Eikonal approximation in the transition radiation theory
Simple variant of eikonal approximation in transition radiation theory is proposed. Spectral-angular density of radiation on fiber-like targets and nanotubes is calculated using Born and eikonal approximations. The conditions of validity of both approximations are considered. Запропоновано простий в...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Cite this: | Eikonal approximation in the transition radiation theory / N.F. Shul’ga and V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 210-213. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | Eikonal approximation in the transition radiation theory / N.F. Shul’ga and V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 210-213. — Бібліогр.: 10 назв. — англ. |
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| description | Simple variant of eikonal approximation in transition radiation theory is proposed. Spectral-angular density of radiation on fiber-like targets and nanotubes is calculated using Born and eikonal approximations. The conditions of validity of both approximations are considered.
Запропоновано простий варіант ейконального наближення в теорії перехідного випромінювання. Розраховано спектрально-кутову густину випромінювання на нитковидних мішенях та нанотрубках з використанням борнiвського та ейконального наближень. Розглянуто умови застосовності обох наближень.
Предложен простой вариант эйконального приближения в теории переходного излучения. Рассчитана спектрально-угловая плотность излучения на нитевидных мишенях и нанотрубках с использованием борновского и эйконального приближений. Рассмотрены условия применимости обоих приближений.
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EIKONAL APPROXIMATION IN THE TRANSITION
RADIATION THEORY
N.F. Shul’ga1, and V.V. Syshchenko2
1A.I. Akhiezer Institute of Theoretical Physics,
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: shulga@kipt.kharkov.ua;
2Belgorod State University, Belgorod, Russian Federation;
e-mail: syshch@bsu.edu.ru
Simple variant of eikonal approximation in transition radiation theory is proposed. Spectral-angular density of
radiation on fiber-like targets and nanotubes is calculated using Born and eikonal approximations. The conditions of
validity of both approximations are considered.
PACS: 41.60.-m
1. INTRODUCTION
Transition radiation (TR) arises, in general case,
when a charged particle moves in a medium with chang-
ing dielectric properties [1-3]. It was demonstrated in
[4] that characteristics of TR (spectral-angular density
and polarization) strongly depend on the details of the
target geometry. However, precise calculations of TR
characteristics could be carried out only in the cases of
the simplest geometry, like the normal incidence of the
particle onto the plane border between substances with
different dielectric properties. To calculate spectral-
angular density of TR in more complex situations we
need some approximate methods.
We use the system of units in which the velocity of
light c = 1.
2. BORN APPROXIMATION
IN TRANSITION RADIATION THEORY
The spectral-angular density of TR could be written
in a form [5]
2
22
2
)8(
Ik ×=
Ε
π
ω
οωdd
d , (1)
where ω and k are the frequency and wave vector of
radiated wave, and
∫ −= − )())(1(3 rErI kr
ωωε
ierd . (2)
Here εω is the dielectric permittivity of the target
substance (we assume that it tends to unit on large dis-
tances from the target), and Eω is time Fourier compo-
nent of the electric field produced in the target sub-
stance by the passing particle. For small radiation angles
we can write (1) in a form
2
22
4
)8( ⊥≈
Ε I
π
ω
ω dod
d , (3) d
dyL
dd
= ∫
where I⊥ is the component of I perpendicular to the par-
ticle velocity v.
For high frequencies the dielectric permittivity could
be written in plasma form:
22 /1 ωωεω p−= , (4)
where ω p is the plasma frequency. If the value (1 - εω )
in (2) could be treated as small perturbation, we can
substitute into (2) non-disturbed Coulomb field of the
moving particle,
=⊥ γ
ωρ
ργ
ω
ω
ω v
Ke
v
e z
v
iC
12
)( 2)( ρrE (5)
(where ρ and z are the longitudinal and transverse com-
ponents of radius-vector r, γ is Lorentz-factor of the par-
ticle, K1(x) is the modified Bessel function of the third
kind), instead of complete electric field Eω in the target
substance. This approach is analogous to Born approxi-
mation in quantum scattering theory (see, e.g., [6]).
Fig. 1. Incidence of a particle onto fiber-like target
Born approximation gives the possibility to consider
TR on targets with rather complex geometry, like di-
electric fibers and nanotubes [5,7]. Let the ultrarelativis-
tic electron incident onto the fiber-like target under
small angle ψ to its axis (Fig. 1). The radiation effi-
ciency (that is the spectral-angular density of radiation
integrated over impact parameters) could be written in
this case in the form
),(2
26
0 ϕθ
ψω
γ
οω
ψ
οω
F
m
nLe
d
ddK e=
Ε , (6)
where L is the total length of the fiber, ne is the number
of electrons per unit length of the fiber, and F is a func-
tion that determines the angular distribution of radiation.
This function is plotted in Fig. 2 for different fibers with
the same characteristic radius R. We can see that the in-
tensity and angular distribution of TR substantially de-
pend on the details of the fiber structure. This fact creates
new possibilities in diagnostics of nanostructures.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 210-213. 210
Fig. 2. The function F (θ,ϕ) for uniform cylindrical
fiber (upper plot), fiber with Gaussian distribution of
the electron density in the plane perpendicular to the
fiber axis (lower plot) and nanotube (middle plot) with
the same characteristic radius, Rω /γ = 0.2ψγ . Here
θ x = θ cosϕ , θ y = θ sinϕ , direction of the particle’s
velocity corresponds to θ x = θ y = 0 , direction of the
fiber’s axis corresponds to θ x = ψ , θ y = 0; ψ = 10 -3,
γ = 2000
However, Born approximation in TR theory is valid
only in the range of very high frequencies of radiated
waves (that will be discussed below). So, we need to
develop another approximate method with wider range
of validity for radiation frequencies. One of possible
ways to do this is to use the eikonal approximation.
3. EIKONAL APPROXIMATION
IN TRANSITION RADIATION THEORY
Previously different authors had proposed their vari-
ants of eikonal approximation in TR theory. Particu-
larly, M.L. Ter-Mikaelyan had done that in his well-
known monograph [2]. However, his method substan-
tially used one-dimensional character of the problem
considered.
Another variant, based on the construction of Green
function for Maxwell equations with charge and current
of the moving particle, was developed by Alikhanyan
and Chechin in [8]. However, their method leads to very
awkward calculations in application to problems with
complex geometry of the target.
In the present article we propose the variant of eiko-
nal approximation based on the method of equivalent
photons (see, e.g., [9]). In this approach we approximate
the Coulomb field of the incident particle (5) by the
packet of plane waves that permits us to describe the
evolution of the field in target substance using free
Maxwell equations without particle charge and current:
( ) ( ) ωωωω εωω EEE −+=+∆ 1divgrad 22 . (7)
According to the idea of eikonal approximation (see,
e.g., [6]), we shall find the solution in the form of plane
wave multiplied by the slowly changing function,
)()( rrE Φ= zie ω
ω , (8) 2
<<
ω
ω p
with the following boundary condition under z → -∞
(that is before interaction with the target):
=⊥ γ
ωρ
ργ
ω ω
ω v
Ke
v
e zi
12
)0( 2)( ρrE .
As a result, we obtain the following expression for
vector I⊥:
∫ ⊥−
⊥ = ƒÏ2
2
4 kI ied
v
ei ρ
γ
γ
ωρ
ρ v
K1
ρ
( )
−
−−× ∫
∞
∞−
1)(1
2
exp dzi rωε
ω . (9)
Let us compare the ranges of validity of both devel-
oped approximations in TR theory. All conditions [10]
are summarized in Table 1. We see that eikonal results
would be valid in more soft range of the spectrum, then
Born ones (condition (I)), and have no restriction on the
target thickness l along the particle trajectory (condition
(II)). On the other hand, validity of the eikonal ap-
proximation needs the target with rather smoothly
changing dielectric properties in transverse direction
(condition (III), where ρe f f is the effective transverse
distance on which dielectric properties of the target are
changing substantially).
Table 1. Conditions of validity of approximations
in TR theory
Condi-
tion Born approximation Eikonal approximation
(I)
Hard range of radia-
tion spectrum:
2
2
2
−<< γ
ω
ω p
More soft range of the
spectrum:
( ) 1, 2
2
22 <<<<−
ω
ω
γθ p
(II)
Thin target:
12
2
<<lp
ω
ω
ω
No restriction
(III) No restriction 1−>> pffe ωρ
(IV) No restriction
Small radiation angles
1
2
2
<<l
θ
ω
For our problem of TR on the fiber-like target these
conditions take the form presented in Table 2.
Consider now two limiting cases. At first, let us find
the range of parameters in which the results of Born and
eikonal approximations coincide. It could be demon-
strated that for this goal we must expand the condition
(II) to the eikonal formulae, the condition (IV) to the
Born ones and also apply the additional condition (V)
(see Table 3).
Table 2. Conditions of validity of approximations
for radiation on fiber-like target
Condi-
tion Born approximation Eikonal approximation
(I) 2
2
2
−<< γ
ω
ω p ( ) 1, 2
2
22 <<<<−
ω
ω
γθ p
(II) 12
2
ψω
R
No restriction
(III) No restriction 1>>pRω
(IV) No restriction 12 <<
ψ
ωθ
R
211
Table 3. Ranges of parameters in which both approxi-
mations give the same results
Соndi-
tion Born approximation Eikonal approximation
(I) 2
2
2
−<< γ
ω
ω p ( ) 1, 2
2
22 <<<<−
ω
ω
γθ p
(II) 12
2
2
<<
ψω
ω
ω
Rp No restriction
(III) No restriction 1>>pRω
(IV) No restriction
12 <<
ψ
ωθ
R
and 12 <<−
ψ
R
ωγ
(V) 1>>ψγ
Fig. 3. The function F for the uniform cylindrical fi-
ber in the case when Born and eikonal approximations
give the same results: Rω /γ = 0.015ψγ , ψ = 0.1,
γ = 2000, ωp = 20 eV (for example, ω = 13 keV,
R = 10 -5 cm)
In this range of parameters the angular distribution
of TR on the uniform cylindrical fiber will look like it is
pictured on Fig. 3. We see that the angular distribution
has the form of empty cone that is characteristic to the
problem on TR under normal incidence of the particle
onto flat dielectric plate. However, the angular distribu-
tion in our case do not possess axial symmetry around
the particle velocity. This fact means that in the given
range of parameters TR just carries some information
about the target geometry.
Another interesting limiting case is the case of very
thick fiber, when its radius is much larger than the char-
acteristic transverse dimension of the relativistically
compressed Coulomb field of the incident particle,
R >> γ / ω (see Fig. 4).
Fig. 4. Incidence of the particle onto thick fiber,
R >> γ / ω
In this case we obtain from (9)
22 )/(
8
γωω
π
+
=
⊥
⊥
⊥
kv
e kI
−
−−× 12
2
exp 2
0
2
2
2
yRi p
ψω
ωω .
This result could be simply interpreted when we com-
pare it to the corresponding formula for the case of
normal incidence of the particle onto flat dielectric plate
of thickness a:
22 )/(
8
γωω
π
+
=
⊥
⊥
⊥ kv
e kI
−
− 1
2
exp 2
2
ai p
ω
ωω .
We can see that very thick fiber looks for the incident
particle like a plate with local thickness 2
0
22 yR −
ψ
.
Hence, in the limit of thick fiber the angular distribution
of TR has a shape like in the case of flat plate (Fig. 5).
The details of the fiber structure could make influence
only to the absolute value of radiation intensity, not to
the shape of angular distribution.
4. CONCLUSION
The simple variant of eikonal approximation in TR
theory based on the method of equivalent photons is
developed.
The results of eikonal and Born approximations co-
incide in a rather wide range of parameters. This fact
permits to expand result obtained using Born approxi-
mation to more soft range of the radiation spectrum.
Fig. 5. The function F for the uniform cylindrical fi-
ber in the case Rω /γ = 0.15ψγ , ψ = 0.1, γ = 2000
Spectral-angular distribution of TR is sensitive to the
details of the target geometry, that could be used in di-
agnostics of nanostructures.
This work is accomplished in the content of the Pro-
gram “Advancement of the scientific potential of high
education” by Russian Ministry of Education and Sci-
ence (project RNP.2.1.1.1.3263), Russian Foundation
for Basic Research (project 05-02-16512) and the inter-
nal grant of Belgorod State University.
REFERENCES
1. V.L. Ginzburg, V.N. Tsytovich. Transition Radia-
tion and Transition Scattering. Bristol, New York:
Adam Hilger Publ., 1990, 360 p.
212
2. M.L. Ter-Mikaelyan. High-Energy Electromagnetic
Processes in Condensed Media. New York:
“Wiley”, 1972, 457 p.
3. G.M. Garibian, C. Yang. X-ray Transition Radia-
tion. Yerevan: Publ. Acad. of Science of Armenia,
1983, 320 p. (in Russian).
4. N.F. Shul’ga, S.N. Dobrovol’skii. Theory of transi-
tion radiation from relativistic electron in thin metal
target //JETP. 2000, v. 90, p. 579-583.
5. N.F. Shul’ga, V.V. Syshchenko. Polarization of tran-
sition radiation on some sorts of targets //Problems
of Atomic Science and Technology. 2001, N6(1),
p. 137-137.
6. R.G. Newton. Scattering theory of waves and parti-
cles. New York: “McGraw-Hill Book Company”,
1968, 608 p.
7. N.F. Shul’ga, V.V. Syshchenko. Transition radiation
of high energy particles on fiber-like targets //Phys.
Lett. A. 2003, v. 313, p. 307-311.
8. A.I. Alikhanyan, V.A. Chechin. Application of eik-
onal approximation in X-ray transition radiation the-
ory //Proc. P.N. Lebedev Phys. Inst. 1982, v. 140,
p. 146-158 (in Russian).
9. J.D. Jackson. Classical Electrodynamics. New York:
“John Wiley & Sons”, 1998.
10. N.F. Shul’ga, V.V. Syshchenko. Transition radiation
by relativistic electrons in inhomogeneous substance
//H. Wiedemann (ed.), Advanced Radiation Sources
and Applications. Dordrecht: “Springer”, 2006,
p. 129-148.
ЭЙКОНАЛЬНОЕ ПРИБЛИЖЕНИЕ В ТЕОРИИ ПЕРЕХОДНОГО ИЗЛУЧЕНИЯ
Н.Ф. Шульга, В.В. Сыщенко
Предложен простой вариант эйконального приближения в теории переходного излучения. Рассчитана
спектрально-угловая плотность излучения на нитевидных мишенях и нанотрубках с использованием бор-
новского и эйконального приближений. Рассмотрены условия применимости обоих приближений.
ЕЙКОНАЛЬНЕ НАБЛИЖЕННЯ В ТЕОРIΪ ПЕРЕХIДНОГО ВИПРОМIНЮВАННЯ
М.Ф. Шульга, В.В. Сищенко
Запропоновано простий варіант ейконального наближення в теорії перехідного випромінювання. Розра-
ховано спектрально-кутову густину випромінювання на нитковидних мішенях та нанотрубках з використан-
ням борнiвського та ейконального наближень. Розглянуто умови застосовності обох наближень.
213
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| id | nasplib_isofts_kiev_ua-123456789-110954 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T19:02:03Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shul’ga, N.F. Syshchenko, V.V. 2017-01-07T14:16:11Z 2017-01-07T14:16:11Z 2007 Eikonal approximation in the transition radiation theory / N.F. Shul’ga and V.V. Syshchenko // Вопросы атомной науки и техники. — 2007. — № 3. — С. 210-213. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/110954 Simple variant of eikonal approximation in transition radiation theory is proposed. Spectral-angular density of radiation on fiber-like targets and nanotubes is calculated using Born and eikonal approximations. The conditions of validity of both approximations are considered. Запропоновано простий варіант ейконального наближення в теорії перехідного випромінювання. Розраховано спектрально-кутову густину випромінювання на нитковидних мішенях та нанотрубках з використанням борнiвського та ейконального наближень. Розглянуто умови застосовності обох наближень. Предложен простой вариант эйконального приближения в теории переходного излучения. Рассчитана спектрально-угловая плотность излучения на нитевидных мишенях и нанотрубках с использованием борновского и эйконального приближений. Рассмотрены условия применимости обоих приближений. This work is accomplished in the content of the Program “Advancement of the scientific potential of high education” by Russian Ministry of Education and Science (project RNP.2.1.1.1.3263), Russian Foundation for Basic Research (project 05-02-16512) and the internal grant of Belgorod State University. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники QED processes in strong fields Eikonal approximation in the transition radiation theory Ейкональне наближення в теорiϊ перехiдного випромiнювання Эйкональное приближение в теории переходного излучения Article published earlier |
| spellingShingle | Eikonal approximation in the transition radiation theory Shul’ga, N.F. Syshchenko, V.V. QED processes in strong fields |
| title | Eikonal approximation in the transition radiation theory |
| title_alt | Ейкональне наближення в теорiϊ перехiдного випромiнювання Эйкональное приближение в теории переходного излучения |
| title_full | Eikonal approximation in the transition radiation theory |
| title_fullStr | Eikonal approximation in the transition radiation theory |
| title_full_unstemmed | Eikonal approximation in the transition radiation theory |
| title_short | Eikonal approximation in the transition radiation theory |
| title_sort | eikonal approximation in the transition radiation theory |
| topic | QED processes in strong fields |
| topic_facet | QED processes in strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110954 |
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