Skin-effect influence on transition radiation
It is considered the transition radiation for the normal incidence of a charged particle on the boundary of plasma medium in the conditions of anomalous skin effect, at the frequencies much less than plasma frequency. The problem is solved in the assumption that electron scattering from the boundary...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Zitieren: | Skin-effect influence on transition radiation / V.I.Miroshnichenko, V.M. Ostroushko2 // Вопросы атомной науки и техники. — 2015. — № 3. — С. 103-108. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859520115306921984 |
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| author | Miroshnichenko, V.I. Ostroushko, V.M. |
| author_facet | Miroshnichenko, V.I. Ostroushko, V.M. |
| citation_txt | Skin-effect influence on transition radiation / V.I.Miroshnichenko, V.M. Ostroushko2 // Вопросы атомной науки и техники. — 2015. — № 3. — С. 103-108. — Бібліогр.: 17 назв. — англ. |
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| description | It is considered the transition radiation for the normal incidence of a charged particle on the boundary of plasma medium in the conditions of anomalous skin effect, at the frequencies much less than plasma frequency. The problem is solved in the assumption that electron scattering from the boundary is partially specular and partially diffusive. The spectral density of the radiated energy is obtained for the cases of uniform particle motion and of the motion with two running across the boundary.
Розглянуто перехідне випромінювання для нормального падіння зарядженої частинки на межу плазмового середовища в умовах аномального скін-ефекту, на частотах, значно менших від плазмової. Задачу розв’язано в припущенні, що відбиття електронів від межі є частково дзеркальним, частково дифузним. Отримано спектральну густину випроміненої енергії для випадків рівномірного руху частинки та руху з дворазовим перетинанням межі.
Рассмотрено переходное излучение для нормального падения заряженной частицы на границу плазменной среды в условиях аномального скин-эффекта, на частотах, значительно меньших, чем плазменная. Задача решена в предположении, что отражение электронов от границы является частично зеркальным, частино диффузным. Получена спектральная плотность излученной энергии для случаев равномерного движения частицы и движения с двукратным пересечением границы.
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| first_indexed | 2025-11-25T20:53:33Z |
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ELECTRODYNAMICS
SKIN-EFFECT INFLUENCE ON TRANSITION RADIATION
V. I.Miroshnichenko1 , V.M.Ostroushko2 ∗
1 Institute of Applied Physics of National Academy of Science, 40030, Sumy, Ukraine;
2National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received April 3, 2015)
It is considered the transition radiation for the normal incidence of a charged particle on the boundary of plasma
medium in the conditions of anomalous skin effect, at the frequencies much less than plasma frequency. The problem
is solved in the assumption that electron scattering from the boundary is partially specular and partially diffusive.
The spectral density of the radiated energy is obtained for the cases of uniform particle motion and of the motion
with two running across the boundary.
PACS: 41.60.-m
1. INTRODUCTION
Whet a particle crosses the plasma boundary the
transition radiation arises. Its characteristics are
studied for different situations. In particular, the
problem was solved with account of spatial disper-
sion [1]. Also, the characteristics of the transition ra-
diation generated by the particle, which crosses the
boundary soon after collision, are studied [2]. With
aim to develop generators of transition radiation the
characteristics of radiation generated by modulated
beams are obtained [3], and experimental investiga-
tions of generation of wide-range transition radiation
with use of pulsed accelerators of direct action are
carried out [4]. Transition radiation as elementary
mechanism is the base of operation of some other de-
vices, in particular, monotron [5].
If plasma medium is metal at low temperature,
the radiation may be realized in the conditions of
anomalous skin effect. Its theory to the consider-
able extent was built in [6] and [7], where the re-
flection of an electromagnetic wave from the plasma
with sharp boundary is considered, and the scattering
of electrons from the boundary is characterized with
some proportion between electrons scattered specu-
larly and diffusely. This proportion depends on the
angle of electron incidence. In [8], the problem is
solved for an arbitrary such dependence, with use
of expansion into Neumann series. But considerable
amount of results, in particular, of exact ones, were
obtained in the assumption that the proportion is
constant. In particular, in such assumption the prob-
lem of normal wave incidence in maximum anomalous
skin effect conditions was solved [9] and the character-
istics of longitudinal field penetration into plasma in
the near conditions are determined [10]. Also, in [11],
the problem of normal incidence, in the assumption
that distribution function of the scattered electrons
is fixed up to the factor, which describes the type of
electron scattering from the boundary, is solved ex-
actly, and in [12], the explicit relationships for plasma
layer are obtained.
The main object of the present work is to ob-
tain the amplitude of radiation for normal particle
incidence on the locally isotropic plasma with the
sharp boundary at the frequencies much greater than
collision frequency, but much less than plasma fre-
quency, in presence of considerable spatial disper-
sion. In the next sections the construction of solving
is described. The method, in comparing with one of
[13] (where an oblique incidence of an electromagnetic
wave on plasma is considered), is somewhat changed,
and some designations are introduced in a different
way. In the section next to the last, the question of
efficiency of generation of wide-range radiation with
use of the pulsed accelerators of direct action is dis-
cussed.
2. INITIAL RELATIONSHIPS
Let a particle with charge Z0e0 moves along OZ axis
with velocity β0ce⃗z, where e⃗z is unit vector of OZ
axis, c is the speed of light, e0 is electron charge,
β0 ∈ (−1, 1), β0 ̸= 0, and the plasma medium is
in the half-space z > 0. Maxwell equations there
may be written in the form rotE⃗ + c−1(∂/∂t)H⃗ = 0,
rotH⃗ − c−1(∂/∂t)E⃗ − 4πc−1(⃗j + j⃗0) = 0, where
j⃗0 = Z0e0δ(x)δ(y)δ(z − β0ct)β0ce⃗z, j⃗ = e0
∫
d3v⃗v⃗f ,
the perturbation, f = f(v⃗, r⃗, t), of electron
distribution function obeys the equation
(∂/∂t)f + v⃗(∂/∂r⃗)f + (e0/m)E⃗(∂/∂v⃗)f0 + νf = 0,
m is electron mass, ν is collision frequency, the un-
perturbed electron distribution function f0 is taken
for the isotropic Fermi distribution with zero tem-
perature, f0 = 3n0(4πv
3
F )
−1 at v < vF , f0 = 0 at
v > vF , vF is electron velocity at Fermi level, n0 is
electron density. Electron flow from the boundary
∗Corresponding author E-mail address: ostroushko-v@kipt.kharkov.ua
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.103-108.
103
into plasma is characterizes by the part, p ∈ (0, 1), of
electrons scattered from the boundary specularly (the
rest is scattered diffusely) and obeys the boundary
condition f(vz) = pf(−vz) for vz > 0 at z = 0. It is
assumed that Fourier transformation with the factor
exp[ı̇ωc−1(ct− kxx− kyy − kzz)] and the integra-
tion over intervals t ∈ (−∞,+∞), x, y ∈ (−∞,+∞),
z ∈ (0,+∞) is applied. Let us put
qλ(η) = 3η−2{(2η)−1log[(1 + η)/(1− η)]− 1},
qτ (η) = 3{1− (2η)−1
(
1− η2
)
×
×log[(1 + η)/(1− η)]}/(2η2),
Qλ(kz) = 1− Ω2qλ(β(k
2
⊥ + k2z)
1/2),
Qτ (kz) = 1− k2⊥ − k2z − Ω2qτ (β(k
2
⊥ + k2z)
1/2),
Ψλ(kz) = ωc−1[k⊥E⊥(kz) + kzEz(kz)],
Ψτ (kz) = ωc−1[k⊥Ez(kz)− kzE⊥(kz)],
Φλ(kz) = kzIz(kz) +
+Qλ(kz)[Ψλ(kz) + pΨλ(−kz)], (1)
Φτ (kz) = k⊥Iz(kz) +
+Qτ (kz)[Ψτ (kz)− pΨτ (−kz)]. (2)
Here k⊥ = |⃗k⊥|, k⃗⊥ = kxe⃗x + ky e⃗y, e⃗x and e⃗y
are unit vectors of the axes OX and OY,
E⊥ is projection of Fourier component of elec-
tric field strength on the vector k⃗⊥ direc-
tion, β = vFω[c(ω + ı̇ν)]−1, Ω = ωe[ω(ω + ı̇ν)]−1/2,
ωe = (4πe20n0/m)1/2, Iz(kz) = Iz0/(kz − kz0),
kz0 = β−1
0 , Iz0 = 4πsign(β0)Z0e0/ω. The func-
tions Ψλ(kz) and Ψτ (kz) (and the functions E⊥(kz)
and Ez(kz)) should be analytical in the half-plane
Imkz < 0 and in the point kz = −kz0. The func-
tions Φλ(kz) and Φτ (kz) correspond to some linear
combinations of the left hand sides of Maxwell equa-
tions written above, and they should be analytical in
the half-plane Imkz > 0 and in the point kz = kz0,
in connection with validity of the equations in the
half-space z > 0. The functions Ψλ(kz) and Ψτ (kz)
should be bounded in the half-plane Imkz < 0, and
also, the equalities Ψ(±ı̇k⊥) = 0 for the function
Ψ(kz) = k⊥Ψλ(kz)− kzΨτ (kz) should be held. With
the method similar to one used in [6], [14], and [15]
for the problem of wave incidence on the medium,
each of the equalities, (1) or (2), together with the
requirements of analyticity, is reduced to Riemann-
Hilbert boundary problem for the pair of functions,
{Φλ,τ (kz), Ψλ,τ (−kz)} and {Φλ,τ (−kz), Ψλ,τ (kz)},
analytical in the different half-planes. For the bound-
ary (at z → 0+) values of relevant field components,
Ẽz(z), Ẽ⊥(z), and H̃φ(z) (the unit vector of φ di-
rection is e⃗φ = [e⃗z, k⃗⊥/k⊥]), which were obtained
with integration only with respect to t, x and y, and
with the factor exp[ı̇ωc−1(ct− kxx− kyy)], one has
the equalities Ẽz(0+) = ı̇Ψλ(∞), Ẽ⊥(0) = −ı̇Ψτ (∞),
and H̃φ(0) = limu→∞{−ı̇u[Ψτ (u)−Ψτ (∞)]} (the
letter u, and the letter w below, sometimes is used
instead of kz as arguments of functions; the values
of Ẽz(0+) and Ẽz(0−) for p ̸=1 may be different,
in connection with existence of infinitely thin charge
layer varying with time at the sharp plasma boundary
[16]). Let us denote A = (βΩ)2/3. It is assumed, that
|β0| ∼ 1, vF ≪ c, and the frequency is considered, for
which ν ≪ ω, |A| ≫ 1 (so, skin effect is close to max-
imum anomalous), moreover, the collision frequency
ν is considered as infinitesimal, and its positive-
ness is used only for ascertaining the rules of path
tracing around singularities in the complex plane.
Also, it is assumed, that k⊥ ∈ (0, 1). The waves
with k⊥ > 1 are not emitted into the free half-space
z < 0, as they decreases there exponentially with
z → −∞. The functions Qλ,τ (kz) have the branch-
ing points ±q, where q = (β−2 − k2⊥)
1/2, Imq → 0+
with ν → 0+. Let the functions Q+
λ,τ (kz) are ana-
lytical in the half-plane Imkz > 0 and near the point
q, and the equalities Q+
λ,τ (kz)Q
+
λ,τ (−kz) = Qλ,τ (kz),
Q+
λ (∞) = 1, and limu→∞[Q+
τ (u)/u] = 1 take place,
and let Q×
λ,τ (kz) = Q+
λ,τ (kz)/Q
+
λ,τ (−kz). If the cut Γ
in the half-plane Imkz > 0 from the point q to infin-
ity is made then analytical extension of the functions
Qλ,τ (kz) with different path tracing around the point
q gives the different values of the functions, so that
for the values at the different cut sides the equalities
Qλ,τ (kz(1− ı̇0))−Qλ,τ (kz(1 + ı̇0)) =
= −Ω2∆λ,τ (β(k
2
z + k2⊥)
1/2)
with ∆λ(η) = −3πı̇η−3, ∆τ (η) = 3πı̇(η−3 − η−1)/2
take place. Denoting Xλ,τ (kz) = Ψλ,τ (−kz)Q
+
λ,τ (kz),
Yλ,τ (kz) = Φλ,τ (kz)/Q
+
λ,τ (kz), one comes to the
equations,
Xλ(kz) + pQ×
λ (kz)Xλ(−kz) =
= Yλ(−kz) + kzIz(−kz)/Q
+
λ (−kz), (3)
Xτ (kz)− pQ×
τ (kz)Xτ (−kz) =
= Yτ (−kz)− k⊥Iz(−kz)/Q
+
τ (−kz), (4)
and to the requirements of analyticity of the functions
Xλ,τ (kz) and Yλ,τ (kz) in the half-plane Imkz > 0 and
in the points q and kz0, and also, the limiting val-
ues of the quantities Xλ(kz), Yλ(kz), Xτ (kz)/kz, and
Yτ (kz)/kz at kz → +ı̇∞ should be bounded.
3. THE EQUATIONS FOR
LONGITUDINAL FIELD
From the equation (3), representing some terms as
sums of the functions analytical in the different half-
planes, Imkz > 0 and Imkz < 0, and transforming the
equation in such a way that each its side is analytical
in one of the half-planes and tends there to zero with
kz → ∞, for Imkz > 0 one can get the equality
Xλ(kz)− Iz0kz0[(kz + kz0))Q
+
λ (kz0)]
−1 +
+(2πı̇)−1p
∫
dw(w − kz)
−1 ×
×[Q×
λ (w)Xλ(−w)−Xλ(∞)]−Xλ(∞) = 0. (5)
The path of integration in (5) is symmetrical with
respect to zero, goes near the real axis in its positive
direction, and the points kz, kz0, and q are to be to
left of the path. Replacing w with −w and moving
the path to the cut Γ, one can get the equation
Xλ(kz)− Iz0kz0[(kz + kz0))Q
+
λ (kz0)]
−1 =
= Xλ(∞)− pK̂λ[kz, w;Xλ(w)], (6)
104
in which an action of the operator K̂ on a function
f(w) is defined with the equalities
K̂λ,τ [u,w; f(w)] =
=
∫
Γ
dw(u+ w)−1Kλ,τ (w)f(w),
Kλ,τ (w) = (2πı̇)−1Ω2[Q+
λ,τ (w)]
−2 ×
×∆λ,τ (β(w
2 + k2⊥)
1/2)
(the designations with index τ are used below). Sim-
ple manipulations give the equation
Xλ(kz)/kz + Iz0[(kz + kz0))Q
+
λ (kz0)]
−1 =
= Xλ(0)/kz + pK̂λ[kz, w;Xλ(w)/w], (7)
solution of which may be given as linear combination,
Xλ(kz) = Xλ(0)X
r
λ(kz)− Iz0[Q
+
λ (kz0)]
−1Xe
λ(kz), of
the solutions of two equations,
Xr
λ(kz)/kz − k−1
z = pK̂λ[kz, w;X
r
λ(w)/w], (8)
Xe
λ(kz)/kz − (kz + kz0)
−1 =
= pK̂λ[kz, w;X
e
λ(w)/w]. (9)
From the integral equations (8) and (9), the values of
Xr,e
λ (kz) at Γ may be found, and then these equations
may be used as explicit formulae for Xr,e
λ (kz) in all
complex plane of kz, except of the cut, symmetrical
to Γ with respect to zero.
4. THE EQUATIONS FOR TRANSVERSE
FIELD
For the known Ẽ⊥(0) and H̃φ(0), the solution of (4)
may be given with the linear combination,
Xτ (kz) = −ı̇X(p; kz)H̃φ(0) +
+Iz0k⊥[Q
+
τ (kz0)]
−1Xe
τ (kz)
+[ı̇kzX(−p; kz)− cτΨτ1X(p; kz)]Ẽ⊥(0).
Here Ψτ1 = ı̇c−1
τ limu→∞[uX(−p;u)−Q+
τ (u)],
cτ = exp(−ı̇π/6)A/β, Xe
τ (kz) and X(±p; kz) are the
solutions of the functional equations
Xe
τ (kz)− pQ×
τ (kz)X
e
τ (−kz) =
= Y e
τ (−kz) + (kz + kz0)
−1
and X(±p; kz)∓ pQ×
τ (kz)X(±p;−kz) = Y (±p;−kz)
with the requirements of analyticity of the functions
Xe
τ (kz), Y e
τ (kz), X(±p; kz), and Y (±p; kz) in the
half-plane Imkz > 0 and in the points q and kz0, and
the requirements Xe
τ (kz) → 0 and X(±p; kz) → 1 for
kz → +ı̇∞. The way similar to one used in deducing
of the equations (6) and (7) leads to the equations
Xe
τ (kz)− (kz + kz0)
−1 = pK̂τ [kz, w;X
e
τ (w)],
X(p; kz)− 1 = pK̂τ [kz, w;X(p;w)], (10)
X(−p; kz)/kz −X(−p; 0)/kz =
= pK̂τ [kz, w;X(−p;w)/w] . (11)
If 1/β ≪ |w| ≪ A/β then Kτ (w) ≈ 1/π, so, the ker-
nels in the equations for the following six func-
tions, Xe
τ (u/β), u−1X(p; ı̇cτ/u), u−1X(−p;u/β),
u−1Xe
τ (ı̇cτ/u), X(p;u/β), and X(−p; ı̇cτ/u), as
the functions of u, for 1 ≪ |w| ≪ A are close to
p/[π(u+ w)]. The possibility to construct the so-
lution of the equation with the kernel p/[π(u+ w)]
explicitly makes it possible to use the method of
semi-inversion. Let us denote κ = π−1arcsin(p) and
consider the equation
X(u) = f(u) +
+ sin(πκ)
∫ ∞
1
dw[π(u+ w)]−1X(w). (12)
Replacing u and w with exp(u) and exp(w), one
transforms it to the integral equation on the inter-
val (0,∞) with the kernel dependent on the differ-
ence u− w. Solving such equation with Wiener-Hopf
method, one gets the equality
X(u) = f(u) +
∫ ∞
1
dwVκ(u,w)f(w), (13)
in which
Vκ(u,w) = π−1(uw)−1/2tan(πκ)×
×{sinh[ln(u/w)(1/2 + κ)]/sinh[ln(u/w)]+
+π−1tan(πκ)
∑∞
m,n=1
[
(−1)m+n−1×
×Λκ,mΛκ,n(σκ,m + σκ,n)
−1×
× exp(−σκ,m lnu− σκ,n lnw)]} ,
σκ,n = n− 1/2 + (−1)nκ, Λκ,n = Λκ(ı̇σκ,n),
Λκ(s) = [1− sin(πκ)]1/2 ×
×
∏∞
n=1
[(1− ı̇s/σκ,n)/(1− ı̇s/σ0,n)].
The difference between the kernels of the equa-
tions for the mentioned six functions and the kernel
p/[π(u+ w)] for 1 ≤ |w| ≪ A depends on w and u
approximately as (u+ w)−1w−1. If after the limit
transition {A → ∞, β → 0} one considers the in-
tegral with this difference as the known function
(although it contains the unknown function) and in-
cludes it into the function f(u) in the equation of the
type (12) then the equality (13) becomes the integral
equation, the kernel of which sufficiently quickly de-
creases with the unbounded increase of variables, and
such equation with simple change of variables may
be transformed to the integral equation with the
bounded kernel on the bounded interval. The func-
tions Xe
τ (u/β), u
−1X(p; ı̇cτ/u), and u−1X(−p;u/β)
at 1 ≪ |u| ≪ A depend on u approximately as uκ−1.
In the equations for the functions u−1Xe
τ (ı̇cτ/u),
X(p;u/β), and X(−p; ı̇cτ/u), at 1 ≪ |u| ≪ A free
terms are relatively small, and the solutions of these
equations are close to ones of relevant homogeneous
equations, which depend on u at 1 ≪ |u| ≪ A ap-
proximately as u−κ, in connection with the equal-
ity π−1 sin(πκ)
∫∞
1
dw(u+ w)−1P−κ(w) = P−κ(u),
where P is Legendre function. After solv-
ing of relevant equations, the estimations
of X(−p;u/β)/X(−p; 0) and X(−p; ı̇cτ/u) at
u = (ı̇βcτ )
1/2 give a possibility to calculate the
value of Fa = X(−p; 0)[A exp(ı̇π/3)]κ. In connec-
tion with the equality X(−p; 0)X(p; 0) = 1 (de-
duced briefly in the next paragraph), the equality
X(−p; 0)/X(p; 0) = F 2
a [A exp(ı̇π/3)]−2κ is held.
The equality X(−p; 0)X(p; 0) = 1 may be ob-
tained from the equalities (10) and (11), which
have the same kernel, K0(u,w) = pKτ (w)/(u+ w),
and the same resolvent R(u,w; p) defined with
the equalities R(u,w; p) =
∑∞
n=0[Kn(u,w)p
n] and
105
Kn+1(u,w) =
∫
Γ
dw′K0(u,w
′)Kn(w
′, w) [17]. Writ-
ing the solutions through the resolvent, for the val-
ues of X(p; 0) and X(−p;∞)/X(−p; 0), with use of
the equality Kτ (u)R(u,w; p) = Kτ (w)R(w, u; p), one
can get the equality X(p; 0) = X(−p;∞)/X(−p; 0)
and take into account the condition X(−p;∞) = 1.
As really the type of the electron scattering from
the boundary depends on electron incidence angle
[8], the question arises how does this type influ-
ences on the degrees in the dependences uκ−1 and
u−κ. Let the dependence p(ϑ), where ϑ is the an-
gle between normal and electron motion direction,
is analytical function in the interval ϑ ∈ (0, π/2),
and for a sufficiently small positive a the equality
limϑ→π/2 {[p(ϑ)− p(π/2)](cosϑ)−a} = 0 takes place.
Then in the solving construction through relevant in-
tegral equations the nonzero p(ϑ)− p(π/2) leads to
the integrals, which may be transformed to ones with
the bounded kernels on the bounded intervals. As a
result, the value of κ in the mentioned degrees has to
correspond to the value of p(π/2).
5. THE RADIATION AMPLITUDE
Let us put
Fλ = β−1 lim
u→0
(∂/∂u)log[Xr
λ(u)/Q
+
λ (u)],
Fτ = β−1 lim
u→0
(∂/∂u)log[X(−p;u)/Q+
τ (u)].
At {β ≪ 1, A ≫ 1}, the values of Fa, Fλ, Fτ ,
and Ψτ1 are close to real numbers (dependent on
p). In the paper [9], in fact, the relationship
Ψτ1 ≈ (π2/48)1/6[sin(α/2)/ sin(α/3)]2 is obtained,
where α = arccos(p). The numerical solving of rele-
vant integral equations by the way described above
shows that the dependences of the quantities Fa,
Fλ(1− p), and Fτ (1− p)1/2 on p, accurate to within
1% are close to linear ones, with the values close to 1,
0.714, and 0.277 at p = 0 and to 0.85, 1.34, and 0.6
at p = 1. Limitedness of the quantities Fλ(1− p) and
Fτ (1− p)1/2 near p = 1 is connected with the analyt-
icity of the function Vκ(u,w) as the function of κ near
the point κ = 1/2 and with the possibility to expand
the resolvent of the symmetrical continuous bounded
kernel of the integral equation on the bounded in-
terval into the series in terms of eigenfunctions with
coefficients, which contain in the denominators the
differences between the factor at integral and relevant
eigenvalue of this factor [17].
If |kz| ≤ 1 then with use of the conditions
Ψ(±ı̇k⊥) = 0 one gets
CEẼ⊥(0)− H̃φ(0)≈BEIz0, (14)
where CE = ı̇cτΨτ1 + βk2⊥(Fλ − Fτ )X
2(−p; 0),
BE = −βΩ−1k⊥(Fλ − Fτ )X(−p; 0). The consider-
ation of field in the half-space z < 0, for the given
current, j⃗0 = Z0e0δ(x)δ(y)δ(z − β0ct)β0ce⃗z, gives
Ẽ⊥(0) + wzH̃φ(0) = ı̇k⊥(wz − kz0)
−1Iz0, (15)
where wz = (1− k2⊥)
1/2. For the boundary val-
ues, H̃r
φ(0−), Ẽr
⊥(0−), and Ẽr
z (0−), of relevant
field components of the wave with wave num-
ber kz = −wz emitted into the half-space z < 0,
one has H̃r
φ(0−) = H̃φ(0) + ı̇k⊥(k
2
z0 − w2
z)
−1Iz0,
Ẽr
⊥(0−) = −wzH̃
r
φ(0−), Ẽr
z (0−) = −k⊥H̃
r
φ(0−),
and the emitted energy may be given with
the integral
∫∞
0
dω
∫ π/2
0
dθ2π sin θW (ω, θ), where
the angle θ is connected with k⊥ through
the equality k⊥ = sin θ, and the function
W (ω, θ) = (2π)−4c−1ω2cos2 θ|H̃r
φ(0−)|2 gives the
spectral density of the radiation into a solid angle.
From (14) and (15) one can find the boundary values,
Ẽ⊥(0) and H̃φ(0), and then get the value of Xλ(0)
and the functions Xλ,τ (kz), Ψλ,τ (kz), and E⊥,z(kz),
through which the field in plasma is described.
One can obtain the relationships |Ẽ⊥(0)| ≪ |H̃φ(0)|,
H̃φ(0)≈ı̇k⊥[wz(wz − kz0)]
−1Iz0, Ẽ⊥(0)≈H̃φ(0)/CE ,
and CE≈ı̇cτΨτ1. So, the amplitude of the transition
radiation in the given frequency range is close to
one, corresponding to the case of ideally conducting
medium in the half-space z > 0. The difference of the
medium from ideally conducting one leads to change
of the emitted wave amplitude on relatively small
amount, and this change may be estimated with use
of nonzero surface impedance, as it was made in
[1]. The second summand in the definition of CE
is relatively small, and it corresponds to the small
contribution into impedance from the existence of
the component normal to the boundary in the field
created at the boundary by the particle motion in
the half-space z < 0 (such summand also appears
with solving of the problem of oblique incidence of
electromagnetic wave on plasma medium [13], and it
is absent in the case of normal incidence). The right
hand side of (14) deals with the field created with
the particle motion in the half-space z > 0.
If ω ≪ ωe then impedance is small, and the spec-
tral density of the emitted energy is nearly indepen-
dent on ω. But if ω ≫ ωe then the spectral density
is quickly decreases with frequency increase, and the
radiated energy is the bounded quantity.
6. THE RADIATION BY THE MOTION
WITH SHORT-TERM GOING OUT FROM
THE MEDIUM
The difference of the given medium from the ideally
conducting one may to have a considerable influence
on the radiation amplitude in the case when the field
created at the boundary in connection with the par-
ticle motion in the half-space z < 0 is small. As an
example of such situation, it may be considered the
case when a particle moving along OZ-axis goes out of
the medium at t = −t0, where t0 > 0, and at t = 0 it
changes the motion direction with opposite one with-
out change of the absolute value of velocity, due to
elastic collision. In this case, the consideration of the
field in the half-space z < 0 gives the equalities
Ẽ⊥(0) + wzH̃φ(0) = −ı̇k⊥Iz2(wz),
H̃r
φ(0−) = H̃φ(0)− 2[wz(β
2
0w
2
z − 1)]−1 ×
×[sin(ωt0β0wz)− β0wz sin(ωt0)]β0k⊥Iz0,
where β0 > 0 is assumed, and
Iz2(kz) = 2Iz0β0(1− β2
0k
2
z)
−1 ×
×[cos(ωt0)− exp(ı̇ωt0β0kz) + ı̇β0kzsin(ωt0)].
106
For the field in the half-space z > 0, taking the lin-
ear combination, with the coefficients ∓ exp(∓ı̇ωt0),
of the solutions of the problems, in which the par-
ticle with velocity ∓β0c crosses the plane z = 0 at
t = 0, one can get in the right hand side of (14) the
additional factor 2ı̇ sin(ωt0).
If ωt0 ≪ 1 and π/2− θ ≫ β/A then for the
solution, which may be obtained from the ap-
proximate equations CEẼ⊥(0)− H̃φ(0)≈2ı̇ωt0BEIz0
and Ẽ⊥(0) + wzH̃φ(0)≈ı̇k⊥β0(ωt0)
2Iz0, one has
|H̃r
φ(0−)/H̃φ(0)− 1|≪1. If β3A−5/2−κ≪ωt0≪1
then the main contribution to radiation gives
the particle motion in the half-space z < 0 and
the relationships H̃r
φ(0−)≈ı̇k⊥β0wz
−1(ωt0)
2Iz0 and
|Ẽ⊥(0)/H̃φ(0)|≪1 are held. But if ωt0≪β3A−5/2−κ
then the main part of transition radiation is gen-
erated due to the particle motion in the half-
space z > 0, and there are held the relationships
Ẽ⊥(0)/H̃φ(0)≈− wz and
H̃r
φ(0−)≈2ı̇ exp[−ı̇π(κ+ 1)/3]×
×β3A−5/2−κk⊥(Fλ − Fτ )Fa(Ψτ1wz)
−1ωt0Iz0.
7. EFFICIENCY OF ENERGY
TRANSFORMATION
If the particle transits from the free half-space into
the plasma then before the transition the particle is
attracted to the plasma medium polarized by the par-
ticle (in the case of ideally-conducting medium the
attraction corresponds to interaction with mirror im-
age having opposite sign of charge). So, for such
motion direction, the transition radiation is accom-
panied with particle acceleration and increase of its
kinetic energy, and the energy source for the acceler-
ation and radiation is the potential energy of inter-
action of the free particle with the plasma medium
polarized by it (and in the case of ideally conduct-
ing medium the radiated energy is formally infinite).
But to become free the particle has to come out of
some medium or accelerator. Such going out is also
accompanied with transition radiation, and the parti-
cle decelerates, attracts to the medium polarized by
it and gives its kinetic energy to increase of poten-
tial energy and to the generation of radiation. For
equal velocities of transition through the boundary
the radiation amplitudes in the cases of going in and
going out at low frequencies (when ω ≪ ωe) are ap-
proximately equal. But the efficiency of energy trans-
formation may be considerable in the case when the
particle losses the considerable part of its kinetic en-
ergy during deceleration and this energy goes mainly
on radiation. If the particle bunch goes out of acceler-
ator exit device ended with antenna then for consid-
erable bunch deceleration during its interaction with
its, conditionally saying, mirror image in antenna the
charge of the bunch has to be considerably large (and
for such charge, in particular, the force, pushing the
bunch apart, is greater than the force, decelerating
the bunch). To be the bunch comparatively compact
during its going out of the pulsed direct action accel-
erator (for generation of wide-range radiation) it is
necessary the accelerating field sufficiently homoge-
neous with respect to the bunch dimension, and for
its forming the electrode dimension has to be greater
than the bunch dimension, and the field strength has
to be greater than one pushing the bunch apart, and
so, the charge at the electrode has to be greater than
the bunch charge. In such a case, the radiation gener-
ated during the quick displacement of such charge to
the electrode is considerably more powerful and not
lesser wide-range than one generated during deceler-
ation of the accelerated bunch in the antenna region.
So, using only the charge displacement in the con-
ductors, which assemble the power supply system of
the pulsed direct action accelerator, without beam
in vacuum, it may be generated much more powerful
wide-range radiation than one, which may be gener-
ated by beam with antenna.
8. CONCLUSIONS
So, it is presented the solving construction of the
problem of transition radiation for the case of nor-
mal incidence of the particle on the locally isotropic
plasma with the sharp boundary and the mixed type
of electron scattering from it at the frequencies much
greater than collision frequency, but much less than
plasma frequency. When the particle motion is uni-
form, the radiation amplitude is close to one, which
may be obtained in the case of crossing the bound-
ary of ideally conducting medium. But if the particle
goes out of the medium on the short time then the
difference of the medium from ideally conducting one
may lead to considerable difference in the radiation
amplitude.
References
1. E.A. Kaner, V.M. Jakovenko. To the theory of
transition radiation // JETP. 1962, v.42, N.2,
p.471–478 (in Russian).
2. N.F. Shul’ga, S.V. Trofymenko. High-energy
wave packets. ‘Half-bare’ electron // Journ. of
Kh. Nat. Univ., phys. series ”Nuclei, Particles,
Fields”. 2013, iss. 1(57), p.59–69.
3. V.A. Balakirev, G.L. Sidel’nikov. Transition ra-
diation of the modulated electron beams in non-
homogeneous plasma. Review. Kharkov, KIPT,
1994, 104 p. (in Russian).
4. V.A. Balakirev, N.I. Gaponenko, A.M. Gorban’,
et al. Excitement TEM-horn antenna by impul-
sive relativistic electron beam // PAST. Series:
Plasma Physics (5). 2000, N3, p.118–119.
5. V.A. Buts, I.K. Koval’chuk. Elementary mecha-
nism of oscillations excitation by electron beam
in a cavity // Ukr. Fiz. Zh. 1999, v.44, N11,
p.1356–1363 (in Ukrainian).
107
6. G.E.H. Reuter, E.H. Sondheimer. The theory of
the anomalous skin effect in metals // Proc. Roy.
Soc. 1949, v.195, p.336–364.
7. M.Ja. Azbel’, E.A. Kaner. Anomalous skin-effect
for arbitrary collision integral // JETP. 1955,
v.29, N6(12), p.876–878 (in Russian).
8. A.V. Latyshev, A.A. Jushkanov. Influence on the
impedance value of the specular scattering coeffi-
cient dependence on the electron incidence angle
// ZhTF. 2010, v.80, N9, p.1–7 (in Russian).
9. L.E. Hartmann, J.M. Lattinger. Exact solution
of the integral equation for the anomalous skin
effect and cyclotron resonance in metals // Phys.
Rev. 1966, v.151, N2, p.430–433.
10. V.M. Gokhfeld, M.I. Kaganov, G.Ja. Ljubarskij.
Anomalous penetration of the varied longitudinal
electric field into degenerate plasma for an arbi-
trary specular reflection parameter // ZhETF.
1987, v.92, N2, p.523–530 (in Russian).
11. A.V. Latyshev, A.A. Jushkanov. Analytical solu-
tion of skin-effect problem for an arbitrary coef-
ficient of accomodation of electrons tangent mo-
mentum // ZhTF. 2000, v.70, N8, p.1–7 (in Rus-
sian).
12. A.N. Kondratenko, V.I. Miroshnichenko. Ki-
netic theory of passing of electromagnetic waves
through plasma layer // ZhTF. 1965, v.35, N12,
p.2154–2159; ZhTF. 1966, v.36, N1, p.25–32 (in
Russian).
13. V.I. Miroshnichenko, V.M. Ostroushko. Scatter-
ing of oblique electromagnetic wave from the
sharp plasma boundary in anomalous skin-effect
condition for the mixed electron scattering from
the boundary // Ukr. Fiz. Zh. 2002, v.47, N2,
p.147–153 (in Ukrainian).
14. V.P. Silin, A.A. Rukhadze. Electromagnetic prop-
erties of plasma and plasma-like media. Moscow:
”Gosatomizdat”, 1961, 244p. (in Russian).
15. V.I. Miroshnichenko. Electromagnetnc properties
of half-infinite plasma for diffusive electron srat-
tering from the boundary // ZhTF. 1966, v.36,
N6, p.1008–1016 (in Russian).
16. V.I. Kurilko, V.A. Popov. To kinetic theory of
longitudinal waves excitation in the bounded
plasma // ZhTF. 1966, v.36, N3, p.466–469 (in
Russian).
17. V.I. Smirnov. A course of higher mathematics,
v.4, part 1. Moscow: ”Nauka”, 1974, 336 p. (in
Russian).
ÂËÈßÍÈÅ ÑÊÈÍ-ÝÔÔÅÊÒÀ ÍÀ ÏÅÐÅÕÎÄÍÎÅ ÈÇËÓ×ÅÍÈÅ
Â.È.Ìèðîøíè÷åíêî , Â.Í.Îñòðîóøêî
Ðàññìîòðåíî ïåðåõîäíîå èçëó÷åíèå äëÿ íîðìàëüíîãî ïàäåíèÿ çàðÿæåííîé ÷àñòèöû íà ãðàíèöó ïëàç-
ìåííîé ñðåäû â óñëîâèÿõ àíîìàëüíîãî ñêèí-ýôôåêòà, íà ÷àñòîòàõ, çíà÷èòåëüíî ìåíüøèõ, ÷åì ïëàç-
ìåííàÿ. Çàäà÷à ðåøåíà â ïðåäïîëîæåíèè, ÷òî îòðàæåíèå ýëåêòðîíîâ îò ãðàíèöû ÿâëÿåòñÿ ÷àñòè÷íî
çåðêàëüíûì, ÷àñòè÷íî äèôôóçíûì. Ïîëó÷åíà ñïåêòðàëüíàÿ ïëîòíîñòü èçëó÷åííîé ýíåðãèè äëÿ ñëó-
÷àåâ ðàâíîìåðíîãî äâèæåíèÿ ÷àñòèöû è äâèæåíèÿ ñ äâóêðàòíûì ïåðåñå÷åíèåì ãðàíèöû.
ÂÏËÈÂ ÑÊIÍ-ÅÔÅÊÒÓ ÍÀ ÏÅÐÅÕIÄÍÅ ÂÈÏÐÎÌIÍÞÂÀÍÍß
Â. I.Ìiðîøíè÷åíêî , Â.Ì.Îñòðîóøêî
Ðîçãëÿíóòî ïåðåõiäíå âèïðîìiíþâàííÿ äëÿ íîðìàëüíîãî ïàäiííÿ çàðÿäæåíî¨ ÷àñòèíêè íà ìåæó ïëàç-
ìîâîãî ñåðåäîâèùà â óìîâàõ àíîìàëüíîãî ñêií-åôåêòó, íà ÷àñòîòàõ, çíà÷íî ìåíøèõ âiä ïëàçìîâî¨.
Çàäà÷ó ðîçâ'ÿçàíî â ïðèïóùåííi, ùî âiäáèòòÿ åëåêòðîíiâ âiä ìåæi ¹ ÷àñòêîâî äçåðêàëüíèì, ÷àñòêîâî
äèôóçíèì. Îòðèìàíî ñïåêòðàëüíó ãóñòèíó âèïðîìiíåíî¨ åíåðãi¨ äëÿ âèïàäêiâ ðiâíîìiðíîãî ðóõó ÷àñ-
òèíêè òà ðóõó ç äâîðàçîâèì ïåðåòèíàííÿì ìåæi.
108
|
| id | nasplib_isofts_kiev_ua-123456789-112103 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-25T20:53:33Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Miroshnichenko, V.I. Ostroushko, V.M. 2017-01-17T16:08:01Z 2017-01-17T16:08:01Z 2015 Skin-effect influence on transition radiation / V.I.Miroshnichenko, V.M. Ostroushko2 // Вопросы атомной науки и техники. — 2015. — № 3. — С. 103-108. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 41.60.-m https://nasplib.isofts.kiev.ua/handle/123456789/112103 It is considered the transition radiation for the normal incidence of a charged particle on the boundary of plasma medium in the conditions of anomalous skin effect, at the frequencies much less than plasma frequency. The problem is solved in the assumption that electron scattering from the boundary is partially specular and partially diffusive. The spectral density of the radiated energy is obtained for the cases of uniform particle motion and of the motion with two running across the boundary. Розглянуто перехідне випромінювання для нормального падіння зарядженої частинки на межу плазмового середовища в умовах аномального скін-ефекту, на частотах, значно менших від плазмової. Задачу розв’язано в припущенні, що відбиття електронів від межі є частково дзеркальним, частково дифузним. Отримано спектральну густину випроміненої енергії для випадків рівномірного руху частинки та руху з дворазовим перетинанням межі. Рассмотрено переходное излучение для нормального падения заряженной частицы на границу плазменной среды в условиях аномального скин-эффекта, на частотах, значительно меньших, чем плазменная. Задача решена в предположении, что отражение электронов от границы является частично зеркальным, частино диффузным. Получена спектральная плотность излученной энергии для случаев равномерного движения частицы и движения с двукратным пересечением границы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика Skin-effect influence on transition radiation Вплив скiн-ефекту на перехiдне випромiнювання Влияние скин-эффекта на переходное излучение Article published earlier |
| spellingShingle | Skin-effect influence on transition radiation Miroshnichenko, V.I. Ostroushko, V.M. Электродинамика |
| title | Skin-effect influence on transition radiation |
| title_alt | Вплив скiн-ефекту на перехiдне випромiнювання Влияние скин-эффекта на переходное излучение |
| title_full | Skin-effect influence on transition radiation |
| title_fullStr | Skin-effect influence on transition radiation |
| title_full_unstemmed | Skin-effect influence on transition radiation |
| title_short | Skin-effect influence on transition radiation |
| title_sort | skin-effect influence on transition radiation |
| topic | Электродинамика |
| topic_facet | Электродинамика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112103 |
| work_keys_str_mv | AT miroshnichenkovi skineffectinfluenceontransitionradiation AT ostroushkovm skineffectinfluenceontransitionradiation AT miroshnichenkovi vplivskinefektunaperehidneviprominûvannâ AT ostroushkovm vplivskinefektunaperehidneviprominûvannâ AT miroshnichenkovi vliânieskinéffektanaperehodnoeizlučenie AT ostroushkovm vliânieskinéffektanaperehodnoeizlučenie |