
harmonics (curve 4 in Fig. 4).
In the case of essentially higher time shifts between four electrons
(6)
(6)
= ∆t34(6) = 0.1 ⋅ π / ω06 ) we have found that there is no radiation at
( ∆t12 = ∆t23
frequencies 5 (4 i − 3) ω06 , 5 ( 4i − 2 ) ω06 , and 5 ( 4i − 1) ω06 , (i = 1,2,…,10) (curve
5 in Fig. 5 and curve 6 in Fig. 6) and at the frequencies 20i (i = 1,2,…,10) the
coherence factor takes the maximum value equal to sixteen.
The influence of the Doppler effect on our spectra is well seen for the first
radiation harmonics until the overlapping between the forward and back radiations
does not become essential.
4. CONCLUSIONS
The calculated radiation spectra for the synchrotron-Cherenkov radiation
have an oscillating behaviour. For small time shifts between four electrons moving
along a spiral in a medium, at low harmonics they radiate as a charged particle with
the charge 4e and the rest mass 4m0 , i.e. by a factor of sixteen higher than a single
electron.
7
Radiation spectrum of electrons moving in spiral in medium
51
For such the electron system in the frequency range of 0 − 50ω02 we have
found the existence of superradiant regime so far as the dimension of this system is
smaller in comparison to the radiation wavelength.
The influence of the Doppler effect determines the band’s boundaries of the
separate harmonics in the radiation spectra for the systems of one and four
electrons.
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