Aharonov-Bohm effect in scattering of high-energy particles
Quantum-mechanical scattering of coherent high-energy charged particles by a magnetic vortex is considered. The vortex core is assumed to be impermeable to scattered particles, and effects of its transverse size are taken into account. The limit of high energies of scattered particles corresponds to...
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| Cite this: | Aharonov-Bohm effect in scattering of high-energy particles / Yu.A. Sitenko, N.D. Vlasii // Вопросы атомной науки и техники. — 2012. — № 1. — С. 121-124. — Бібліогр.: 9 назв. — англ. |
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| citation_txt | Aharonov-Bohm effect in scattering of high-energy particles / Yu.A. Sitenko, N.D. Vlasii // Вопросы атомной науки и техники. — 2012. — № 1. — С. 121-124. — Бібліогр.: 9 назв. — англ. |
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| description | Quantum-mechanical scattering of coherent high-energy charged particles by a magnetic vortex is considered. The vortex core is assumed to be impermeable to scattered particles, and effects of its transverse size are taken into account. The limit of high energies of scattered particles corresponds to the quasiclassical limit, and we show that in the scattering the Aharonov-Bohm effect persists in this limit owing to the Fraunhofer diffraction in the forward direction. The issue of the experimental detection of the Fraunhofer diffraction peak and the Aharonov-Bohm effect in the quasiclassical limit is discussed.
Рассматривается квантово-механическое рассеяние высокоэнергетических заряженных частиц магнитным вихрем. Ядро вихря предполагается непроницаемым для рассеиваемых частиц, и учитываются эффекты его поперечных размеров. Предел высоких энергий рассеиваемых частиц соответствует квазиклассическому пределу, и мы показываем, что эффект Ааронова-Бома в рассеянии в этом пределе выживает благодаря дифракции Фраунгофера в направлении вперед. Обсуждаются вопросы экспериментального детектирования пика Фраунгоферовой дифракции и эффекта Ааронова-Бома в квазиклассическом пределе.
Розглядається квантово-механічне розсіяння високоенергетичних заряджених частинок магнітним вихором. Припускається, що ядро вихора є непроникним для розсіюваних частинок, та враховуються ефекти його поперечних розмірів. Границя високих енергій розсіюваних частинок відповідає квазікласичній границі, і ми показуємо, що ефект Ааронова-Бома в розсіянні в цій границі виживає завдяки дифракції Фраунгофера в напрямку вперед. Обговорюються питання експериментального детектування піка Фраунгоферової дифракції та ефекта Ааронова-Бома в квазікласичній границі.
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AHARONOV-BOHM EFFECT IN SCATTERING OF
HIGH-ENERGY PARTICLES
Yu.A. Sitenko1∗and N.D. Vlasii1,2
1Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 03680, Kyiv, Ukraine
2Physics Department, Taras Shevchenko National University of Kyiv, 01601, Kyiv, Ukraine
(Received November 2, 2011)
Quantum-mechanical scattering of coherent high-energy charged particles by a magnetic vortex is considered. The
vortex core is assumed to be impermeable to scattered particles, and effects of its transverse size are taken into
account. The limit of high energies of scattered particles corresponds to the quasiclassical limit, and we show that
in the scattering the Aharonov-Bohm effect persists in this limit owing to the Fraunhofer diffraction in the forward
direction. The issue of the experimental detection of the Fraunhofer diffraction peak and the Aharonov-Bohm effect
in the quasiclassical limit is discussed.
PACS: 03.65.Nk, 03.65.Ta, 41.75.Fr, 75.70.Kw
1. INTRODUCTION
Starting from its discovery in 1959, the Aharonov-
Bohm effect [1] is considered in the framework of two
somewhat different, although closely related, setups.
The first one concerns the fringe shift in the interfer-
ence pattern due to two coherent particle beams un-
der the influence of the impermeable magnetic vortex
placed between the beams. The second one deals with
scattering of a particle beam directly on an imperme-
able magnetic vortex. All experiments are performed
in the first setup, though the second setup is more
elaborate from the theoretical point of view. The
reason of this somewhat a paradoxical situation lies
in the simple fact that the efforts in the elaboration of
scattering theory for the Aharonov-Bohm effect were
mostly concentrated on the case of long-wavelength
scattered particles, when the transverse size of the
impermeable magnetic vortex was neglected. Since a
direct scattering experiment is hard to perform with
long-wavelength (slow-moving) particles, thus elabo-
rated theory remained actually unverified. The aim
of the present paper is to extend the theory to the
case of short-wavelength (fast-moving) particles and
to reach the realm where the experimental verifica-
tion is quite feasible (see also [2–4]).
2. DOUBLE-SLIT INTERFERENCE
EXPERIMENT
But, first, let us recall briefly the setup which
is conventionally used for the verification of the
Aharonov-Bohm effect (see, e.g., [5, 6]). It involves
the observation of the interference patterns result-
ing from the two coherent electron beams bypassing
from different sides an impermeable magnetic vor-
tex which is orthogonal to the plane defined by the
beams. This is a so called double-slit interference ex-
periment, although in reality an electrostatic biprism
is used to bend the beams and to direct them on the
detection screen. Let the detection screen be parallel
to the screen with slits, L be the distance between
the screens, and d be the distance between the slits;
otherwise, in the biprism setting, the line connecting
images of a source is parallel to the detection screen,
L is the distance between the line and the screen,
and d is the distance between the images. The in-
terference pattern on the detection screen consists of
equally spaced fringes which are in the same direction
as the magnetic vortex,
I(y) = 4I0(y) cos2
[(
yd
λL
+
Φ
Φ0
)
π
]
, (1)
where y is the coordinate which is orthogonal to the
fringes on the detection screen (y = 0 corresponds
to the point which is symmetric with respect to the
slits), I0(y) is the intensity in the case when either of
the slits is closed, λ is the electron wavelength, Φ is
the flux of the impermeable magnetic vortex placed
just after the screen with slits (otherwise, after the
biprism), Φ0 = hce−1 is the London flux quantum.
Intensity I(y) (1) is oscillating with period
Δy = λLd−1 (2)
and the enveloping function given by 4I0(y) which is
a Gaussian centred at y = 0. At the centre of the
detection screen one gets
I(0) = 4I0(0) cos2
(
Φ
Φ0
π
)
. (3)
If L� d and λ, then one can use dimensionless (an-
gular) variable ϕ = y/L. The period of oscillations
∗Corresponding author E-mail address: yusitenko@bitp.kiev.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 121-124.
121
in this variable is
Δϕ = λd−1. (4)
Evidently, the period of oscillation decreases with
the decrease of wavelength λ. The distance resolu-
tion of the detector should be at least as high as 1
2Δy,
that is why the observation of the interference pattern
becomes more complicated in the short-wavelength
limit. Since the enveloping function takes a form of a
narrow peak in this limit, it is appropriate to define
the visibility of the central point
V =
∣∣I(0) − I
(± 1
2Δy
)∣∣
I(0) + I
(± 1
2Δy
) . (5)
Assuming I0
(− 1
2Δy
)
= I0
(
1
2Δy
)
, one finds immedi-
ately
V = (6)
=
∣∣∣I0(0)−I0
(
1
2Δy
)
+
[
I0(0)+I0
(
1
2Δy
)]
cos
(
2 Φ
Φ0
π
)∣∣∣
I0(0)+I0
(
1
2Δy
)
+
[
I0(0)−I0
(
1
2Δy
)]
cos
(
2 Φ
Φ0
π
) .
3. DIRECT SCATTERING EXPERIMENT
We start with the Schrödinger equation for the
wave function describing the stationary scattering
state
Hψ(r, ϕ) =
h̄2k2
2m
ψ(r, ϕ), (7)
where m is the particle mass and k is the absolute
value of the particle wave vector (k = 2π/λ); the im-
permeable magnetic vortex is assumed to be directed
orthogonally to the plane with polar coordinates r
and ϕ, and we confine ourselves to the particle mo-
tion in this plane, since the motion along the vortex
is free. Out of the vortex core the Schrödinger hamil-
tinian takes form
H = − h̄2
2m
[
r−1∂rr∂r + r−2
(
∂ϕ − iΦΦ−1
0
)2
]
, (8)
and we impose condition
lim
r→∞ eikrψ(r, ±π) = 1, (9)
signifying that the incident wave comes from the far
left; the forward direction is ϕ = 0, and the backward
direction is ϕ = ±π.
Without a loss of generality we assume that the
vortex has a shape of cylinder of radius rc and impose
the Robin boundary condition on the wave function:
{[cos(ρπ) + rc sin(ρπ)∂r ]ψ(r, ϕ)}|r=rc
= 0. (10)
The solution to (7) with Hamiltonian (8), which sat-
isfies (9) and (10), takes the following form
ψ(r, ϕ) =
∑
n∈Z
einϕei(|n|−
1
2 |n−μ|π)
[
J|n−μ|(kr)
−Υ(ρ)
|n−μ|(krc)H
(1)
|n−μ|(kr)
]
, (11)
where Z is the set of integer numbers, μ = ΦΦ−1
0 ,
Jα(u) and H(1)
α (u) are the Bessel and first-kind Han-
kel functions of order α, and
Υ(ρ)
α (u) =
Jα(u) + tan(ρπ)u∂uJα(u)
H
(1)
α + tan(ρπ)u∂uH
(1)
α (u)
. (12)
Thus, wave function (11) consists of two parts: the
one which will be denoted by ψ0(r, ϕ) is independent
of rc, and the other one which will be denoted by
ψc(r, ϕ) is dependent on rc.
Taking the asymptotics of the first part at large
distances from the vortex, one can get (see [7])
ψ0(r, ϕ) = eikr cos ϕ+iμϕ
{
cos(μπ) − isgn(ϕ) sin(μπ)
×
[
1 − ei( 1
2+ν−μ)ϕerfc
(
e−iπ/4
√
2kr
∣∣∣sin ϕ
2
∣∣∣)]}
,
(13)
where it is implied that −π < ϕ < π, erfc(z) =
2√
π
∞∫
z
du e−u2
is the complementary error function,
ν = [[μ]] is the integer part of μ, and sgn(u) ={
1, u > 0
−1, u < 0
}
. In the case
√
kr
∣∣sin ϕ
2
∣∣ � 1, one
gets
ψ0(r, ϕ) = eikr cos ϕeiμ[ϕ−sgn(ϕ)π]+f0(k, ϕ)
ei(kr+π/4)
√
r
,
(14)
where
f0(k, ϕ) = i
ei(ν+ 1
2 )ϕ
√
2πk
sin(μπ)
sin(ϕ/2)
(15)
is the scattering amplitude which was first obtained
by Aharonov and Bohm [1]. In the case kr � 1 but√
kr
∣∣sin ϕ
2
∣∣ � 1, one gets
ψ0(r, ϕ) = eikr cos(μπ). (16)
Taking the large-distance asymptotocs of the rc-
dependent part of the wave function, one gets
ψc(r, ϕ) = fc(k, ϕ)
ei(kr+π/4)
√
r
, (17)
where
fc(k, ϕ) = i
√
2
πk
∑
n∈Z
einϕei(|n|−|n−μ|π)Υ(ρ)
|n−μ|(krc).
(18)
In the low-energy (long-wavelength) limit amplitude
fc (18) is suppressed by powers of krc as compared to
amplitude f0 (15); however this limit is not feasible
to experimental measurements. In the higt-energy
(short-wavelength) limit, amplitude f0 (15) is sup-
pressed and wave function ψ0 (13) is actually reduced
to an incident wave, eikr cos ϕ, which is distorted by
the flux-dependent factors, see first term in (14) and
(16). Amplitude fc (18) in this limit is prevailing, and
we obtain the corresponding differential cross section,
see [2],
dσ
dϕ
= |fc(k, ϕ)|2
= 4rcΔ 1
2krc
(ϕ) cos2
(
1
2
krcϕ+ μπ
)
+
rc
2
∣∣∣sin ϕ
2
∣∣∣ , (19)
122
where
Δx(ϕ) =
1
4πx
sin2(xϕ)
sin2(ϕ/2)
(−π < ϕ < π) (20)
is a strongly peaked at ϕ = 0 and x � 1 function
which can be regarded as a regularization of the an-
gular delta-function,
lim
x→∞Δx(ϕ) =
1
2π
∑
n∈Z
einϕ, Δx(0) =
x
π
. (21)
Thus, the first term on the right-hard side of (19)
describes the forward peak of the Fraunhofer diffrac-
tion on the vortex, while the second term describes
the classical reflection from the vortex. Using nota-
tions of the vortex diameter, d = 2rc, and the particle
wavelength, λ = 2π/k, we rewrite (19) in a form sim-
ilar to (1):
dσ
dϕ
= 2dΔ dπ
2λ
(ϕ) cos2
[(
ϕd
2λ
+
Φ
Φ0
)
π
]
+
d
4
∣∣∣sin ϕ
2
∣∣∣ ;
(22)
the differential cross section of the Fraunhofer diffrac-
tion is oscillating with period, cf. (4),
Δϕ = 2λd−1. (23)
In the strictly forward direction one gets, cf. (3),
dσ
dϕ
∣∣∣∣
ϕ=0
=
d2
λ
cos2
(
Φ
Φ0
π
)
. (24)
Defining the visibility of the central point in the dif-
ferential cross section as
V =
|dσ|ϕ=0 − dσ|ϕ=± 1
2Δϕ|
dσ|ϕ=0 + dσ|ϕ=± 1
2Δϕ
, (25)
we get
V =
∣∣∣1 − 4
π2 +
(
1 + 4
π2
)
cos
(
2 Φ
Φ0
π
)∣∣∣
1 + 4
π2 +
(
1 − 4
π2
)
cos
(
2 Φ
Φ0
π
) . (26)
The maximal visibility (V = 1) is attained for the flux
which is quantized in the units of the Abrikosov vor-
tex flux
(
Φ = 1
2nΦ0
)
; the minimal visibility (V = 0)
is attained at
Φ=
{
n± 1
4
± 1
2π
arcsin
[(
1− 4
π2
) (
1+
4
π2
)−1
]}
Φ0.
4. SUMMARY AND DISCUSSION
We have shown that the fringe shift emerging un-
der the influence of a magnetic vortex in the diffrac-
tion pattern in the forward direction in a direct scat-
tering experiment with high-energy particles is com-
pletely analogous to that in the interference pattern
in a double-slit experiment. It should be emphasized
that permeability of the magnetic vortex does not af-
fect the diffraction pattern (first term in (22)), only
the classical reflection (second term in (22)) is af-
fected. The latter circumstance facilitates the obser-
vation of the scattering Aharonov-Bohm effect in the
quasiclassical limit.
Certainly, the Fraunhofer diffraction (i.e. the dif-
fraction in almost parallel rays) is a well-known phe-
nomenon of wave optics. Poisson was the first who
predicted theoretically in the early nineteenth cen-
tury the emergence of a spot of brightness in the cen-
tre of a shadow of an opaque disc; the prediction was
in a contradiction with the laws of geometric (ray)
optics. It is curious that Poisson used his prediction
as an argument to disprove wave optics: this demon-
strates how unexpected and unbelievable was Pois-
son’s result at that time. Nevertheless, the bright-
ness spot in the centre of the disc shadow was soon
observed; the decisive experiments were performed
by Arago and Fresnel. The diffraction on the opaque
disc bears the name of Poisson, and the brightness
spot in the shadow centre bears the name of Arago,
see [8]. The same effect persists for scattering of light
on an opaque ball and other obstacles. However, in
the case of obstacles in the form of a long strip or
cylinder, the streak of brightness in the centre of a
shadow of such obstacles is elusive to experimental
measurements: as is noted in the well-known trea-
tise [9], it seems more likely that the measurable
quantity is the classical cross section, although the
details of this phenomenon depend on the method of
measurement. Almost six decades have passed from
the time when this assertion was made by Morse and
Feshbach, and experimental facilities have improved
enormously since then. It is a challenge for exper-
imentalists to reconsider the Fraunhofer diffraction
on the cylindrical obstacles. In the present paper
we draw attention to this long-standing problem by
pointing at the circumstances when the detection of
the forward diffraction peak will be the detection of
the Aharonov-Bohm effect persisting in the quasiclas-
sical limit.
Acknowledgements
The work was supported in part by the Ukrainian-
Russian SFFR-RFBR project F40.2/108 “Applica-
tion of string theory and field theory methods to non-
linear phenomena in low-dimensional systems”.
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3. Yu.A. Sitenko, N.D. Vlasii. Scattering theory
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123
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|
| id | nasplib_isofts_kiev_ua-123456789-107009 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:46:32Z |
| publishDate | 2012 |
| publisher | Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine |
| record_format | dspace |
| spelling | Sitenko, Yu.A. Vlasii, N.D. 2016-10-10T20:34:58Z 2016-10-10T20:34:58Z 2012 Aharonov-Bohm effect in scattering of high-energy particles / Yu.A. Sitenko, N.D. Vlasii // Вопросы атомной науки и техники. — 2012. — № 1. — С. 121-124. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 03.65.Nk, 03.65.Ta, 41.75.Fr, 75.70.Kw https://nasplib.isofts.kiev.ua/handle/123456789/107009 Quantum-mechanical scattering of coherent high-energy charged particles by a magnetic vortex is considered. The vortex core is assumed to be impermeable to scattered particles, and effects of its transverse size are taken into account. The limit of high energies of scattered particles corresponds to the quasiclassical limit, and we show that in the scattering the Aharonov-Bohm effect persists in this limit owing to the Fraunhofer diffraction in the forward direction. The issue of the experimental detection of the Fraunhofer diffraction peak and the Aharonov-Bohm effect in the quasiclassical limit is discussed. Рассматривается квантово-механическое рассеяние высокоэнергетических заряженных частиц магнитным вихрем. Ядро вихря предполагается непроницаемым для рассеиваемых частиц, и учитываются эффекты его поперечных размеров. Предел высоких энергий рассеиваемых частиц соответствует квазиклассическому пределу, и мы показываем, что эффект Ааронова-Бома в рассеянии в этом пределе выживает благодаря дифракции Фраунгофера в направлении вперед. Обсуждаются вопросы экспериментального детектирования пика Фраунгоферовой дифракции и эффекта Ааронова-Бома в квазиклассическом пределе. Розглядається квантово-механічне розсіяння високоенергетичних заряджених частинок магнітним вихором. Припускається, що ядро вихора є непроникним для розсіюваних частинок, та враховуються ефекти його поперечних розмірів. Границя високих енергій розсіюваних частинок відповідає квазікласичній границі, і ми показуємо, що ефект Ааронова-Бома в розсіянні в цій границі виживає завдяки дифракції Фраунгофера в напрямку вперед. Обговорюються питання експериментального детектування піка Фраунгоферової дифракції та ефекта Ааронова-Бома в квазікласичній границі. The work was supported in part by the UkrainianRussian
 SFFR-RFBR project F40.2/108 “Application
 of string theory and field theory methods to nonlinear
 phenomena in low-dimensional systems”. en Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine Вопросы атомной науки и техники Section B. QED Processes at High Energies Aharonov-Bohm effect in scattering of high-energy particles Эффект Ааронова-Бома в рассеянии высокоэнергетических частиц Ефект Ааронова-Бома в розсіянні високоенергетичних частинок Article published earlier |
| spellingShingle | Aharonov-Bohm effect in scattering of high-energy particles Sitenko, Yu.A. Vlasii, N.D. Section B. QED Processes at High Energies |
| title | Aharonov-Bohm effect in scattering of high-energy particles |
| title_alt | Эффект Ааронова-Бома в рассеянии высокоэнергетических частиц Ефект Ааронова-Бома в розсіянні високоенергетичних частинок |
| title_full | Aharonov-Bohm effect in scattering of high-energy particles |
| title_fullStr | Aharonov-Bohm effect in scattering of high-energy particles |
| title_full_unstemmed | Aharonov-Bohm effect in scattering of high-energy particles |
| title_short | Aharonov-Bohm effect in scattering of high-energy particles |
| title_sort | aharonov-bohm effect in scattering of high-energy particles |
| topic | Section B. QED Processes at High Energies |
| topic_facet | Section B. QED Processes at High Energies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/107009 |
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