Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence
Using the computer experiment methods directional effects of relativistic electrons’ coherent reflection from crystal surface at glancing incidence were studied in conditions when it is due to multiple transversal scattering of particles by atomic chains (axial surface channeling). Directional depen...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1103952017-01-05T03:02:25Z Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence Dyuldya, S.V. Ядернo-физические методы и обработка данных Using the computer experiment methods directional effects of relativistic electrons’ coherent reflection from crystal surface at glancing incidence were studied in conditions when it is due to multiple transversal scattering of particles by atomic chains (axial surface channeling). Directional dependencies of backscattering coefficients, ranges and depths of reflected electrons’ penetration in crystal and their angular distributions have been calculated. It has allowed to elicit the directional effects of strings that lead to reflection at grazing angles close to the beam incident angle with respect to atomic chain as well as kinetic effects of surface plane that result in specular reflection and dominate at large beam misalignments with respect to low-index crystallographic directions. Методами комп'ютерного експерименту досліджені орієнтаційні ефекти у когерентному відбитті релятивістських електронів від поверхні кристалів за умов ковзного падіння та аксіального поверхневого каналювання, коли відбиття визначається азимутальним багаторазовим розсіюванням частинок на атомних ланцюжках. Розраховані орієнтаційні залежності коефіцієнтів відбиття, довжин пробігу, глибин проникнення у кристал та кутових розподілів зворотно розсіяних частинок. Їх аналіз дозволив виявити орієнтаційні ефекти ланцюжків, що ведуть до відбиття під ковзними кутами, близькими до кута орієнтації вісі пучка до напрямку атомного ланцюжка, та кінетично обумовлені ефекти площини, що призводять до дзеркального відбиття та домінують за великих азимутальних разорієнтацій пучка до низькоіндексних кристалографічних напрямків. Методами компьютерного эксперимента исследованы ориентационные эффекты в когерентном отражении релятивистских электронов от поверхности кристаллов при скользящем падении в условиях аксиального поверхностного каналирования, когда отражение определяется азимутальным многократным рассеянием частиц на атомных цепочках. Рассчитаны ориентационные зависимости коэффициентов отражения, длин пробега, глубин проникновения в кристалл и угловых распределений обратно рассеянных частиц. Их анализ позволил выявить ориентационные эффекты цепочек, проявляющиеся в отражении под углами скольжения, близкими к углу ориентации оси пучка к направлению атомной цепочки, и кинетически обусловленные эффекты плоскости, приводящие к зеркальному отражению и доминирующие при больших азимутальных разориентациях пучка к низкоиндексным кристаллографическим направлениям. 2007 Article Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence / S.V. Dyuldya // Вопросы атомной науки и техники. — 2007. — № 5. — С. 98-105. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 02.70.Uu, 07.05.Tp, 29.27.Eg, 41.75.Ht, 61.85.+p, 68.49.Jk, 68.49.-h http://dspace.nbuv.gov.ua/handle/123456789/110395 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. Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence Вопросы атомной науки и техники |
| description |
Using the computer experiment methods directional effects of relativistic electrons’ coherent reflection from crystal surface at glancing incidence were studied in conditions when it is due to multiple transversal scattering of particles by atomic chains (axial surface channeling). Directional dependencies of backscattering coefficients, ranges and depths of reflected electrons’ penetration in crystal and their angular distributions have been calculated. It has allowed to elicit the directional effects of strings that lead to reflection at grazing angles close to the beam incident angle with respect to atomic chain as well as kinetic effects of surface plane that result in specular reflection and dominate at large beam misalignments with respect to low-index crystallographic directions. |
| format |
Article |
| author |
Dyuldya, S.V. |
| author_facet |
Dyuldya, S.V. |
| author_sort |
Dyuldya, S.V. |
| title |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| title_short |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| title_full |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| title_fullStr |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| title_full_unstemmed |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| title_sort |
directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
| topic_facet |
Ядернo-физические методы и обработка данных |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110395 |
| citation_txt |
Directional effects in albedo and angular distributions of relativistic electrons reflected from single crystals at grazing incidence / S.V. Dyuldya // Вопросы атомной науки и техники. — 2007. — № 5. — С. 98-105. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT dyuldyasv directionaleffectsinalbedoandangulardistributionsofrelativisticelectronsreflectedfromsinglecrystalsatgrazingincidence |
| first_indexed |
2025-07-08T00:33:57Z |
| last_indexed |
2025-07-08T00:33:57Z |
| _version_ |
1837036825271599104 |
| fulltext |
DIRECTIONAL EFFECTS IN ALBEDO AND ANGULAR
DISTRIBUTIONS OF RELATIVISTIC ELECTRONS
REFLECTED FROM SINGLE CRYSTALS
AT GRAZING INCIDENCE
S.V. Dyuldya∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received May 23, 2006)
Using the computer experiment methods directional effects of relativistic electrons’ coherent reflection from crystal
surface at glancing incidence were studied in conditions when it is due to multiple transversal scattering of particles by
atomic chains (axial surface channeling). Directional dependencies of backscattering coefficients, ranges and depths
of reflected electrons’ penetration in crystal and their angular distributions have been calculated. It has allowed to
elicit the directional effects of strings that lead to reflection at grazing angles close to the beam incident angle with
respect to atomic chain as well as kinetic effects of surface plane that result in specular reflection and dominate at
large beam misalignments with respect to low-index crystallographic directions.
PACS: 02.70.Uu, 07.05.Tp, 29.27.Eg, 41.75.Ht, 61.85.+p, 68.49.Jk, 68.49.-h
It is well known that at grazing incidence of swift
charged particles on a crystalline surface at small an-
gles with respect to major crystallographic directions
the directional effects of crystal orientation are ob-
served. They are due to correlations of particle colli-
sions with atoms of surface atomic chains. These col-
lective correlated (or coherent) reflection mechanisms
are well-studied for ions [1] and result in specific fea-
tures of energy spectra and angular distributions of
reflected particles (ion focusing, semichanneling and
surface channeling).
Similar mechanisms of coherent reflection are ex-
pected to take place also for relativistic electrons and
positrons although currently no experimental data
for high energy region are available and the inves-
tigations are limited with theoretical estimations and
modeling [2, 3, 4, 5]. The significance of these stud-
ies along with fundamental value of investigations of
new mechanisms of particle-surface interactions is de-
termined by the opportunity of their application for
high energy beams control [6].
From this point of view the mechanisms of co-
herent reflections of relativistic electrons are of the
greatest interest. Due to negative charge of electrons
at highly relativistic energies they differ from those
for positive ions or positrons. In particular the neg-
ative charge prohibits their coherent specular reflec-
tion from the surface plane [1, 6], the planar surface
semichanneling.
In Ref. [2] we had proposed the mechanism of re-
flection of electrons from single crystal surfaces that is
due to azimuthal coherent scattering of particles by
an atomic chain (the axial surface semichanneling).
The backscattering coefficient (albedo) had been cal-
culated at angles θ with respect to axial direction less
then the critical angle θa of axial channeling (Lind-
hard angle).
For small θ ¿ θa even single azimuthal scatter-
ing by atomic chain leads to high efficiency of coher-
ent reflection [2]. However as the grazing angle ap-
proaches the Linhdard angle the efficiency of the axial
semichanneling mechanism rapidly decreases and the
effects of multiple scattering by atomic strings [7, 8]
(the axial surface channeling) become important [4].
The theoretical investigation of coherent reflec-
tion of electrons at axial surface channeling was car-
ried out in Ref. [5] by means of computer simulation
methods for broad range of grazing angles of beam in-
cidence. The modeling results have been interpreted
within the scope of the phenomenological diffusion
model based on the assumption of rapid isotropiza-
tion of the flux of above-barrier electrons in the axial
surface channel transversal plane at multiple coherent
scattering by atomic chains.
In the present paper in order to complete the pic-
ture of directional effects at coherent reflection from
axial surface channels the directional dependencies of
its integral and differential characteristics are studied
as functions of the crystal target tilt and rotation an-
gles.
1. PROBLEM SETUP
Considered is the glancing incidence of a colli-
mated beam of swift electrons with energy E onto
an ideal surface of a single crystal plate of length L
at grazing angle ψin ¿ 1 to surface plane and the
∗Corresponding author. E-mail address: sdul@kipt.kharkov.ua
98 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5.
Series: Nuclear Physics Investigations (48), p.98–105.
azimuthal angle ϑin ¿ 1 between the beam axis sur-
face projection and the direction of one of low-index
crystallographic axes (see Fig.1,a).
Both ψin and ϑin are assumed to be small enough
so that the angle θin =
√
ψ2
in + ϑ2
in between the
beam axis and the atomic chain is of order of mag-
nitude of Lindhard angle θa =
√
2Ua/E where
Ua = 2Ze2/d, Z is the crystal atomic number, d is
the chain interatomic distance. The initial transver-
sal projection of the beam axis forms the azimuthal
angle φin = arctan(ψin/ϑin) with the surface plane
(see Fig.1,b).
For highly relativistic energies at angles θin <
102 · θa the coherent interaction of electrons with
atoms of chains is described by the averaged contin-
uum potential U(r) of the axial channel that depends
only on 2D vector r(x, y) of the particle’s transversal
coordinates. In the first approximation the longitu-
dinal motion of particle is free, only the transver-
sal coherent scattering of particles occurs and their
transversal energy is conserved. Thus in the axial
case the dynamics the particles surface channeling is
per se two-dimensional. The longitudinal coordinate
z = v‖ · t is in the first approximation proportional
to the time t of coherent interaction.
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
9
10
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
in
/
a
in
/
a
in = 0.5· a
in = 1.5· a
in = 4.0· a
(b)
in
,
de
g
Fig.1. The geometry of a beam glancing incidence
onto a crystal surface (a) and the geometrical
interrelations between different angles of a beam
axis orientation with respect to a surface and to the
close-packed atomic chain (b)
By definition, the coherent reflection events occur
when at such interaction electrons trajectories cross
the surface plane y = 0 and exit the crystal at
emergency angles ψout with respect to the plane and
ϑout between the surface projection of emergency di-
rection and the crystallographic axis. If total energy
losses are neglected the emergency angle with respect
to the axis is conserved in view of the conservation of
transversal energy E⊥:
θout =
√
ψ2
out + ϑ2
out
∼= θin. (1)
The problem consists in the calculation of integral
coherent reflection coefficient R for the plate of length
L as a function of incident angles ψin and ϑin. Also
of interest are the differential characteristics of reflec-
tion such as angular distributions or the distributions
of emergency points over the longitudinal coordinate
z (the reflection length distributions). Energy depen-
dencies of these characteristics are universal because
the equations of transversal motion indicate that for
coherent interaction all these quantities depend only
on the ratios ψin/θa and ϑin/θa. Thus the depen-
dency of, e.g., R(ψin/θa, ϑin/θa) can be easily scaled
to the dependency R(ψin, ϑin) for any value of Lind-
hard angle θa ∝ E–−(1/2).
To solve the problem the approach of Ref. [5] is
used. This publication contains the detailed descrip-
tion of the applied statistical modeling method and
code.
Transversal trajectories of surface channeling are
modeled as the sequences of binary collisions of parti-
cles with atomic strings till the reflection event takes
place or the plate length L is passed. The collisions
are assumed to be uncorrelated, so the ordered struc-
ture of the channel’s transversal plane is neglected.
Since the specific goal of this work consists in the
investigation namely of the coherent mechanism of
reflection we knowingly ignore in our model the ef-
fects of incoherent scattering of particles by atomic
thermal vibrations and crystal electrons as well as
the total energy losses due to ionization and radiation
stopping and the incident beam divergence effects (all
of them will be studied in future).
In Ref. [5] we limited ourselves with the degen-
erated case ϑin = 0 when the beam axis lays in the
plane orthogonal to the surface plane. It corresponds
to the normal transversal incidence (φin = 90◦)
when the characteristics of coherent reflection depend
only on grazing angle ψin. In this case the coherent
reflection requires the multiple transversal scattering
by angles ∆φ ≈ π.
However unlike for the conventional incoherent re-
flection the directional effects of coherent reflection
are not limited with the dependence on ψin. The
presence of the preferential direction of close-packed
atomic chain has to result in the dependencies of re-
flection characteristics on the azimuthal crystal rota-
tion angle ϑin. Only the united description of these
dependencies describes the effect in whole. Small ra-
tios of angles ψin and ϑin correspond to oblique di-
rection of the incident transversal momentum (down
to the grazing transversal incidence; see Fig.1,b). At
fixed ψin the increase of ϑin also results in the in-
crease of the total angle θin with respect to the axis
and, hence, to the increase of E⊥ ∝ θ2
in.
To study directional effects we varied both ψin
and ϑin angles in two ways. In the first one the defi-
99
nite constant transversal energy E⊥ was maintained
and the incident angle φin was altered from normal
(φin = 90◦) to grazing (φin → 0◦) incidence in the
transversal plane. Another way adequately repro-
duces experimental setups when crystals are rotated
by the angle ϑin at constant ψin; thus θin and E⊥
increase while φin decreases.
Similar to Ref. [5] the model case of reflection
of a narrow beam of 5.43 GeV electrons from the
{011} surface of 1 cm long and 1 mm thick Silicon
plate was considered. Electrons experienced the co-
herent interaction with 〈001〉 atomic chains that was
described by continuum potential in the Moliere ap-
proximation at temperature 300 K. Typical statistics
of Monte Carlo modeling was about 104 of particle
histories that has provided the integral statistical
uncertainties less then 1%.
2. INTEGRAL REFLECTION
PARAMETERS
2.1. COHERENT REFLECTION
PROBABILITY
As it can be seen from Fig.2,a at fixed transver-
sal energy of electrons the total albedo R of coherent
reflection increases as φin decreases. The rate and
the relative growth of R(φin) are greater at larger
θin (and E⊥).
90° 80° 70° 60° 50° 40° 30° 20° 10° 0°
50
60
70
80
90
100
(a)
in = 1.5· a
in = 4.0· a
R,
%
in
, deg
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
90
100
(b)
in = 0.5· a
in = 1.5· a
in = 4.0· a
in = 0
in
/
a
R,
%
in
/
a
Fig.2. The directional dependencies of total albedo
R of coherent reflection upon the angle φin of
transversal incidence at fixed initial transversal
energy (a) and on the azimuthal crystal rotation
angle ϑin at fixed grazing angles ψin (b). For
comparison the dependency R(ψin) at ϑin = 0◦ ,
φin = 90◦ [5] is also depicted
This effect discovers the close analogy of reflec-
tion mechanisms between the conventional incoher-
ent backscattering of particles from amorphous me-
dia and the coherent reflection subject to the fact
that the latter one is determined by the scattering in
the transversal space.
In fact, in the transversal plane the angle φin has
the meaning of a grazing angle of particle incidence.
Its reduction facilitates the reflection of the beam
fraction coherently scattered toward the surface plane
(for its reflection the scattering by angles ∆φ ∼ φin
is required). The reflection probability approaches
unity at conditions of grazing transversal incidence
φin → 0◦. Therefore the observed effect is similar to
the enhancement of conventional incoherent reflec-
tion at grazing incidence that is more pronounced at
higher energies of particles [1].
The above-mentioned factor also substantially af-
fects the azimuthal dependencies of coherent reflec-
tion probability on the angle ϑin with respect to the
axial surface channel direction at fixed grazing angle
ψin. In Fig.2,b they are compared with the depen-
dency R(ψin) at conditions of normal transversal in-
cidence: ϑin = 0◦, φin = 90◦ [5]. In the latter case
the increase of ψin up to 8 × θa reduces the coher-
ent reflection coefficient by one order of magnitude
because the growth of transversal energy effectively
prohibits the reflection when it requires the transver-
sal backscattering by angles ∼ π. As ϑin increases
the transversal energy is also increased (and particles
have the opportunity to sink deeper into the plate)
but again the correspondent reduction of φin facil-
itates the coherent reflection. These two competi-
tive factors (among which the growth of E⊥ is dom-
inant) form weaker dependency R(ϑin) that in the
same range of transversal energies reduces only by
20 . . . 40%.
2.2. COHERENT REFLECTION LENGTH
Other descriptive integral parameter of coherent
reflection is the mean length 〈z〉 that particles travel
in the longitudinal direction of a plate till the reflec-
tion occurs.
It determines the reflected electrons’ life time in
a crystal and therefore at grazing incidence has the
meaning of the effective crystal thickness for all sec-
ondary processes of interaction (e.g. for coherent
electromagnetic radiation) [5].
The directional dependencies of 〈z〉 are depicted
in Fig.3 and the details of their forming are illustrated
by histograms of the reflection probability distribu-
tions over z shown in Figs.4 and 5.
As it can be seen from Fig.3,a,b the transition
to the grazing transversal incidence sharply reduces
the mean length of coherent reflection. The degree of
reduction increases with the increase of transversal
energy. It means that even at very large transver-
sal energies the beam fraction that reflects at small
ranges in crystal appears as the angle φin decreases.
100
90° 80° 70° 60° 50° 40° 30° 20° 10° 0°
0
500
1000
1500
2000
2500
3000
3500
4000
(a)
in = 1.5· a
in = 4.0· a
<
z >
,
m
in
, deg
0 1 2 3 4 5 6 7 8
0
500
1000
1500
2000
2500
3000
3500
5000
6000
7000
(b)
in = 0.5· a in = 1.5· a
in = 4.0· a in = 0
in
/
a
<
z >
,
m
in
/
a
Fig.3. Directional dependencies of mean length
〈z〉 of coherent reflection at the same modeling
conditions that those of Fig.2,a,b
Fig.4. The distributions fz(z) of coherently
reflected electrons over the reflection length z as
functions of transversal azimuthal angle φin at
different E⊥
The existence of such a fraction is observed in dif-
ferential distributions shown in Fig.4. At θin = 1.5·θa
it is tracked even at normal transversal incidence:
φin = 90◦. However for this case it is completely
absent at θin = 4 · θa where the broad maximum
of the distribution function ranges to large reflec-
tion lengths. As φin decreases the more particles
are reflected at small z. For θin = 4 · θa even
the threshold effect occurs: the distribution function
maximum sharply shifts to the region of small z at
φin = 50◦ − 40◦.
The directional dependencies of 〈z〉 on angles ϑin
and ψin depicted in Fig.3,b demonstrate the some-
what unexpected saturation effect. Unlike for the
normal transversal incidence (ϑin = 0◦) where the
rapid increase of 〈z〉 occurs [5] the azimuthal de-
pendencies of mean reflection lengths saturate at
ϑin ∼ (3 . . . 4) · θa and practically do not depend
on ϑin up to θin = 8 · θa, the upper limit of the
modeled azimuthal misalignments. Because the total
coherent reflection probability R(ϑin) is nevertheless
decreasing within this range of ϑin (see Fig.2,b) it
means that a kind of complex compensative effect
takes place: the rapid increase of population of parti-
cle fraction having small reflection lengths competes
with the rapid growth of reflection length for those
particles that have the chance to sink deeply in a
plate and substantially contribute to 〈z〉 due to large
z values.
The normalized distribution over reflection
lengths:
fz(z) =
1
R(L; ψin, ϑin)
· dR(z;ψin, ϑin)
dz
(2)
at large angles ϑin of beam misalignment with re-
spect to the surface channel axis direction is weakly
dependent on ϑin (see Fig.5,b-d). This fact agrees
with the effect of 〈z〉 saturation and is satisfied if
R ∝ R1(ϑin) ·R2(L).
Therefore in this case the properly directional
(that depends on ϑin) and the kinetic (that depends
on L) components of coherent reflection probability
R are effectively factorized. As now it has to be con-
sidered as a fact observed in computer experiment
because we have not found a simple qualitative ex-
planation of this feature.
Fig.5. The distributions fz(z) of coherently
reflected electrons over the reflection length z at
normal transversal incidence (a) and as functions
of crystal rotation angle ϑin at various fixed grazing
angles ψin (b–d)
2.3. COHERENT REFLECTION DEPTH
Similar directional effects have been found for an-
other integral characteristic of coherent reflection, the
mean reflection depth 〈y〉 defined in Ref. [5] as an
averaged over all reflected particles maximal depth
y that each particle has reached in the near-surface
101
layer of a crystal plate before the reflection event.
This quantity describes the lateral thickness of a crys-
tal layer that contributes to the particles backscatter-
ing.
The directional dependencies of 〈y〉 and the dis-
tributions of reflected electrons over maximal depth
y are shown in Figs. 6 to 8. The behavior of the 〈y〉
directional dependencies and the distributions over y
are quite similar to those observed for the reflection
length z (though evidently the range of y is much
smaller then that of z; rough estimation shows that
〈y〉 ∼ 〈z〉 · θin).
Hence we should only notice that similar satu-
ration of 〈y〉 at large ϑin is observed as well as the
stationary behavior of normalized distribution func-
tions fy(y) that become independent on the crystal
rotation angle ϑin.
3. ANGULAR DISTRIBUTIONS OF
COHERENTLY REFLECTED ELECTRONS
From the point of view of the investigation of
mechanisms and kinetics of coherent reflection the an-
gular distributions of backscattered relativistic elec-
trons are of great interest because they can be mea-
sured experimentally.
Basing on modeling data we have calculated the
normalized distributions of reflected particles over
grazing emergency angles ψout with respect to the
surface plane and over azimuthal angles φout =
arctan(ψout/ϑout) in the transversal space of a sur-
face channel.
Concerning the azimuthal distributions the basic
effect that has been found in modeling consists in
the transition from the symmetric distribution for
normal transversal incidence to drastically asymmet-
ric distributions for oblique and grazing directions
of initial transversal momentum of electrons. This
effect is observed at the increase of the rotation angle
ϑin.
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
(b)
in = 0.5· a
in = 1.5· a
in = 4.0· a
in = 0
in
/
a
<
y
>,
m
in
/
a
Fig.6. Directional dependencies of mean depth 〈y〉
that coherently reflected electrons reach at the same
modeling conditions that those of Fig. 2
Fig.7. The distributions fy(y) of coherently
reflected electrons over the reflection depth y as
functions of transversal azimuthal angle φin at
different constant transversal energies
Therefore the coherent reflection displays the
same trend in the directional dependencies of an-
gular distributions that are peculiar for the con-
ventional three-dimensional incoherent backscatter-
ing from amorphous media. But here they are man-
ifested only in the transversal 2D space of surface
channel. The above-mentioned transition evidently
follows from the data of Fig.9 shown for fixed ini-
tial transversal energies. On can see that at oblique
initial azimuthal angles φin < 20◦ the peaks at def-
inite emergency angles φout appear that are close to
the directions of specular reflection in the transversal
plane: φout ≈ 180◦ − φin.
Fig.8. The distributions fy(y) of coherently
reflected electrons over the reflection depth y at
normal transversal incidence (a)and as functions of
crystal rotation angle ϑin at different fixed grazing
angles ψin (b-–d)
The comparison of angular distributions of
Fig.9,a,b for different total incident angles θin shows
that the effect is more pronounced at large transver-
sal energies. But in all cases the azimuthal angular
distribution of grazing transversal reflection is broad
enough to state that it is originated from multiple
collisions of electrons with atomic strings. At normal
transversal incidence (see Fig.10,a) azimuthal angu-
lar distributions have broad maxima at the emer-
gency azimuth φout = 90◦ i.e. at normal direction
of transversal momentum. The distributions become
102
slightly narrower as the incident grazing angle ψin
increases when the particle reflection at grazing az-
imuthal angles φout → (0◦, 180◦) is blocked more ef-
fectively.
Fig.9. Azimuthal angular distributions of
coherently reflected electrons in the transversal
plane as functions of the transversal incidence angle
φin at fixed transversal energies
The transition from normal to oblique transversal
incidence with simultaneous growth of transversal en-
ergy is realized with the increase of the rotation angle
ϑin. This is demonstrated by distributions shown in
Fig.10,b–d.
The asymmetry is the most acute at small ψin
because at such grazing angles in accordance with
the curves of Fig.1,b the angle φin decreases with the
increase of ϑin more rapidly. It is a kind of pure geo-
metric effect. As ψin increases certain broadening of
azimuthal distributions takes place (see Fig.10,c,d).
Fig.10. Azimuthal angular distributions of
coherently reflected electrons at normal transversal
incidence (a) and as functions of angle ϑin of beam
orientation with respect to axial surface channel
direction at fixed grazing angles ψin (b-–d)
Angular distributions of coherently reflected elec-
trons over angles ψout with respect to surface plane
are shown in Fig.11. Due to the conservation of
transversal energy at pure coherent scattering the
total emergency angle θout = θin and according to
Eq. (1) the reflection of electrons at angles ψout > θin
is impossible. This is evidently remarkable in Fig.11.
Also one can see that at normal transversal inci-
dence (ϑin = 0◦, see Fig.11,a) sharp maxima of dis-
tributions are located near the incidence angle ψin;
thus in this case the coherent reflection is close to the
specular one. The width of distribution curves grows
with the increase of ψin.
The angular distributions directional dependen-
cies on the rotation angle ϑin (see Fig.11,b-d) can be
qualitatively interpreted within the scope of certain
asymptotically bi-fractional model.
From the one hand the quasi-specularly reflected
electrons at emergency angles ψout ≈ ψin are ob-
served. In view of the conservation of E⊥ for these
electrons the correlation |ϑout| ∼ |ϑin| has to be sat-
isfied. This implies that π − φout ∼ φin for this
fraction of reflected beam. Thus the peak of spec-
ular reflection is formed by electrons that have ex-
perienced quasi-specular reflection in the transversal
plane. Their contribution is more expressed at small
ψin (see Fig.11,b) and blurs as ψin increases.
Another fraction of electrons emerges at graz-
ing angles ψout that are close to θin, the total angle
with respect to axis. The equation (1) implies that
ϑout → 0 and therefore φout → 90◦ for this fraction.
The transversal projections of such electrons’ mo-
menta are close to the surface normal. This fraction
of the coherently reflected beam is due to electrons
that have experienced strong multiple transversal
scattering by atomic strings and practically have lost
information on the initial incident azimuth.
Fig.11. Angular distributions of coherently
reflected electrons over the grazing emergency angle
ψout at normal transversal incidence (a) and as
functions of angle ϑin of beam orientation with
respect to axial surface channel direction at fixed
ψin (b—d)
The behavior of these fractions follows from the
dependencies of angular distributions over ψout on
φin shown in Fig. 12. In the broad range of azimuthal
angles φin near the surface normal the reflection at
angles ψout → θin dominates that is due to the mul-
tiple axial scattering. It spreads toward smaller ψout
when the transversal incidence becomes oblique. At
103
grazing one (φin < 20◦) the expressed peak appears
at small ψout. It describes the coherent reflection at
the specular angle ψout ≈ ψin = θin · sin φin with
respect to the crystal surface.
Fig.12. Angular distributions of coherently
reflected electrons over grazing emergency angles
ψout as functions of the transversal incidence angle
φin at fixed transversal energies
Indeed quantitatively the partitioning of reflected
beam onto these two fractions looks somewhat vol-
untary because in fact there exist noticeable amount
of reflected electrons that occupy intermediate po-
sitions between these two limiting cases of coherent
reflection.
4. DISCUSSION AND CONCLUSIONS
As a result of computer experiments the compre-
hensive description of directional effects in the clas-
sical coherent reflection of electrons at axial surface
channeling has been obtained for the first time. It
has been shown that this mechanism of particle re-
flection from crystalline surfaces differs qualitatively
from the conventional backscattering from disordered
media being dependent not only on the beam graz-
ing incidence angle but on the supplementary direc-
tional parameter, the angle with respect to surface
low-index atomic chains.
As it is evident from angular distributions, it
demonstrates either the effect of pure axial scattering
(”string effect”) or the effect determined from the sur-
face planarity (”plane effect”). These are the major
qualitative directional effects that are supposed to be
observed in the glancing backscattering experiments
at high energies.
The former one is tracked in the existence of par-
ticles reflected at grazing angles close to the beam in-
cident angle to atomic chains at any angles of glanc-
ing incidence. The latter one is responsible for the
quasi-specular reflection at grazing angles close to the
incident grazing angle itself. It dominates for large
beam misalignments with respect to chains and cor-
responds to the glancing incidence in the transversal
plane. Thus even for negatively charged electrons the
crystal surface can effectively behave as a specularly
reflected plane though (opposite to the case of planar
coherent reflection of ions and positrons) this fact is
due to the transversal scattering in a thick enough
near-surface layer of crystal.
The modeling results have confirmed high effi-
ciency of coherent reflection mechanism in a broad
angular range of incidence as compared with the
Lindhard angle. They indicate that coherent re-
flection is determined by the complex kinetics of
transversal multiple scattering of electrons and there-
fore constitutes the physical effect that has to be de-
scribed within the generalization of the known theory
of directional effects in multiple scattering [7, 8] to the
case of semi-infinite media.
Finally one should notice that such a theory as
well as further computer simulation efforts have to
take into account certain important effects that are
currently omitted in our model not assigned to repro-
duce all actual experimental conditions. It concerns
the incoherent multiple and single nuclear scattering
that leads to the non-conservation of transversal en-
ergy and gives rise to the incoherently reflected frac-
tion of electrons [3]. The total energy losses of parti-
cles shall be included that can be significant because
certain fraction of reflected electrons travel in crystal
by macroscopic distances. Also it is expected that
supplementary directional effects in angular distribu-
tions of reflected electrons can arise from the ordered
structure of atomic chains lattice [4] as well as from
the morphology of real surfaces.
REFERENCES
1. 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).
2. V.V. Gann, V.I. Vit’ko, S.V. Dyuldya, N.N. Na-
sonov, V.V. Rozhkov. Coherent scattering and
radiation of relativistic electrons at grazing in-
cidence on a single crystal surface // Problems
of Atomic Science and Technology. Series: Ra-
diation Damage Physics and Radiation Material
Science. 1983, N5(28), p.72-80 (in Russian).
3. E.G. Vyatkin, V.A. Dolgikh, S.A. Vorobiev.
Glancing scattering of relativistic electrons by a
crystal surface — computer simulation // Radi-
ation Effects. 1986, v.100, N1-2, p.39-50.
4. V.V. Rozhkov, S.V. Dyuldya. Small-angle reflec-
tion of relativistic electrons from single crystals
due to multiple interactions with atomic chains
and planes // Abstr. of 3rd all-union conference
on radiation of relativistic particles in crystals.
Naltchik, KBGU, 1988, p.67-68 (in Russian).
5. S.V. Dyuldya. Surface channeling and coherent
reflection of relativistic electrons // Problems of
Atomic Science and Technology. Series: Radi-
ation Damage Physics and Radiation Material
Science. 1998, N6(72), p.9-24 (in Russian).
6. M.A. Kumakhov, F.F. Komarov. Reflection of
particles and quanta from solid surfaces and reg-
104
ulation of their trajectories // Radiation Effects.
1985, v.90, N3-4, p.269-281.
7. V.V. Beloshitskij, M.A. Kumakhov. Directional
effects at passage of charged particles in two-
dimensional lattice of crystal atomic chains //
Physica Tverdogo Tela. 1973, v.15, N5, p.1588-
1592 (in Russian).
8. N.F. Shulga, V.I. Truten, S.P. Fomin. Directional
effects at interaction of high energy particles with
atomic strings of a crystal // JETP. 1984, v.87,
N1, p.250-263 (in Russian).
ОРИЕНТАЦИОННЫЕ ЭФФЕКТЫ В АЛЬБЕДО И УГЛОВЫХ РАСПРЕДЕЛЕНИЯХ
РЕЛЯТИВИСТСКИХ ЭЛЕКТРОНОВ, ОТРАЖЕННЫХ ОТ МОНОКРИСТАЛЛОВ
ПРИ СКОЛЬЗЯЩЕМ ПАДЕНИИ
С.В. Дюльдя
Методами компьютерного эксперимента исследованы ориентационные эффекты в когерентном от-
ражении релятивистских электронов от поверхности кристаллов при скользящем падении в условиях
аксиального поверхностного каналирования, когда отражение определяется азимутальным многократ-
ным рассеянием частиц на атомных цепочках. Рассчитаны ориентационные зависимости коэффици-
ентов отражения, длин пробега, глубин проникновения в кристалл и угловых распределений обратно
рассеянных частиц. Их анализ позволил выявить ориентационные эффекты цепочек, проявляющиеся
в отражении под углами скольжения, близкими к углу ориентации оси пучка к направлению атомной
цепочки, и кинетически обусловленные эффекты плоскости, приводящие к зеркальному отражению и
доминирующие при больших азимутальных разориентациях пучка к низкоиндексным кристаллогра-
фическим направлениям.
ОРIЄНТАЦIЙНI ЕФЕКТИ В АЛЬБЕДО ТА КУТОВИХ РОЗПОДIЛАХ
РЕЛЯТИВIСТСЬКИХ ЕЛЕКТРОНIВ, ВIДБИТИХ ВIД МОНОКРИСТАЛIВ
ЗА УМОВ КОВЗНОГО ПАДIННЯ
С.В. Дюльдя
Методами комп’ютерного експерименту дослiдженi орiєнтацiйнi ефекти у когерентному вiдбиттi
релятивiстських електронiв вiд поверхнi кристалiв за умов ковзного падiння та аксiального поверхне-
вого каналювання, коли вiдбиття визначається азимутальним багаторазовим розсiюванням частинок
на атомних ланцюжках. Розрахованi орiєнтацiйнi залежностi коефiцiєнтiв вiдбиття, довжин пробiгу,
глибин проникнення у кристал та кутових розподiлiв зворотно розсiяних частинок. Їх аналiз дозволив
виявити орiєнтацiйнi ефекти ланцюжкiв, що ведуть до вiдбиття пiд ковзними кутами, близькими до
кута орiєнтацiї вiсi пучка до напрямку атомного ланцюжка, та кiнетично обумовленi ефекти площини,
що призводять до дзеркального вiдбиття та домiнують за великих азимутальних разорiєнтацiй пучка
до низькоiндексних кристалографiчних напрямкiв.
105
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