Spectral method in the quantum theory of axial channeling
The quantization of the transverse motion energy in the continuous potentials of atomic strings and planes can take place under passage of fast charged particles through crystals. The energy levels for electron moving in axial channeling regime in a system of parallel [110] atomic strings of a silic...
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Shul'ga, N.F. Syshchenko, V.V. Neryabova, V.S. 2015-04-18T14:51:23Z 2015-04-18T14:51:23Z 2014 Spectral method in the quantum theory of axial channeling / N.F. Shul'ga, V.V. Syshchenko, V.S. Neryabova // Вопросы атомной науки и техники. — 2014. — № 5. — С. 111-114. — Бібліогр.: 7 назв. — анг. 1562-6016 PACS: 02.60.Cb, 03.65.Ge, 05.45.Mt, 05.45.Pq, 61.85.+p https://nasplib.isofts.kiev.ua/handle/123456789/80493 The quantization of the transverse motion energy in the continuous potentials of atomic strings and planes can take place under passage of fast charged particles through crystals. The energy levels for electron moving in axial channeling regime in a system of parallel [110] atomic strings of a silicon crystal are found for the electron energy of order of several hundreds of MeV, when a total number of energy levels becomes large. High resolution of the spectral method permits its usage for investigation of quantum chaos problem. При прохождении быстрых заряженных частиц через кристалл может иметь место квантование энергии поперечного движения в непрерывных потенциалах атомных цепочек и плоскостей. Найдены уровни энергии электрона, движущегося в режиме аксиального каналирования в системе параллельных цепочек [110] кристалла кремния, для случая электрона с энергией порядка нескольких сотен МэВ, когда число уровней энергии становится велико. Высокая разрешающая способность метода позволяет использовать его для исследования проблемы квантового хаоса. При проходженнi швидких заряджених часток крiзь кристал може бути наявним квантування енергiї поперечного руху в неперервних потенцiалах атомних ланцюжкiв та площин. Виявлено рiвнi енергiї електрона, що рухається у режимi аксiального каналювання у системi паралельних ланцюжкiв [110] кристала кремнiю, для випадка електрона з енергiєю близько кiлькох сотень МеВ, коли число рiвнiв енергiї стає великим. Висока роздiльна здатнiсть методу дозволяє використовувати його для дослiдження проблеми квантового хаосу. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика Spectral method in the quantum theory of axial channeling Спектральный метод в квантовой теории аксиального каналирования Спектральний метод в квантовiй теорii аксиального каналювання Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Spectral method in the quantum theory of axial channeling |
| spellingShingle |
Spectral method in the quantum theory of axial channeling Shul'ga, N.F. Syshchenko, V.V. Neryabova, V.S. Электродинамика |
| title_short |
Spectral method in the quantum theory of axial channeling |
| title_full |
Spectral method in the quantum theory of axial channeling |
| title_fullStr |
Spectral method in the quantum theory of axial channeling |
| title_full_unstemmed |
Spectral method in the quantum theory of axial channeling |
| title_sort |
spectral method in the quantum theory of axial channeling |
| author |
Shul'ga, N.F. Syshchenko, V.V. Neryabova, V.S. |
| author_facet |
Shul'ga, N.F. Syshchenko, V.V. Neryabova, V.S. |
| topic |
Электродинамика |
| topic_facet |
Электродинамика |
| publishDate |
2014 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Спектральный метод в квантовой теории аксиального каналирования Спектральний метод в квантовiй теорii аксиального каналювання |
| description |
The quantization of the transverse motion energy in the continuous potentials of atomic strings and planes can take place under passage of fast charged particles through crystals. The energy levels for electron moving in axial channeling regime in a system of parallel [110] atomic strings of a silicon crystal are found for the electron energy of order of several hundreds of MeV, when a total number of energy levels becomes large. High resolution of the spectral method permits its usage for investigation of quantum chaos problem.
При прохождении быстрых заряженных частиц через кристалл может иметь место квантование энергии поперечного движения в непрерывных потенциалах атомных цепочек и плоскостей. Найдены уровни энергии электрона, движущегося в режиме аксиального каналирования в системе параллельных цепочек [110] кристалла кремния, для случая электрона с энергией порядка нескольких сотен МэВ, когда число уровней энергии становится велико. Высокая разрешающая способность метода позволяет использовать его для исследования проблемы квантового хаоса.
При проходженнi швидких заряджених часток крiзь кристал може бути наявним квантування енергiї поперечного руху в неперервних потенцiалах атомних ланцюжкiв та площин. Виявлено рiвнi енергiї електрона, що рухається у режимi аксiального каналювання у системi паралельних ланцюжкiв [110] кристала кремнiю, для випадка електрона з енергiєю близько кiлькох сотень МеВ, коли число рiвнiв енергiї стає великим. Висока роздiльна здатнiсть методу дозволяє використовувати його для дослiдження проблеми квантового хаосу.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80493 |
| citation_txt |
Spectral method in the quantum theory of axial channeling / N.F. Shul'ga, V.V. Syshchenko, V.S. Neryabova // Вопросы атомной науки и техники. — 2014. — № 5. — С. 111-114. — Бібліогр.: 7 назв. — анг. |
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| fulltext |
ELECTRODYNAMICS
SPECTRAL METHOD IN THE QUANTUM THEORY OF
AXIAL CHANNELING
N.F.Shul’ga1, V.V.Syshchenko2∗, V.S.Neryabova2
1National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine;
2Belgorod State University, 308015, Belgorod, Russian Federation
(Received April 6, 2013)
The quantization of the transverse motion energy in the continuous potentials of atomic strings and planes can
take place under passage of fast charged particles through crystals. The energy levels for electron moving in axial
channeling regime in a system of parallel [110] atomic strings of a silicon crystal are found for the electron energy
of order of several hundreds of MeV, when a total number of energy levels becomes large. High resolution of the
spectral method permits its usage for investigation of quantum chaos problem.
PACS: 02.60.Cb, 03.65.Ge, 05.45.Mt, 05.45.Pq, 61.85.+p
1. INTRODUCTION
The motion of a fast charged particle in a crystal near
one of crystallographic axes or planes is determined
mainly by the uniform potential that is the potential
of the crystal lattice averaged along the axis or plane
near which the motion takes the place. The longitudi-
nal component of the particle’s momentum p∥ parallel
to the crystallographic axis or plane is conserved in
such field. So, the problem on the particle’s motion in
the crystal is reduced to the two-dimensional problem
of its motion in the transverse plane. The finite mo-
tion in the potential wells formed by the uniform po-
tentials of the atomic axes and planes is known as the
axial or planar channeling, respectively (see [1, 2] and
references therein). The quantum effects can mani-
fest themselves during such motion. Particularly, the
quantization of the transverse motion energy can take
the place. The potential wells formed by the uni-
form string or plane potentials have the complicated
shape that does not permit the analytical integration
of the Schrödinger equation and makes necessary the
development of numerical methods for searching the
transverse motion energy levels and other quantum
characteristics of the particle’s motion in the uni-
form potentials of the atomic planes and strings of
the crystal.
In the present article we apply the so-called spec-
tral method [3] to the search of energy eigenvalues
in the two-dimensional case of axial channeling (for
[110] strings of the silicon crystal as an example).
This method had been successfully used earlier in the
one-dimensional problem concerning planar channel-
ing [4].
2. SPECTRAL METHOD
The spectral method of searching the energy eigen-
values of the quantum system [3] is based on the
computation of correlation function for the time de-
pendent wave functions of the system at the initial
and current time momenta, Ψ(x, y, 0) and Ψ(x, y, t):
P (t) =
∫ ∞
−∞
∫ ∞
−∞
Ψ∗(x, y, 0)Ψ(x, y, t) dxdy. (1)
Fourier transform of this correlation function,
PE =
∫ ∞
−∞
P (t) exp(iEt/h̄) dt, (2)
contains information about the energy eigenval-
ues. Indeed, every solution of the time-dependent
Schrödinger equation
ĤΨ(x, y, t) = ih̄
∂
∂t
Ψ(x, y, t) (3)
could be expressed as the superposition
Ψ(x, y, t) =
∑
n,j
An,jun,j(x, y) exp(−iEnt/h̄) (4)
of the Hamiltonian’s eigenfunctions un,j(x, y),
Ĥun,j(x, y) = Enun,j(x, y),
where the index j is used to distinguish the degen-
erate states corresponding to the energy En. Com-
putation of the correlation function (1) for the wave
function of the form (4) gives
P (t) =
∑
n,n′,j,j′
exp(−iEn′t/h̄)A∗
n,jAn′,j′
×
∫ ∞
−∞
∫ ∞
−∞
u∗n,j(x, y)un′,j′(x, y)dxdy =
∗Corresponding author E-mail address: syshch@yandex.ru
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.111-114.
111
=
∑
n,n′,j,j′
exp(−iEn′t/h̄)A∗
n,jAn′,j′δnn′δjj′ =
=
∑
n,j
|An,j |2 exp(−iEnt/h̄). (5)
Fourier transformation of (5) leads to the expression
PE = 2πh̄
∑
n,j
|An,j |2 δ(E − En). (6)
We see that the Fourier transformation of the cor-
relation function looks like a series of δ-form peaks,
positions of which indicate the energy eigenvalues.
So, the computation of the energy levels for the
given system consists of the following steps:
1. Choosing the arbitrary initial wave function
Ψ(x, y, 0). The only conditions of the choice
are:
— tendency to zero under x, y → ±∞, neces-
sary for every bound state;
— wide spectrum that covers the depth of the
potential well;
— absence of any symmetry which could lead
to the lack of some eigenfunctions in the super-
position (4).
Asymmetric Gaussian waveform would be a
good choice for the most cases.
2. Numerical integration of the time-dependent
Schrödinger equation (3) with the initial value
Ψ(x, y, 0) for the discrete series of the time mo-
menta; the value of the time step ∆t as well
as other computational details are discussed in
[3, 4, 5, 6].
3. Computation of the integral (1) for every dis-
crete time momentum from t = 0 to some max-
imal t = T . Subsequent integration of the ob-
tained correlation function P (t) with the expo-
nent in (2) is carried out over the finite time
interval:
PE =
∫ T
0
P (t) exp(iEt/h̄) dt. (7)
As a result, we obtain a series of peaks of fi-
nite width (inverse proportional to T ) instead
of infinitely narrow δ-like peaks (6).
3. STATISTICAL PROPERTIES OF THE
ENERGY LEVELS
The motion of the fast charged particle in a crys-
tal under small angle ψ to the crystallographic axis
densely packed with atoms could be (with good ac-
curacy) described as a motion in the uniform string
potential (e.g. the potential of the atomic string av-
eraged along its axis) [1, 2]. The longitudinal (e.g.
parallel to the string axis) component of the particle’s
momentum p∥ is conserved in such a field. The mo-
tion in the transverse plane will be described in this
case by the two-dimensional analog of Schrödinger
equation [1, Ch. 7, §53]{
− h̄2
2E∥/c2
(
∂2
∂x2
+
∂2
∂y2
)
+ U(x, y)
}
Ψ(x, y, t) =
= ih̄
∂
∂t
Ψ(x, y, t), (8)
in which the value E∥/c
2 plays the role of the parti-
cle’s mass (where E∥ =
√
m2c4 + p2∥c
2 ).
The uniform string potential could be approxi-
mated by the formula [1, Ch. 6, §41]
U1(x, y) = −U0 ln
(
1 +
βR2
x2 + y2 + αR2
)
, (9)
where for the [110] string of silicon U0 = 60.0 eV,
α = 0.37, β = 3.5, R = 0.194 Å (Thomas-Fermi ra-
dius); the least distance between two parallel strings
is a/4 = 5.431/4 Å (where a is the lattice period). So,
the uniform potential, in which the electron’s trans-
verse motion takes the place, will be described by the
two-well function (Fig. 1)
U(x, y) = U1(x, y + a/8) + U1(x, y − a/8) (10)
(neglecting the influence of far-away strings). The
finite motion of the electron in such potential (cor-
responding to negative values of the transverse mo-
tion energy E⊥) is known as axial channeling [1].
−2
−1
0
1
2
−2 −1 0 1 2
−140
−120
−100
−80
−60
−40
−20
0
y , Angstrom
x , Angstrom
U
(x
, y
)
,
eV
Fig.1. Potential energy (10) of the electron in the
field of uniform potentials of two neighboring atomic
strings [110] of a silicon crystal
Classical motion of the particle in the axial sym-
metric potential (9) of the single string is regular
because of the presence of two integrals of motion:
transverse energy and angular momentum. The clas-
sical motion in the potential (10) for the particle
with the transverse energy above the potential saddle
point is chaotic [1].
The studying of quantum chaos means investiga-
tion of the behavior of quantum systems, which clas-
sical analogs allow the dynamical chaos [7]. One of
the signatures of the quantum chaos phenomenon are
112
the statistical properties of the large massive of the
energy levels of the system in the semiclassical do-
main, where the levels lie densely to each other.
As it stated in [7], for regular motion the differ-
ent energy levels do not interact with each other, that
leads to the random distribution of the distances s be-
tween neighboring energy levels with the exponential
distribution function
p(s) =
1
D
exp(−s/D), (11)
where D is the average distance between neighboring
energy levels on the spectral interval under consider-
ation. For the chaotic motion the phenomenon of re-
pulsion of levels takes the place, that in typical cases
leads to Wigner distribution
p(s) =
πs
2D2
exp(−πs2/4D2). (12)
Such behavior has been observed for the particle in
billiard with mirror walls and for excited states of
some nuclei.
The distributions of inter-level distances for single
(9) and double (10) string potentials are presented on
Fig.2. We see that the behavior of levels corresponds
on a certain extent to the predictions (11) and (12).
0 0.05 0.1 0.15 0.2 0.25
0
5
10
15
20
25
30
35
40
45
50
s , eV
N
um
be
r
of
le
ve
ls
0 0.05 0.1 0.15
0
10
20
30
40
50
60
s , eV
N
um
be
r
of
le
ve
ls
Fig.2. Distributions of the distances between
neighboring levels in the single (upper plot) and
double (lower plot) potential wells on the interval
−11 ≤ E⊥ ≤ 2.5 eV for the electron’s longitudinal
energy E∥ = 200 MeV, in comparison with the
exponential (11) (dashed line) and Wigner (12)
(solid line) distribution functions. The resolution of
levels is not worse than s = 0.02 eV
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
5
10
15
20
25
30
35
40
45
50
s , eV
N
um
be
r
of
le
ve
ls
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0
5
10
15
20
25
30
35
40
45
50
s , eV
N
um
be
r
of
le
ve
ls
Fig.3. The same as on Fig.2 for even states
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0
5
10
15
20
25
30
35
40
45
50
s , eV
N
um
be
r
of
le
ve
ls
0 0.05 0.1 0.15
0
5
10
15
20
25
30
35
40
45
50
s , eV
N
um
be
r
of
le
ve
ls
Fig.4. The same as on Fig.2 for odd states
However, it is emphasized in [7] that only the lev-
els corresponding the eigenstates with the same sym-
metry could interact with each other. The distribu-
113
tions of inter-level distances for the states with pos-
itive and negative parity are plotted on Figs. 3 and
4. We see that the obtained distributions are rather
far from the predictions (11) and (12).
6. CONCLUSIONS
The distributions of inter-level distances between lev-
els of the transverse motion energy for the axial chan-
neling electron in the uniform potential of the single
[110] atomic string of silicon crystal as well as the po-
tential of two neighboring strings are calculated using
the spectral method of the energy levels searching.
The results demonstrate that statistical properties of
the massive of energy levels in this case do not agree
to the predictions of [7] for regular and chaotic mo-
tion.
This work is supported in part by internal grant of
Belgorod State University, and the government con-
tract 2.2694.2011 dated 18.01.2012.
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ÑÏÅÊÒÐÀËÜÍÛÉ ÌÅÒÎÄ Â ÊÂÀÍÒÎÂÎÉ ÒÅÎÐÈÈ ÀÊÑÈÀËÜÍÎÃÎ
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Ïðè ïðîõîæäåíèè áûñòðûõ çàðÿæåííûõ ÷àñòèö ÷åðåç êðèñòàëë ìîæåò èìåòü ìåñòî êâàíòîâàíèå ýíåð-
ãèè ïîïåðå÷íîãî äâèæåíèÿ â íåïðåðûâíûõ ïîòåíöèàëàõ àòîìíûõ öåïî÷åê è ïëîñêîñòåé. Íàéäåíû óðîâ-
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öåïî÷åê [110] êðèñòàëëà êðåìíèÿ, äëÿ ñëó÷àÿ ýëåêòðîíà ñ ýíåðãèåé ïîðÿäêà íåñêîëüêèõ ñîòåí ÌýÂ,
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èñïîëüçîâàòü åãî äëÿ èññëåäîâàíèÿ ïðîáëåìû êâàíòîâîãî õàîñà.
ÑÏÅÊÒÐÀËÜÍÈÉ ÌÅÒÎÄ Â ÊÂÀÍÒÎÂIÉ ÒÅÎÐI�I ÀÊÑÈÀËÜÍÎÃÎ
ÊÀÍÀËÞÂÀÍÍß
Ì.Ô.Øóëüãà, Â.Â.Ñèùåíêî, Â.Ñ.Íåðÿáîâà
Ïðè ïðîõîäæåííi øâèäêèõ çàðÿäæåíèõ ÷àñòîê êðiçü êðèñòàë ìîæå áóòè íàÿâíèì êâàíòóâàííÿ åíåðãi¨
ïîïåðå÷íîãî ðóõó â íåïåðåðâíèõ ïîòåíöiàëàõ àòîìíèõ ëàíöþæêiâ òà ïëîùèí. Âèÿâëåíî ðiâíi åíåðãi¨
åëåêòðîíà, ùî ðóõà¹òüñÿ ó ðåæèìi àêñiàëüíîãî êàíàëþâàííÿ ó ñèñòåìi ïàðàëåëüíèõ ëàíöþæêiâ [110]
êðèñòàëà êðåìíiþ, äëÿ âèïàäêà åëåêòðîíà ç åíåðãi¹þ áëèçüêî êiëüêîõ ñîòåíü ÌåÂ, êîëè ÷èñëî ðiâíiâ
åíåðãi¨ ñò๠âåëèêèì. Âèñîêà ðîçäiëüíà çäàòíiñòü ìåòîäó äîçâîëÿ¹ âèêîðèñòîâóâàòè éîãî äëÿ äîñëiä-
æåííÿ ïðîáëåìè êâàíòîâîãî õàîñó.
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