Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators
Using the novel method of macroscopic computer simulation of the reflection of charged particles from solid surfaces at grazing incidence the kinetics of planar macrochanneling, the directed transport of particles between surfaces separated by macroscopic distance, has been studied for swift electro...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators / S.V. Dyuldya, M.I. Bratchenko, M.A. Skorobogatov // Вопросы атомной науки и техники. — 2007. — № 5. — С. 90-97. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1103942017-01-05T03:02:22Z Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators Dyuldya, S.V. Bratchenko, M.I. Skorobogatov, M.A. Ядернo-физические методы и обработка данных Using the novel method of macroscopic computer simulation of the reflection of charged particles from solid surfaces at grazing incidence the kinetics of planar macrochanneling, the directed transport of particles between surfaces separated by macroscopic distance, has been studied for swift electrons and ions. The dechanneling functions at macrochanneling have been calculated for the first time. The modeling of angular distributions of particles has confirmed the effect of beam angular splitting at the exit of macrochannel. The method of the taking into account of surface atomic discreteness at modeling of coherent effects of ion planar semichanneling has been developed. It has been shown that due to stochastic effects of particles reflection the macrochanneling of electrons in long channels is actually reduced to the effects of beam collimation while for ions the coherent effect allows to stabilize the macrochanneling and to achieve the beam transportation at long distances without considerable losses of intensity. Оригінальним методом макроскопічного математичного моделювання відбиття заряджених частинок від поверхні твердого тіла досліджена кінетика площинного макроканалювання швидких електронів та іонів — орієнтованого транспорту частинок між поверхнями, розділеними макроскопічною відстанню. Вперше розраховані функції деканалювання частинок при макроканалюванні. Моделювання кутових розподілів підтвердило існування ефекту розділення потоків частинок. Розроблено метод урахування атомної дискретності поверхні в моделюванні когерентних ефектів площинного напівканалювання іонів. Показано, що з-за стохастичності процесу відбиття макроканалювання електронів в каналах великої довжини ефективно визначається механізмами колімації пучка, тоді як для іонів когерентний ефект дозволяє стабілізувати макроканалювання та досягти транспортування та розділення пучка на великих відстанях без значних втрат інтенсивності. Оригинальным методом макроскопического математического моделирования отражения заряженных частиц от поверхности твердого тела исследована кинетика плоскостного макроканалирования быстрых электронов и ионов — ориентированного транспорта частиц между поверхностями, разделенными макроскопическим расстоянием. Впервые рассчитаны функции деканалирования частиц при макроканалировании. Моделирование угловых распределений подтвердило существование эффекта разделения потоков частиц. Разработан метод учета атомной дискретности поверхности при моделировании когерентных эффектов плоскостного полуканалирования ионов. Показано, что ввиду стохастичности процесса отражения макроканалирование электронов в каналах большой длины эффективно определяется механизмами коллимации пучка, тогда как для ионов когерентный эффект позволяет стабилизировать макроканалирование и достичь транспортировки и разделения пучка на больших расстояниях без существенных потерь интенсивности. 2007 Article Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators / S.V. Dyuldya, M.I. Bratchenko, M.A. Skorobogatov // Вопросы атомной науки и техники. — 2007. — № 5. — С. 90-97. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 02.70.Uu, 07.05.Tp, 29.27.Eg, 61.85.+p, 68.49.Jk, 68.49.Sf, 41.75.Ht, 41.85.Ct, 41.85.Ja, 41.85.Si http://dspace.nbuv.gov.ua/handle/123456789/110394 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядернo-физические методы и обработка данных Ядернo-физические методы и обработка данных |
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Ядернo-физические методы и обработка данных Ядернo-физические методы и обработка данных Dyuldya, S.V. Bratchenko, M.I. Skorobogatov, M.A. Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators Вопросы атомной науки и техники |
| description |
Using the novel method of macroscopic computer simulation of the reflection of charged particles from solid surfaces at grazing incidence the kinetics of planar macrochanneling, the directed transport of particles between surfaces separated by macroscopic distance, has been studied for swift electrons and ions. The dechanneling functions at macrochanneling have been calculated for the first time. The modeling of angular distributions of particles has confirmed the effect of beam angular splitting at the exit of macrochannel. The method of the taking into account of surface atomic discreteness at modeling of coherent effects of ion planar semichanneling has been developed. It has been shown that due to stochastic effects of particles reflection the macrochanneling of electrons in long channels is actually reduced to the effects of beam collimation while for ions the coherent effect allows to stabilize the macrochanneling and to achieve the beam transportation at long distances without considerable losses of intensity. |
| format |
Article |
| author |
Dyuldya, S.V. Bratchenko, M.I. Skorobogatov, M.A. |
| author_facet |
Dyuldya, S.V. Bratchenko, M.I. Skorobogatov, M.A. |
| author_sort |
Dyuldya, S.V. |
| title |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| title_short |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| title_full |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| title_fullStr |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| title_full_unstemmed |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| title_sort |
modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Ядернo-физические методы и обработка данных |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110394 |
| citation_txt |
Modeling of kinetics of macrochanneling of fast electrons and ions in planar gap collimators / S.V. Dyuldya, M.I. Bratchenko, M.A. Skorobogatov // Вопросы атомной науки и техники. — 2007. — № 5. — С. 90-97. — Бібліогр.: 12 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T00:33:52Z |
| last_indexed |
2025-07-08T00:33:52Z |
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1837036819719389184 |
| fulltext |
MODELING OF KINETICS OF MACROCHANNELING
OF FAST ELECTRONS AND IONS
IN PLANAR GAP COLLIMATORS
S.V. Dyuldya∗, M.I. Bratchenko, M.A. Skorobogatov
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received May 23, 2006)
Using the novel method of macroscopic computer simulation of the reflection of charged particles from solid surfaces
at grazing incidence the kinetics of planar macrochanneling, the directed transport of particles between surfaces
separated by macroscopic distance, has been studied for swift electrons and ions. The dechanneling functions at
macrochanneling have been calculated for the first time. The modeling of angular distributions of particles has
confirmed the effect of beam angular splitting at the exit of macrochannel. The method of the taking into account
of surface atomic discreteness at modeling of coherent effects of ion planar semichanneling has been developed.
It has been shown that due to stochastic effects of particles reflection the macrochanneling of electrons in long
channels is actually reduced to the effects of beam collimation while for ions the coherent effect allows to stabilize the
macrochanneling and to achieve the beam transportation at long distances without considerable losses of intensity.
PACS: 02.70.Uu, 07.05.Tp, 29.27.Eg, 61.85.+p, 68.49.Jk, 68.49.Sf, 41.75.Ht, 41.85.Ct, 41.85.Ja, 41.85.Si
1. INTRODUCTION
Macrochanneling, the steered motion of charged
particles between deflective solid surfaces separated
by macroscopic distance, is the macroscopic analogue
of the particles’ channeling in single crystals. The ef-
fect had been discovered experimentally [1] in course
of investigations of the penetration of MeV energies
electron beams through targets with extended inho-
mogeneities.
Unlike for the bulk channeling in crystals that
is controlled by continuum potentials of atomic
planes and/or chains [2] the basic event forming the
macrochanneled particles trajectories is the reflection
from solid surface. It has large probability at grazing
incidence and, depending on the sort of particle, the
state of surface and the angle of beam incidence, can
be originated either from multiple atomic collisions of
a particle inside the near-surface region or from the
scattering by surface atoms.
Due to the fact that between successive reflection
events particles move in a free space without scatter-
ing the macrochanneling is supposed to provide the
beam transport at large distances with small energy
losses [3, 4]. Similarly, the transport through bent
macrochannels or collimators of appropriate geome-
try can lead to the relativistic beams turning [3, 5],
splitting or focusing.
These applications stimulate the interest to the
effect both from the point of view of the development
of the accelerator related beam shapers and irradia-
tion devices as well as of the progress of experimen-
tal technique in high energy physics. The studies
of macrochanneling can also clarify certain aspects of
particle channeling in nanotubes [6] that in fact occu-
pies an intermediate position on the scale of transver-
sal dimensions of channels.
However no attempts can be found in the litera-
ture to build the quantitative kinetic theory of the ef-
fect. In the present paper the kinetics of macrochan-
neling of relativistic electrons and fast ions in planar
gap collimators is studied using the ”macroscopic”
computer simulation of multiple reflections of parti-
cles from solid surfaces. The work is based on the
novel simulation method developed in Ref. [7].
2. MODELING METHODS
For any definite geometry the modeling of
macrochanneling per se comes to the following itera-
tive procedure: the modeling of reflection event alter-
nates to the 3D geometric calculation of the particle
free motion between its collisions with surfaces. The
latter can be easily carried out using the ray tracing
algorithms while the main problem consists in the de-
velopment of adequate and computationally effective
methods of the simulation of the small-angle surface
scattering.
Currently the charged particles’ backscattering
from solid surfaces is well studied both experimen-
tally and theoretically either for ions [8, 9] or for swift
electrons [10]. The reflection coefficient R (integral
albedo) typically decreases with the increase of par-
ticle energy E but increases with the reduction of
the angle ψin between the beam axis and the surface
plane (see Fig. 1).
∗Corresponding author. E-mail address: sdul@kipt.kharkov.ua
90 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5.
Series: Nuclear Physics Investigations (48), p.90-97.
As ψin decreases the angular spectra of backscat-
tered particles demonstrate the transformation from
the cosine-shaped distributions for normal incidence
toward the distributions concentrated near the direc-
tion of specular reflection. For heavy particles (ions)
and for electrons at sufficiently high energies (that
allow to neglect quantum diffraction effects) this
behavior can be described within the scope of con-
ventional mechanism of particle backscattering due
to the sequence of uncorrelated (incoherent) atomic
collisions. The target lattice structure is completely
neglected in these models and the reflection mecha-
nisms are only graded by the multiplicity of atomic
scattering events that yield to the backscattering.
For normal incidence major contribution is given by
single strong Rutherford scattering while for graz-
ing incidence (ψin ¿ 1) substantial contribution is
due to the multiple small-angle scattering that is de-
scribed by boundary problems of kinetic equations of
Fokker-Planck type [10].
Fig.1. The geometry of small-angle reflection of
particles from the solid surface. Indicated are the
incident (ψin = 90◦ − θin) and emergency
(ψout, χout, φout) angles that describe the
backscattering event
Qualitatively the mechanisms of small-angle back-
scattering at grazing incidence are described by the
non-dimensional parameters σ∗ and σ [10].
The scattering parameter:
σ∗(E) =
R0(E)
ltr(E)
(1)
is the ratio of particle range R0 and the transport
scattering length ltr = [n · σtr(E)]−1 where n is
the target atomic concentration, σtr = 2π
∫
(1 −
cos θ)dσel(θ) is the transport cross-section, dσel/dθ
is the differential cross-section of elastic scattering.
The greater is σ∗ the stronger is scattering as com-
pared with stopping.
Integral albedo R, angular distributions and en-
ergy spectra of reflected particles are qualitatively de-
termined by the non-dimensional reflection parame-
ter σ:
σ(E, ψin) =
σ∗(E)
1− cos ψin
≈ R0(E) · 〈θ2
MS(E)〉
ψ2
in
, (2)
where ψin ¿ 1 and 〈θ2
MS(E)〉 is the mean squared
angle of multiple scattering per unit of range. Large
σ À 1 causes large R → 1 and quasi-elastic reflec-
tion. The total albedo R(E,ψin) for arbitrary an-
gle of incidence is represented by the weighted sum
of small-angle (R1(E,ψin)) and diffuse (R2(E,ψin))
backscattering coefficients:
R ≈ R1 + (1−R1) ·R2. (3)
In Ref. [10] explicit parametrizations of R1,2(E, ψin)
functions are given; one should note that the reflect-
ing material’s properties in these formulae are ex-
pressed only via the values of reduced parameters σ∗
and σ.
The correlated energy-angular distribution of re-
flected particles is described by the expression [10]:
Rψχ∆(ψr, χr, ∆) =
2
√
3 · ψr
(πσ∆)3/2∆
· erf
(
2
√
3ψr
σ∆
)
× exp
[
− 4
σ∆
·
(
ψ2
r − ψr + 1 +
χ2
r
4
)]
,
(4)
where ψr = ψout/ψin, χr = χout/ψin, and ∆ =
(E − Eout)/E is the normalized energy loss.
The equations (3–4) are universal in the mean-
ing that they describe the incoherent reflection of
either light or heavy particles of high energies pro-
vided the scattering parameter σ∗ is calculated prop-
erly according to the definite laws of their stopping
and scattering.
In Ref. [7] we have developed the Monte Carlo
method of sampling of angular and energy variables
from the distribution (4). Along with the sampling of
reflection event with the probability (3) this method
allows efficient modeling of multiple reflections of par-
ticles at macroscopic level (i.e. without the require-
ment to model microscopic atomic collisions). Below
this method is applied to the Monte Carlo simulation
of macrochanneling of electrons as well for that of
ions at sufficiently large ψin. The method enhance-
ments developed in order to describe coherent effects
of surface semichanneling of ions at smaller ψin are
introduced in Section 4.
We limited ourselves with the case of planar
macrochannel (collimator) formed by parallel sur-
faces of Copper separated by the distance wch=3 mm
that is much smaller then the lateral dimensions of
the device. For simplicity the surface roughness was
neglected and it was treated as a geometrical plane.
In calculations the macroscopic length L of the col-
limator was varied in a broad range from 10 mm up
to 2 m.
For each modeling case more then 106 histories
of primary particles have been sampled. It allowed
us to score both integral and differential character-
istics of macrochanneling with acceptable statistical
accuracy.
We considered expedient to describe the
macrochanneling within the scope of the same kinetic
concept that is accepted in the conventional theory
of channeling in crystals. Hence we have introduced
the dechanneling function Pch(z) that represents the
probability to find moving particle at depth z of beam
transport through the channel. Evidently Pch(0) = 1
while T = Pch(L) is the macrochannel transmittance
factor (the transparency).
91
Other quantities under investigation were the
statistics of particle reflections from the walls of
channel and angular distributions of particles passed
through it. In toto this gives the complete descrip-
tion of the effect and allows us to make conclusions
concerning its mechanisms and the peculiarities of its
practical applications.
3. PLANAR MACROCHANNELING OF
RELATIVISTIC ELECTRONS
In Fig.2 the obtained macro-dechanneling func-
tions of 3 MeV monodirectional electron beam are
shown.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
10
20
30
40
50
60
70
80
90
100
P ch
(z
/L
),
%
z/L
L = 50 mm
L = 100 mm
L = 150 mm
L = 200 mm
L = 500 mm
3 MeV e- / Cu
wch = 3 mm
in=5°
(a)
0 100 200 300 400 500
0
10
20
30
40
50
60
70
80
90
100
(b)
P ch
(z
),
%
z, mm
in=5.0°
in=2.0°
in=1.0°
in=0.5°
3 MeV e- / Cu
wch = 3 mm
Fig.2. Dechanneling functions of relativistic
electrons in planar macrochannels of different
length L (a) and at different incident angles ψin at
the entrance (b)
According to Eqs. (1-2) for E=3 MeV the scat-
tering parameter σ∗ = 2.55 and for |ψin| = 5◦ the
reflection parameter σ = 670 À 1. Thus the re-
flection is close to elastic one but has comparatively
broad angular distribution near the specular reflec-
tion angle |ψout| = |ψin|.
The characteristic length that describes the depth
dependencies at macrochanneling is the distance z
between successive collisions with surfaces:
∆z(ψ) =
wch
tanψ
≈ wch
ψ
. (5)
At fixed ψ it estimates the half-wavelength of zigzag
trajectory; note that in our case ∆z(5◦) = 34.3 mm.
The curves of Fig.2,a are plotted at fixed ψin ver-
sus the reduced depth (z/L) and in fact demonstrate
different stages of the same depth dependence. It
is clear that the behavior of the dechanneling func-
tion considerably changes with the increase of the
depth as compared with ∆z. At small z < ∆z(ψin)
where the points of first collisions with surface are dis-
tributed the dechanneling function decreases rapidly
due to the absorption of particles by the channel
wall. In our case of the single surface collision albedo
R(5◦) = 0.72 the small-depth asymptotic form of
Pch(z):
Pch(z) ≈ 1− [1−R(ψin)] · z
∆z(ψin)
(6)
demonstrates much slower decrease then the rate ob-
served in modeling (see the dashed line in Fig.2,a).
It means that even at z < ∆z(ψin) multiple surface
collisions of particles can take place due to the devi-
ation of the reflection law from the specular one.
At z ≈ ∆z(ψin) the dechanneling function has the
discontinuity of derivative and the rate of decrease
of Pch(z) is considerably reduced. According to the
intuitively obvious model of specular reflection with
constant absorption probability A(ψin) = 1−R(ψin)
the dechanneling function becomes exponential at
large z:
Pch(z) = exp
(
− z
Rch
)
; Rch =
wch
ln R−1 · tan ψin
.
(7)
Here Rch has the meaning of dechanneling length.
However the calculation of Rch(z) =
−z/ ln Pch(z) according to the modeled Pch(z) data
has shown that at large z the dechanneling length in-
creases with depth; therefore the dechanneling func-
tion decreases slower then the exponent. This fact
is also due to the non-specular mode of backscat-
tering: certain part of electrons reflects at angles
|ψout| < |ψin| and their length of free motion be-
tween successive reflections increases as compared
with ∆z(ψin). Just these particles form the long-
range non-exponential tail of dechanneling function.
The directional dependence of the dechanneling
function shown in Fig.2,b for the macrochannel of
fixed length L = 500 mm indicates that at smaller
incident grazing angles ψin electrons are transported
in the macrochannel more effectively. Mainly it is
due to the increase of free path length between the
walls (that is proportional to ψ−1) and to the cor-
responding decrease of the number of collisions with
surfaces.
The distributions of the number Nrefl of reflec-
tion events shown in Fig.3 indicate that at large L
it becomes substantially smaller then the asymptotic
estimation Nrefl ≈ (L · tan ψin)/wch that follows
from the simple specular reflection model (in this
model at ψin = 5◦, Nrefl ∼ 3 for L=100 mm and
approaches to 14 for L=500 mm).
92
0 1 2 3 4 5 6 7 8 9 10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a)
3 MeV e- / Cu
wch = 3 mm
in = 5°
f N
(N
re
fl
),
re
l.
un
.
N
refl
L = 10 mm
L = 50 mm
L = 100 mm
L = 200 mm
L = 500 mm
0 100 200 300 400 500
1
10
100
0
1
2
3
4
(b)
3 MeV e- / Cu
wch = 3 mm
in = 5°
T(
L)
, %
L, mm
<
N
re
fl >
Fig.3. The distributions of electrons over the
number Nrefl of reflection events (a) and the depen-
dencies of the transmittance factor T and the mean
number of reflections 〈Nrefl〉 (b) in collimators of
various lengths
The rate of increase of the mean number 〈Nrefl〉
substantially reduces with the increase of L and
〈Nrefl〉 practically saturates to the value of 4 at
large L (in the specular reflection model the value
Nrefl ≈ 4 corresponds to L ≈ 140 mm). Accord-
ingly, the transmittance factor T becomes marginal
at large L À ∆z(ψin).
Therefore one can conclude that for relativistic
electrons the deviation of reflection probability R
from unity and the non-specular mode of backscatter-
ing substantially restrict the range of depths where
the particles dynamics looks like the quasi-periodic
motion between the walls (that is peculiar to the pla-
nar channeling in crystals). At large L macrochannels
operate as conventional collimators where the trans-
mittance factor is determined only by the geometric
ratio of width and length.
This conclusion agrees with the results of the
modeling of fine structure of angular distributions
of electrons at the exit of macrochannels of various
lengths. These results are illustrated by Figs. 4 and 5.
At sufficiently small L the maps of angular distri-
butions of Fig.4,a-c are essentially asymmetric in the
vertical direction that corresponds to the emergency
angle ψout with respect to the channel wall.
At L=10 mm that is less then ∆z(ψin) about
80% of electrons pass the channel without collisions
with surfaces (their contribution shrinks to the point
at the vertical axis of lower half plane of Fig.4,a).
The residuary 20% of electrons form a typical an-
gular distribution of single backscattering at grazing
incidence [10, 7].
Fig.4. The maps (projected onto the transversal
plane of planar macrochannel) of angular
distributions of 3 MeV electrons passed through the
macrochannels of various length L at the angle of
incidence ψin = 5◦. The center of gray circle with
angular aperture 20 mrad (1.15◦) indicates the
direction of specular reflection of particles from the
lower wall of macrochannel
Fig.5. The angular distributions of 3 MeV
electrons passed through the macrochannels of
different length L over the grazing emergency angle
ψout to the channel wall (a) and the azimuthal
angle φout in the transversal plane (b, φout = 0◦
corresponds to the beam incidence plane)
93
As L increases the contribution of electrons that
have experienced even numbers of reflection events
arises; they form the distribution in the lower half
plane of angular maps. At the same time the flux
of electrons having odd numbers of reflections and
emerging in the upper half plane is hardly collimated.
At large L the angular distributions tend to be-
come vertically symmetrical (see Fig.4,d-f).
The results of statistical analysis of obtained an-
gular maps are shown in Fig.5. The data depicted in
Fig.5,a demonstrate the effect of beam angular split-
ting first observed experimentally in Ref. [1].
The effect consists in the reduction of the trans-
mittance of a planar macrochannel in directions par-
allel to its walls. On the angular maps of Fig.4,c-f the
splitting manifests itself as a clear band close to the
horizontal axis ψout = 0 and is tracked up to large
lengths of macrochannels.
The separated peaks in angular distributions over
the grazing angle ψout are formed by electrons expe-
rienced even and odd numbers of surface collisions.
The positions of angular peaks are rather close to
the collimation angle ψcol = wch/L of the macrochan-
nel.
Azimuthal distributions of electrons depicted in
Fig.5,b describe the gradual unfolding of a beam in
the lateral plane of the macrochannel and its localiza-
tion nearby the directions φout → ±90◦ parallel to the
channel walls (however the directions φout = ±90◦
themselves remain blocked due to the same splitting
effect).
Hence in long planar gaps the monodirectional
beam transforms into the ribbon-type one rotated
azimuthally by 90◦. This effect is peculiar to pla-
nar geometry only; in macrochannels of other shapes
(e.g. square or cylindrical) azimuthal distributions
are more axially symmetric.
4. COHERENT EFFECTS IN PLANAR
MACROCHANNELING OF IONS
Macrochanneling of positively charged particles
(protons, heavy ions or positrons) differs qualitatively
from the marcochanneling of electrons due to the fact
that it can be affected by coherent effects of corre-
lated interaction of particles with surface plane [4, 5].
Similarly to the description of planar channeling in
crystals this interaction can be described with the
repulsive continuum surface potential U(y) that de-
pends only on the distance y to the surface. The
continuum potential can be introduced not only for
crystalline surface but also for planar surface of amor-
phous medium [11].
If the grazing angle ψin of a positively charged
particle does not exceed the critical angle ψp =√
Up/E of planar channeling [2] (here Up =
2πZ1Z2e
2aTF /a2 is the surface potential barrier, Z1,2
are the atomic number of incident particle and target
medium with the mean interatomic distance a, aTF is
the Thomas-Fermi screening length) then due to the
homogeneity of U(y) along the surface the transver-
sal energy of reflected particle is conserved. Thus
the specular reflection occurs that is due to coher-
ent interaction of a particle with surface atoms with
strong correlation of impact parameters. This mode
of backscattering, the planar surface semichanneling,
has dynamical (but not kinetic) nature and can facili-
tate the beam transportation in long macrochannels.
Obviously for ψin ≥ ψp surface semichanneling be-
comes impossible and the reflection mode returns to
the incoherent one that is described by the distribu-
tion (4).
However it must be taken into account that even
for ψin < ψp stochastic deviations of surface poten-
tial from the averaged U(y) lead to incoherent mul-
tiple scattering that results in the non-conservation
of transversal energy. In the bulk of crystals this
factor causes the dechanneling; for surface scatter-
ing it results in non-specular reflection and has to be
introduced into models designed to describe the ion
macrochanneling quantitatively.
The dominating dechanneling factor at planar
channeling of ions is the discreteness of atomic plane
[2]. The mean-squared angles of incoherent multi-
ple scattering per unit of range z of a particle that
moves at distance y from the discrete atomic plane
with randomly distributed atoms were estimated in
Ref. [12].
For the standard Lindhard potential of particle-
atom interaction [2] the orthogonal (y-) and lateral
(x-) components of this quantity can be written as
follows:
〈
∆θ2
y
∆z
〉
=
45ψ4
p
64π
a2a2
TF
y5
;
〈
∆θ2
x
∆z
〉
=
1
5
〈
∆θ2
y
∆z
〉
.
(8)
To estimate the mean-squared angle
〈
∆θ2
〉
of in-
coherent scattering per single reflection event one has
to integrate the expressions (8) upon the trajectory
y(z) of unperturbed motion governed by the contin-
uum surface potential. The simplest analytical esti-
mation follows from the ”corner-type” trajectory rep-
resentation:
y(z) = ymin(ψ) + |z| · ψ, (9)
where ymin is the distance of the closest approach of
the particle to the surface. For the Lindhard contin-
uum potential ymin(ψ) ≈ 1.5 ·aTF ·(ψ/ψp). Then the
y-component of variance is expressed by the formula:
σ2
y =
〈
∆θ2
y
〉
=
45ψ4
p
128π
a2a2
TF
y4
min(ψ) · ψ =
5
72π
a2
a2
TF
ψ3
(10)
and, again, the x-component is 5 times smaller.
Within this model σx,y only depend on the sorts of
particle and target material and on the angle of in-
cidence while do not depend on particle energy. One
should note that the ratio (σy/ψp)2 ∝ ψ3. Hence
94
the conditions of coherent specular reflection at pla-
nar semichanneling are improved as the grazing angle
of incidence decreases. At critical value ψ = ψp they
are also improved with the increase of particle energy
(because ψp ∝ E−1/2).
At our Monte Carlo modeling of ion macrochan-
neling for ψ < ψp the emergency angles of backscat-
tering around the direction of specular reflection were
sampled from the normal distribution with zero mean
value and variances σx,y obtained above; we also ac-
cepted that R(ψ < ψp) = 1 and neglected the particle
energy losses. At ψ > ψp the same algorithm of inco-
herent reflection was applied that has been used for
electrons.
To clarify the role of coherent effects we also con-
sidered the case of pure incoherent reflection of ions;
for such a purpose we formally set ψp = 0.
The case of macrochanneling of 1 MeV protons
in the same Copper collimator geometry described in
Section 3 was considered. For this case the critical
angle ψp of planar surface semichanneleng equals to
0.5◦.
For the evaluation of reduced parameters σ∗ and
σ according to Eqs. (1)-(2) the proton range Rp =
6.8 µm had been obtained using the SRIM code.
Combining this value with the value of the trans-
port cross-section σtr calculated within the Firsov
model [10] we have calculated the scattering param-
eter σ∗ = 0.017 ¿ 1. It means that these protons
effectively reflect from the surface only at grazing in-
cidence. However at ψin = 0.4◦ the incoherent reflec-
tion parameter (2) for 1 MeV protons σ ≈ 700 À 1
is close to the value σ ≈ 670 for 3 MeV electrons at
ψin = 5◦, the case discussed in Sec.3.
The effect of the coherent reflection mode on the
dechanneling functions is illustrated by Fig.6.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1
10
100
0 1 2 3 4 5 6 7 8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
in=1.0°
L = 1 m
P ch
(z
/L
),
%
z / L
in=0.4°
L = 2 m
1 MeV H+ / Cu
w
ch
= 3 mm
p
= 0.5°
N
refl
f N
(N
re
fl ),
r
el
. u
n
Fig.6. Dechanneling functions of 1 MeV protons in
Copper macrochannels of different lengths with
(solid curves) and without (dashed curves) the
account of coherent reflection (planar surface
semichanneling) mechanism. The inset plot
represents the particle distributions over the number
Nrefl of reflection events
At conditions of a sub-barrier incidence of protons
(ψin = 0.4◦ = 0.8·ψp) the effect of coherent reflection
at planar surface semichanneling is very strong: the
2 m long macrochannel transmits about 70% particles
while for pure incoherent reflection mode the trans-
mittance factor is less then 4%. The most probable
number of reflection events in the case of semichan-
neling shifts toward the theoretical estimation of
specular reflection model: Nrefl ∼ L·ψin/wch ≈ 4.65.
In the case of above-barrier incidence of particles
(ψin = 1◦ = 2 · ψp) the effect of volume capture of
protons into the mode of surface semichanneling is
observed. This effect results in sharp deceleration of
the Pch(z) dependency starting from the depth z ≈
60 cm.
Angular distributions of ions (see Fig.7 and 8)
also demonstrate strong qualitative effects of coher-
ent reflection. At sub-barrier incidence (see Fig.7,a,d)
the introduction of surface semichanneling leads to
drastic changes of the distribution: the symmetri-
cal gauss-shaped spots appear. As it is clear from
Fig.8,a they are close to the specular reflection direc-
tions |ψout| ≈ 0.4◦ and are formed by well-channeled
protons. Azimuthal distribution over φout depicted
in Fig.8,b also indicates the concentration of coher-
ently transported protons nearby the incidence plane
(φout = 0◦ and ±180◦). Horizontal bands in Fig.7,d
are formed by dechanneled particles that have expe-
rienced the incoherent reflection.
If coherent effects are completely ignored the dis-
tribution in this case is qualitatively similar to the an-
gular distribution of electrons at large L (see Fig.4).
At conditions of above-barrier proton incidence
(see Fig.7,b,e) the characteristic effect of surface
semichanneling consists in the splitting of angular
distribution in the upper half plane of transversal an-
gular map. It indicates the separated contributions
of proton flux fractions that are either incoherently
backscattered or captured into the coherent reflection
mode.
The formation of gauss-shaped spots is suppressed
in this case because particles have time to experience
substantial lateral scattering before the volume cap-
ture event occurs.
Fig.7. The maps of angular distributions of
1 MeV protons (a,b,d,e) and 28 keV Ne ions (c,f)
passed through the 3 mm wide Copper
macrochannels of different lengths at different
angles of incidence. Upper maps describe the pure
incoherent mode of reflection while the lower row of
maps corresponds to the incorporation of the
coherent surface semichanneling mode
95
-1.0° -0.8° -0.6° -0.4° -0.2° 0.0° 0.2° 0.4° 0.6° 0.8° 1.0°
0.0
0.5
1.0
1.5
2.0
(a)1 MeV H+ / Cu
wch = 3 mm
L = 2000 mm
in = 0.4°
p = 0.5°
f(
ou
t),
re
l.
un
.
out, deg
-180°-150°-120° -90° -60° -30° 0° 30° 60° 90° 120° 150° 180°
0.000
0.005
0.010
0.015
0.020
1 MeV H+ / Cu
wch = 3 mm
L = 2000 mm
in = 0.4°
p = 0.5°
f(
ou
t),
re
l.
un
.
out, deg
(b)
Fig.8.Angular distributions over the grazing
emergency angle ψout (a) and the azimuthal angle
φout in the transversal plane (b) for protons passed
through the 2 m long planar macrochannel. Dashed
curves — incoherent reflection mode (see Fig.7a),
solid curves — coherent surface semichanneling
mode (see Fig.7,d)
The data of Fig.7,c,f correspond to the case of
macrochanneling of heavier ions (28 keV Ne) at sub-
barrier incidence (ψin = 5◦). In this case ψp = 8.9◦,
σ∗ = 1.2, σ ≈ 315. One can see that for heavy ions
the effect of surface semichanneling is also remark-
able. However due to large value of the incident angle
ψin the destructive effect of surface atomic discrete-
ness becomes very pronounced (this fact directly
follows from Eq.(10)). Thus the incoherent multi-
ple scattering results in the smearing of the spots
of coherent reflection and in the relative increase of
contributions of effects of geometrical collimation of
ion beam.
5. CONCLUSIONS
The computer simulation has shown that the
macrochanneling of relativistic negatively charged
particles between the planar solid surfaces differs con-
siderably from the regular oscillatory motion due
to substantially stochastic nature of small-angle re-
flection of electrons at glancing incidence. In long
channels it practically comes to the beam collima-
tion within the angular range determined by the
channel geometry and makes questionable the long-
range transportation of electron beams without sig-
nificant losses of intensity. From the other hand
the macrochannels of certain optimal length can be
applied for electron beam angular splitting and for
the enhancement of yields of secondary processes of
particle-surface interactions.
It has been shown that for positively charged par-
ticles (ions or positrons) the coherent effect of surface
semichanneling allows to stabilize the macrochannel-
ing trajectories, to reduce the particle losses and to
achieve the transport and splitting of high energy
beams.
The methods of macroscopic mathematical mod-
eling of the macrochanneling are supposed to be effec-
tively applied for calculations of ion-optical systems
that use the particles interaction with solid surfaces
for beam control and shaping. For example one of
the prospective areas of the macrochanneling appli-
cations is the preparation of small-diameter focused
ion beams for the solid surface microanalysis.
REFERENCES
1. V.I. Bojko, V.V. Yevstigneyev, B.A. Kononov
et al. Anisotropy of electron fluxes behind the ex-
tended inhomogeneities in medium // Atomnaya
energiya. 1976, v.41, N5, p.363-365 (in Russian).
2. J. Lindhard. Influence of crystal lattice on mo-
tion of energetic charged particles // Kgl. Dan.
Viden. Selsk. Mat.-Fys. Medd. 1965, v.34, N14.
3. V.I. Bojko, Ye.A. Gorbachov, V.V. Yevstigneyev
et al. Turn and transportation of a beam of
swift electrons based on macrochanneling effect
// JTP. 1981, v. 51, N5, p.1042-1044 (in Rus-
sian).
4. M.A. Kumakhov. Radiation of charged particles
in crystals, Moscow: ”Energoatomizdat”, 1986,
160p. (in Russian).
5. M.A. Kumakhov, F.F. Komarov. Reflection of
particles and quanta from solid surfaces and reg-
ulation of their trajectories // Radiation Effects.
1985, v.90, N3-4, p.269-281.
6. N.K. Zhevhago, V.I. Glebov. Diffraction and
channeling in nanotubes // JETP. 2000, v.118,
N3, p.579-591 (in Russian).
7. S.V. Dyuldya, M.I. Bratchenko. Monte Carlo
method of macroscopic modeling of small-angle
reflection of fast changed particles from solid sur-
face // Problems of Atomic Science and Technol-
ogy. Series: Radiation Damage Physics and Ra-
diation Material Science. 2001, N4(80), p.53-56
(in Russian).
8. V.A. Kurnayev, Ye.S. Mashkova,
V.A. Molchanov. Reflection of light ions
from solid surface. Moscow: ”Energoatomizdat”,
1985, 192p. (in Russian).
96
9. E.S. Parilis, N.Yu. Turayev, F.F. Umarov et al.
Theory of scattering of medium energies atoms
by solid surface. Tashkent: ”Fan”, 1987, 212p.
(in Russian).
10. M.I. Ryazanov, I.S. Tilinin. Surface studies by
particle backscattering. Moscow: ”Energoatomiz-
dat”, 1985, 152p. (in Russian).
11. M.A. Kumakhov. The phenomenon of specular
reflection of charged particles from amorphous
and polycrystalline surfaces // DAN USSR. 1984,
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МОДЕЛИРОВАНИЕ КИНЕТИКИ МАКРОКАНАЛИРОВАНИЯ БЫСТРЫХ
ЭЛЕКТРОНОВ И ИОНОВ В ЩЕЛЕВЫХ КОЛЛИМАТОРАХ
С.В. Дюльдя, М.И. Братченко, М.А. Скоробогатов
Оригинальным методом макроскопического математического моделирования отражения заряжен-
ных частиц от поверхности твердого тела исследована кинетика плоскостного макроканалирования
быстрых электронов и ионов –— ориентированного транспорта частиц между поверхностями, разде-
ленными макроскопическим расстоянием. Впервые рассчитаны функции деканалирования частиц при
макроканалировании. Моделирование угловых распределений подтвердило существование эффекта
разделения потоков частиц. Разработан метод учета атомной дискретности поверхности при модели-
ровании когерентных эффектов плоскостного полуканалирования ионов. Показано, что ввиду стоха-
стичности процесса отражения макроканалирование электронов в каналах большой длины эффектив-
но определяется механизмами коллимации пучка, тогда как для ионов когерентный эффект позволяет
стабилизировать макроканалирование и достичь транспортировки и разделения пучка на больших рас-
стояниях без существенных потерь интенсивности.
МОДЕЛЮВАННЯ КIНЕТИКИ МАКРОКАНАЛЮВАННЯ ШВИДКИХ
ЕЛЕКТРОНIВ ТА IОНIВ У ЩIЛИННИХ КОЛIМАТОРАХ
С.В. Дюльдя, М.I. Братченко, М.О. Скоробогатов
Оригiнальним методом макроскопiчного математичного моделювання вiдбиття заряджених части-
нок вiд поверхнi твердого тiла дослiджена кiнетика площинного макроканалювання швидких елек-
тронiв та iонiв –— орiєнтованого транспорту частинок мiж поверхнями, роздiленими макроскопiчною
вiдстанню. Вперше розрахованi функцiї деканалювання частинок при макроканалюваннi. Моделюван-
ня кутових розподiлiв пiдтвердило iснування ефекту роздiлення потокiв частинок. Розроблено метод
урахування атомної дискретностi поверхнi в моделюваннi когерентних ефектiв площинного напiвка-
налювання iонiв. Показано, що з-за стохастичностi процесу вiдбиття макроканалювання електронiв в
каналах великої довжини ефективно визначається механiзмами колiмацiї пучка, тодi як для iонiв коге-
рентний ефект дозволяє стабiлiзувати макроканалювання та досягти транспортування та роздiлення
пучка на великих вiдстанях без значних втрат iнтенсивностi.
97
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