The effect of a magnetic field on the motion of electrons for the field emission process description
The Schrödinger equation is solved for the wave function of an electron moving in a superposition of external constant and uniform electric and magnetic fields at an arbitrary angle between the field directions. The changing of the potential barrier under influence of the magnetic field parallel to...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2015 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/112233 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The effect of a magnetic field on the motion of electrons for the field emission process description / S.O. Lebedynskyi, V.I. Miroshnichenko, R.I. Kholodov, V.A. Baturin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 62-66. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860194074419527680 |
|---|---|
| author | Lebedynskyi, S.O. Miroshnichenko, V.I. Kholodov, R.I. Baturin, V.A. |
| author_facet | Lebedynskyi, S.O. Miroshnichenko, V.I. Kholodov, R.I. Baturin, V.A. |
| citation_txt | The effect of a magnetic field on the motion of electrons for the field emission process description / S.O. Lebedynskyi, V.I. Miroshnichenko, R.I. Kholodov, V.A. Baturin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 62-66. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The Schrödinger equation is solved for the wave function of an electron moving in a superposition of external constant and uniform electric and magnetic fields at an arbitrary angle between the field directions. The changing of the potential barrier under influence of the magnetic field parallel to the metal surface is shown.
Розв’язується рівняння Шрьодінгера для хвильової функції електрона, що рухається в суперпозиції зовнішніх постійних і однорідних електричному і магнітному полях під довільним кутом між їх напрямками. Показано зміну потенційного бар'єру під впливом магнітного поля, що паралельне поверхні.
Решается уравнение Шредингера для волновой функции электрона, движущегося в суперпозиции внешних постоянных и однородных электрическом и магнитном полях под произвольным углом между их направлениями. Показано изменение потенциального барьера под влиянием магнитного поля, параллельного поверхности
|
| first_indexed | 2025-12-07T18:07:56Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 62
THE EFFECT OF A MAGNETIC FIELD ON THE MOTION
OF ELECTRONS FOR THE FIELD EMISSION PROCESS DESCRIPTION
S.O. Lebedynskyi, V.I. Miroshnichenko, R.I. Kholodov, V.A. Baturin
Institute of Applied Physics NASU, Sumy, Ukraine
E-mail: lebedynskyi.s@gmail.com
The Schrödinger equation is solved for the wave function of an electron moving in a superposition of external
constant and uniform electric and magnetic fields at an arbitrary angle between the field directions. The changing of
the potential barrier under influence of the magnetic field parallel to the metal surface is shown.
PACS: 79.70.+q, 03.65.Ge
INTRODUCTION
The researchers and designers of accelerating struc-
tures for the compact linear accelerator under the CLIC
project were faced with high vacuum RF breakdowns
accompanying the electromagnetic power input that
produces the accelerating field gradients as high as
100 MV/m.
Toward this end, CERN workers have built a DC-
spark facility [1] for use in experiments to elucidate
how the frequency of a high-vacuum breakdown occur-
ring in the gap between the electrodes is related to vari-
ous factors, e.g. material of the electrodes in the accel-
erating gap, electrode surface conditioning procedures,
influence of other circumstances (e.g. external magnetic
field in the gap between the electrodes, etc.).
The influence of the external magnetic field on the
field emission current density was first studied theoreti-
cally by F.J. Blatt [2] and later on in experiments [3 - 9].
The motivation for these works was a goal to determine
the ways of increasing the field emission current, so
they confined themselves to the examination of a con-
figuration of the collinear electric and magnetic fields.
However, the central issue, viz, the determination of
the electron barrier-penetration coefficient at the metal-
vacuum interface was dealt with by the author [2], as he
himself admitted, under the assumption that this coeffi-
cient was independent of the magnetic field in the con-
figuration of the collinear electric and magnetic fields of
interest. Moreover, he advanced neither theoretical
arguments nor experimental evidence to support his
assumption. The present paper is an attempt to make a
first step towards the elucidation of this point, namely,
to describe the quantum-mechanical motion of an elec-
tron in external constant and uniform electric and mag-
netic fields, with the angle between the field directions
being arbitrary.
It is worthy of note that a study of the magnetic field
effect on the field emission is important to perform not
only with the aim to prevent high vacuum breakdowns
occurring in modern colliders, but also to probe a wide
area extending from astrophysics observations of the
electron field emission from magnetized neutron stars
[10 - 12] to investigations into the field emission in
carbon nanotubes [13 - 14].
1. PROBLEM FORMULATION
AND SOLUTION
In our treatment, to describe the electron quantum-
mechanical motion we choose a Cartesian system de-
picted in Fig. 1, with the electric field strength vector
0E
and the magnetic field induction vector 0B
directed
as indicated in the figure.
Fig. 1. Configuration of the electric and magnetic fields
As is seen, the components of the electric field
strength 0E
, of the magnetic induction 0B
, the form of
the electric potential ( )rϕ , the electron potential ener-
gy and the expression for the vector potential can be
written as
( )0 0 0sin , cos , 0 ,E E Eα α− −
( )0 00, ,0 ,B B
( ) ( )0, ,x y E xsin ycosϕ α α= + (1)
( ) ( ) ( )0, , sin cos ,U x y e x y eE x yϕ α α= − = − +
( )00,0, .A B x= −
The input equation for the description of the electron
motion in the fields mentioned above is the Schrödinger
equation for the electron wave function ( ),r tψ :
( ) ( )ˆ,
, ,
r t
i H r t
t
ψ
ψ
∂
=
∂
(2)
where the Hamilton operator is
( )
22 2
2 2
02 2
0
1
2
ˆ
,
H i eB x
m zx y
eE xsin ycosα α
∂ ∂ ∂ = − − + − − ∂∂ ∂
− +
(3)
and m is the electron mass, -e is the electron charge.
As follows from the explicit form of the Hamiltoni-
an operator Ĥ (3), it does not explicitly depend on time
and does not include the z – coordinate in the explicit
form, so it is commutative with the operator of the z-th
component of the momentum we can get the equation
for the wave function component ( ),x yψ that describes
the electron motion in the ( ) ,x y plane
mailto:lebedynskyi.s@gmail.com
ISSN 1562-6016. ВАНТ. 2015. №4(98) 63
( )
( )
2 2
22 2
02 2
0
1
2
sin cos
zp eB x
m x y
eE x y
ψ εψ
α α
∂ ∂
− − + − ∂ ∂ × =
− +
. (4)
The differential operator in Eq. (4) is an algebraic
sum of two operators, each depending on only one vari-
able either x or y
( ) ( ) ( )ˆ, , ˆ ˆ
x yH x y H x H y= + (5)
( ) ( )
2
22
0 02
1 sˆ in ,
2x zH x p eB x eE x
m x
α
∂
= − + − − ∂
(6)
( )
2 2
02
ˆ cos ,
2yH y eE y
m y
α∂
= − −
∂
(7)
( ) ( ){ } ( ), ,ˆ ˆ
x yH x H y x yψ εψ+ = (8)
x yε ε ε+ = , (9)
with the constants xε and yε determine the possible
spectrum of the electron energy related to its movement
along either x- or y-axis.
Considering the additive nature of the Hamiltonian
operator with respect to the dependence on the x, y co-
ordinates, we search for the electron wave function,
( ) ,x yψ in the form of a product of two functions, each
being dependent on only one variable
( ) ( ) ( ),x y X x Y yψ = . (10)
To define the X(x) function the quantum oscillator
differential equation is derived
( )
( ) ( )
2
2
2 0,x
d X
X
d
ξ
ε ξ ξ
ξ
+ − = (11)
where we introduce a dimensionless coordinate ξ in
accordance with the expressions
0
0 0 2
sin
, z
B B
eEpx x x x
m m
α
ω ω
= +′− = ,
1
2
B
a
mω
=
,
1
2
0
2
i
,
s nB z
B B
eEm px a x
m m
αω
ξ ξ
ω ω
= = − −
′
, (12)
0
B
eB
m
ω = is the cyclotron frequency of the electron
rotation in Larmor orbit in the classical case and the
constant xε is represented by the expression
2
0
22
0 sin s1 in2
2x x z
B B B
e E eE
p
m m
α α
ε ε
ω ω ω
= + +
. (13)
Equation (11) is intended for Hermitian functions
[15] and has a finite solution only for a discrete series of
xε values:
( )2 1 , where 0,1,2,x n nε = + = … (14)
The expressions (13) and (14) determine a possible
range of energy xε which is related to the electron mo-
tion in the plane normal to the magnetic induction vec-
tor and can be represented as
2
0 02
,1
2
d
x B
mv
n eE xε ω = + + −
(15)
where 0
0
sind
E
v
B
α= is a drift velocity of the electron
Larmor orbit center in crossed electric and magnetic
fields in the classical description.
The part of the electron wave function that describes
the electron motion in the plane normal to the magnetic
field, can be represented in the explicit form as
( ) ( )
21, ,
22 !
z
nn
ip zx z exp H exp
n
ξψ ξ
π
= −
(16)
where ( )nH ξ is the Hermitian polynomial of n-th or-
der.
Eq. (7) that defines the electron motion along the
magnetic field can be reduced to
2
2 0, d Y Y
d
η
η
− = (17)
where the dimensionless η coordinate is linked to the y
coordinate by the following relation
1
3
0
0
cos
co
2
.
s
yEmeE
y
eE
α
η
α
= −
(18)
The solution of (17) can be reduced to the solution
of the equation for the Bessel function of order 1/3 [15]:
( ) ( ) ( ) ( ) ( )1 2
1 1
3 3
,Y AH BHη η η= + (19)
where ( ) ( )1
1
3
H η and ( ) ( )2
1
3
H η are the Hankel functions of
the 1st- and 2d-order, respectively [15]. The range of
electron energies yε is the characteristic of the electron
motion along the magnetic field; it assumes a continu-
ous series of values.
2. BARRIER-PENETRATION
COEFFICIENT IN METAL IN THE
PRESENCE OF COLLINEAR MAGNETIC
AND ELECTRIC FIELDS
We turn here to the problem of the dependence or
independence of the barrier-penetration coefficient of a
metal electron under electron field emission in the case
of parallel electric and magnetic fields 0 0( || )E B
.
Choosing, as before, the form of the magnetic field
vector potential to be ( )00,0,A eB x= −
we can write the
explicit form of the Hamiltonian operator of an electron
penetrating the potential barrier in metal due to the
external electric field as follows
( ) ( )
( )
22 2
2 2
02 2
2
0
0
1
2
sin cos (20)
16 sin cos
sin cos
ˆ
_
,
H i eB x
m zx y
eC eE x y
x y
x y
α α
πε α α
σ α α
∂ ∂ ∂ = − − + − − ∂∂ ∂
+ − + −
+
× +
where ( )
0 0
1 0
α
σ α
α
<
= ≥
.
Eq. (18) implies the same coordinate system as that
shown in Fig. 1 and includes the following symbols: C
is the potential barrier height; ( )0 sin cos eE x yα α− + is
ISSN 1562-6016. ВАНТ. 2015. №4(98) 64
the potential energy of an electron present in the exter-
nal electric field of given configuration;
( )2
0/16 sin cose x yπε α α+ is the electron potential
energy related to the interaction of the electron with its
”positive” mirror image.
Examine the form of the Hamiltonian operator in a
special case of the electric and the magnetic fields being
collinear, i.e. at the angle α = 0 (as discussed in [2]). In
this case the Hamiltonian operator has additive nature,
that is, it can be represented as a sum of two operators,
each acting on only one variable, either x or y
( ) ( ) ( )ˆ ,ˆ,ˆ
x yH x y H x H y= + (21)
( ) ( )
2 2
2
02 ,
2
ˆ
x zH x p eB x
m x
∂
= − + − ∂
(22)
( ) ( )
2 2 2
02
0
.
2 16
ˆ
y
eH y C eE y y
m yy
σ
πε
∂
= − + − − ∂
(23)
Here the Schrödinger equation for the electron wave
function ( ),x yψ can be written as
( ) ( ){ } ( ), .ˆ ˆ
x yH x H y x yψ εψ+ = (24)
Considering the additive nature of the differential
operator in (21)-(23), the expression for the component
of the ( ),x yψ wave function is sought for in the multi-
plicative form
( ) ( ) ( ), .x y X x Y yψ = (25)
Substitution of (24) into (25) makes it necessary to
solve two independent differential equations
( ) ( ) ( ) ( ), ˆ , x xH x X x X x x ºε= −∞ +∞ , (26)
( ) ( ) ( ) ( ), ,ˆ ,y yH y Y y Y y y ºε= −∞ +∞ (27)
where the quantities xε and yε determine possible
electron energy ranges along the magnetic field and in
the perpendicular thereto plane. The relations (26) and
(27) reveal a possibility of an independent description of
the electron motion along the x- and y-axis as the elec-
tron penetrates the potential barrier at the metal-vacuum
interface in the case of the collinear electric and mag-
netic fields. Proceeding from the above considerations,
we can conclude that the barrier-penetration coefficient
is independent of the magnetic field for the case of par-
allel electric and magnetic fields as was rightly sup-
posed but not proved in [2].
3. THE CHANGING OF THE POTENTIAL
BARRIER UNDER THE MAGNETIC FIELD
INFLUENCE
To better understand the influence of the external
constant magnetic field on the field emission process
let's consider its effect on the form of the potential step
at the metal-vacuum surface in the case when the mag-
netic field is parallel to the surface. The Schrödinger
equation for this case takes the following form
( )
( ) ( )
2
22 2
02
2
0
0
1
2
,
16
y z
d p p eB x
m dx
x x
eeE x
x
ψ εψ
πε
− + + −
=
+ − − −
(28)
where the Hamilton operator is
22 2
2 2
02 2
2
0
0
ˆ 1
2
.
16
H i eB x
m zx y
eeE x
xπε
∂ ∂ ∂ = − − − + − − ∂∂ ∂
− −
As follows from the explicit form of the Hamiltoni-
an operator Ĥ , it does not explicitly depend on time
and does not include the y- and z-coordinates in the
explicit form, so it is commutative with the operator of
the y-th and z-th components of the momentum we can
get the equation for the wave function component
( )xψ that describes the electron motion
2 22 2
02 2
0
2 ' 0,
2 16
B x md m eeE x
xdx
ωψ ε ψ
πε
− − − − =
(29)
where
2 2
2 2
y z
p p
m m
ε ε +′ = + . And the effective potential
energy ( )V x of an electron near a metal surface is
described as following
( )
2 2 2
0
0
.
2 16
B x m eV x eE x
x
ω
πε
= − − (30)
Fig. 2 in different scales shows a comparison of the
potential barrier near the metal surface in the absence of
a magnetic field and in the case of an external uniform
magnetic field parallel to the surface of the metal.
a
b
Fig. 2. Effective potential energy V(x) of an electron
near a metal surface, as given by eq. (30) in different
scales: from 0 to 6·10-8 m (a); from 0 to 1.2·10-3 m (b).
The line is the case of E=100 MV/m, B=0.
By dots shows the case E=100 MV/m, B=1 T
From the figure we can see that near the surface the
form of potential step remains intact. But at some dis-
tance from the metal surface the potential step becomes
infinite. As result we expect that the field emission
process in presence of the external magnetic field paral-
lel to the metal surface will occur only for a limited
interelectrode distance.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 65
4. THE POSSIBLE EXPERIMENTS
The experimental studying of the effect of the mag-
netic field on the field emission current is planned in
Institute of Applied Physics, National Academy of Sci-
ences of Ukraine. The Fig. 3 shows the experimental
setup, which is built to study the high vacuum high
gradient breakdowns, but it can operate for researching
the field emission current.
Fig. 3. Schematic drawing of the experimental setup
4
The composition of the experimental setup consists
of: high-vacuum chamber with the sample-fixing mech-
anism that allows to control motion of the samples dur-
ing the experiment, monopole mass spectrometer for
control the composition of the atmosphere in a vacuum,
system of registration the current before breakdown and
directly the breakdown, the system of heating vacuum
chamber and computer control system installation.
This setup allows to set the gap from 10 microns to
1 mm and apply voltage up to 50 kV. The setup has all
necessary equipment, that allows to measure the field
emission current down to 0.1 nA. These parameters
allows to investigate current in wide regions of gradi-
ents and gaps.
Theoretically field emission current is well de-
scribed by the Fowler-Nordheim equation which in-
cludes image forces gives the following expression for
the current density:
3
3 2 324 2 exp
8 3
e E m e Ej v
h eE
ϕ
π ϕ ϕ
=
, (31)
where ϕ is the work function of electrons, ( )v y is
Nordheim function that has been evaluated for repre-
sentative values of y.
The explicit form of the expression for the field
emission current-density contains the electric field
strength and we assume that it is possible to find evalua-
tion of the influence of the magnetic field using Lorentz
covariance. For case the same electromagnetic invari-
ants in the presence of the magnetic field the electric
field strength will change as following
2 2
*
21 B cE E
E
= − . (32)
As can be seen from the equation the influence of
the electric field should be reduced in presence of the
magnetic field.
Fig. 4. The working chamber of the experimental setup
The working chamber of the experimental setup
(Fig. 4) allows puts inside the magnet for studying the
magnetic field influence on the field emission current. It
is possible to conduct a study the influence of the mag-
netic field magnitude of 1.5 T. According to the prelim-
inary estimates (Eq. (32)) for the field emission current
order of hundreds nA the influence of the magnetic
field will be about 20% of current without magnetic
field. Studying the possibility of locking the field emis-
sion current by external magnetic field also exists in this
experimental setup.
5. SUMMARY
The authors propose a solution for the problem of
electron quantum-mechanical motion in external con-
stant and uniform electric and magnetic fields crossing
at an arbitrary angle. The electron wave function has
been derived in the explicit form for an electron moving
in thus superimposed fields.
It is shown that in the case of collinear electric and
magnetic fields the coefficient of the potential barrier
penetration by the electron does not depend on the mag-
netic field. This fact supports the supposition made by
F.J. Blatt [2] which, to our knowledge, has so far re-
ceived neither theoretical no experimental confirmation.
As is apparent from the form of differential equation
the electron barrier penetration coefficient depends in
the general case on the magnetic field.
The form of the potential step at the metal-vacuum
surface in the case when the magnetic field is parallel to
the surface is shown. Hence the field emission current
can be controlled by the external magnetic field.
The preliminary estimates for the field emission cur-
rent under the influence of the magnetic field were
done.
The estimation of the barrier penetration coefficient
and field emission current in the presence of the mag-
netic field will be a subject of further investigations.
ACKNOWLEDGEMENTS
Publication is based on the research provided by the
grant support of the State Fund For Fundamental Re-
search (project № Φ58/174-2014) as well as by the
National Academy of Sciences of Ukraine (NASU)
under the program of cooperation between NASU,
CERN and JINR Prospective Research into High-
Energy and Nuclear Physics under Contract № ЦO-5-
1/2014).
ISSN 1562-6016. ВАНТ. 2015. №4(98) 66
REFERENCES
1. Morten Kidemo. New spark-test device for material
characterization // Nucl. Instrum. and Methods A.
2004, v. 530, p. 596-606.
2. F.J. Blatt. Field emission in a magnetic field // Phys.
Rev. 1963, v. 131, p. 166-169.
3. R.F. Waites, H.A. Schwetman. Field emission from
bismuth and tungsten in a magnetic field // Phys.
Rev. B. 1973, v. 8, p. 2420-2425.
4. P.J. Kennedy, A.Y. Muir. Modification of field-
emission currents from tungsten by external magnet-
ic fields // Solid State Commun. 1978, v. 27, p. 279-
281.
5. D.J. Flood. Field emission in high magnetic fields //
J. Phys. Chem. Solids. 1970, v. 31, p. 1649-1650.
6. G.A. Gogadze, F.I. Itskovich, I.O. Kulik. Quantum
oscillations of the cold emission current from metals
in a magnetic field // Zh. Eksp. Teor. Fiz. 1964,
v. 46, p. 913-919.
7. I. Buribaev, B.B. Shishkin. Field emission of a tung-
sten in a magnetic field // Fiz. Tverd. Tela. 1970,
v. 12, p. 3309-3311 (in Russian).
8. G.N Fursey, V.E. Ptitsyn, N.V. Egorov. The influ-
ence of the magnetic field on the field emission pro-
cess from W // Pis’ma Zh. Tekh. Fiz. 1979, v. 5,
p. 1161-1164.
9. V.E. Ptitsyn, G.N. Fursey, N.V. Egorov. Anomalies
of the field emission process in a magnetic field //
Pis’ma Zh. Tekh. Fiz. 1980, v. 31, p. 733-737.
10. A. Ghosh, S. Chakrabarty. The work function asso-
ciated with ultra-relativistic electron emission from
strongly magnetized neutron star surface //
J. Astrophys. Astr. 2011, v. 32, p. 377-390.
11. A. Ghosh, S. Chakrabarty. Fowler-Nordheim elec-
tron cold emission formalism in presence of strong
magnetic field // Mon. Not. Astron. Soc. 2012, p. 1-9.
12. A.K. Harding, D. Lai. Physics of strongly magnet-
ized neutron stars // Rep. Prog. Phys. 2006, v. 69,
p. 2631-2708.
13. S.D. Liang, N.Y. Huang, S.Z. Deng, N.S. Xu. Chiral
and quantum size effects of single-wall carbon nano-
tubes on field emission // Appl. Phys. Lett. 2004,
v. 85, p. 813.
14. S.D. Liang, N.Y. Huang, S.Z. Deng, N.S. Xu. Quan-
tum effect in the field emission of carbon nanotubes
// J. Vac. Sci. Technol. 2006, B 24, p. 2.
15. G.N. Watson. A treatise on the theory of Bessel
functions. Cambrige: At the university press, 1922.
16. L.D. Landau, E.M. Lifshitz. Quantum Mechanics
(Volume 3 of A Course of Theoretical Physics). Per-
gamon Press, 1965.
Article received 30.04.2015
ВЛИЯНИЕ МАГНИТНОГО ПОЛЯ НА ДВИЖЕНИЕ ЭЛЕКТРОНОВ ДЛЯ ОПИСАНИЯ ПРОЦЕССА
ПОЛЕВОЙ ЭМИССИИ
С.А. Лебединский, В.И. Мирошниченко, Р.И. Холодов, В.А. Батурин
Решается уравнение Шредингера для волновой функции электрона, движущегося в суперпозиции внеш-
них постоянных и однородных электрическом и магнитном полях под произвольным углом между их
направлениями. Показано изменение потенциального барьера под влиянием магнитного поля, параллельно-
го поверхности.
ВПЛИВ МАГНІТНОГО ПОЛЯ НА РУХ ЕЛЕКТРОНІВ ДЛЯ ОПИСУ ПРОЦЕСУ ПОЛЬОВОЇ ЕМІСІЇ
С.О. Лебединський, В.І. Мирошніченко, Р.І. Холодов, В.А. Батурін
Розв’язується рівняння Шрьодінгера для хвильової функції електрона, що рухається в суперпозиції зов-
нішніх постійних і однорідних електричному і магнітному полях під довільним кутом між їх напрямками.
Показано зміну потенційного бар'єру під впливом магнітного поля, що паралельне поверхні.
|
| id | nasplib_isofts_kiev_ua-123456789-112233 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:07:56Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Lebedynskyi, S.O. Miroshnichenko, V.I. Kholodov, R.I. Baturin, V.A. 2017-01-18T19:55:46Z 2017-01-18T19:55:46Z 2015 The effect of a magnetic field on the motion of electrons for the field emission process description / S.O. Lebedynskyi, V.I. Miroshnichenko, R.I. Kholodov, V.A. Baturin // Вопросы атомной науки и техники. — 2015. — № 4. — С. 62-66. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 79.70.+q, 03.65.Ge https://nasplib.isofts.kiev.ua/handle/123456789/112233 The Schrödinger equation is solved for the wave function of an electron moving in a superposition of external constant and uniform electric and magnetic fields at an arbitrary angle between the field directions. The changing of the potential barrier under influence of the magnetic field parallel to the metal surface is shown. Розв’язується рівняння Шрьодінгера для хвильової функції електрона, що рухається в суперпозиції зовнішніх постійних і однорідних електричному і магнітному полях під довільним кутом між їх напрямками. Показано зміну потенційного бар'єру під впливом магнітного поля, що паралельне поверхні. Решается уравнение Шредингера для волновой функции электрона, движущегося в суперпозиции внешних постоянных и однородных электрическом и магнитном полях под произвольным углом между их направлениями. Показано изменение потенциального барьера под влиянием магнитного поля, параллельного поверхности en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нерелятивистская электроника The effect of a magnetic field on the motion of electrons for the field emission process description Вплив магнітного поля на рух електронів для опису процесу польової емісії Влияние магнитного поля на движение электронов для описания процесса полевой эмиссии Article published earlier |
| spellingShingle | The effect of a magnetic field on the motion of electrons for the field emission process description Lebedynskyi, S.O. Miroshnichenko, V.I. Kholodov, R.I. Baturin, V.A. Нерелятивистская электроника |
| title | The effect of a magnetic field on the motion of electrons for the field emission process description |
| title_alt | Вплив магнітного поля на рух електронів для опису процесу польової емісії Влияние магнитного поля на движение электронов для описания процесса полевой эмиссии |
| title_full | The effect of a magnetic field on the motion of electrons for the field emission process description |
| title_fullStr | The effect of a magnetic field on the motion of electrons for the field emission process description |
| title_full_unstemmed | The effect of a magnetic field on the motion of electrons for the field emission process description |
| title_short | The effect of a magnetic field on the motion of electrons for the field emission process description |
| title_sort | effect of a magnetic field on the motion of electrons for the field emission process description |
| topic | Нерелятивистская электроника |
| topic_facet | Нерелятивистская электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112233 |
| work_keys_str_mv | AT lebedynskyiso theeffectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT miroshnichenkovi theeffectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT kholodovri theeffectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT baturinva theeffectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT lebedynskyiso vplivmagnítnogopolânaruhelektronívdlâopisuprocesupolʹovoíemísíí AT miroshnichenkovi vplivmagnítnogopolânaruhelektronívdlâopisuprocesupolʹovoíemísíí AT kholodovri vplivmagnítnogopolânaruhelektronívdlâopisuprocesupolʹovoíemísíí AT baturinva vplivmagnítnogopolânaruhelektronívdlâopisuprocesupolʹovoíemísíí AT lebedynskyiso vliâniemagnitnogopolânadviženieélektronovdlâopisaniâprocessapolevoiémissii AT miroshnichenkovi vliâniemagnitnogopolânadviženieélektronovdlâopisaniâprocessapolevoiémissii AT kholodovri vliâniemagnitnogopolânadviženieélektronovdlâopisaniâprocessapolevoiémissii AT baturinva vliâniemagnitnogopolânadviženieélektronovdlâopisaniâprocessapolevoiémissii AT lebedynskyiso effectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT miroshnichenkovi effectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT kholodovri effectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription AT baturinva effectofamagneticfieldonthemotionofelectronsforthefieldemissionprocessdescription |