Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field
It is a model theoretical work of the applied character in which: outside the framework of the dipole approximation (with an accuracy of about v/c) the effective interaction force between two atoms of hydrogen at their ionization in a pulsed field of two opposite laser wave is theoretically studie...
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irk-123456789-1070372016-10-12T03:02:26Z Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field Starodub, S.S. Roshchupkin, S.P. Section C. Theory of Elementary Particles. Cosmology It is a model theoretical work of the applied character in which: outside the framework of the dipole approximation (with an accuracy of about v/c) the effective interaction force between two atoms of hydrogen at their ionization in a pulsed field of two opposite laser wave is theoretically studied. It is shown that the effective interaction force between hydrogen ions (atoms after ionization), can become an attractive force on certain time intervals in the presence of the pulsed laser field. As a result the pulsed laser field can slow down backward motion of ions in 6 times. Вне рамок дипольного приближения (с учетом поправок v/c) теоретически изучена эффективная сила взаимодействия между двумя атомами водорода при их ионизации в присутствии внешнего импульсного поля двух лазерных волн, распространяющихся навстречу друг другу. Исследуется возможность максимального сближения атомов водорода и удержание атомов за счет импульсного внешнего поля. Поза рамками дипольного наближення (з урахуванням поправок v/c) теоретично вивчена ефективна сила взаємодії між двома атомами водню за їх іонізації в присутності зовнішнього імпульсного поля двох лазерних хвиль, що розповсюджуються назустріч одна одній. Досліджується можливість максимального зближення атомів водню і утримання їх за рахунок зовнішнього імпульсного поля. 2012 Article Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field / S.S. Starodub, S.P. Roshchupkin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 153-156. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 41.20.Jb, 41.75.Fr http://dspace.nbuv.gov.ua/handle/123456789/107037 en Вопросы атомной науки и техники Institute of Applied Physics NAS of Ukraine |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
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Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology |
| spellingShingle |
Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology Starodub, S.S. Roshchupkin, S.P. Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field Вопросы атомной науки и техники |
| description |
It is a model theoretical work of the applied character in which: outside the framework of the dipole approximation
(with an accuracy of about v/c) the effective interaction force between two atoms of hydrogen at their ionization in a
pulsed field of two opposite laser wave is theoretically studied. It is shown that the effective interaction force between
hydrogen ions (atoms after ionization), can become an attractive force on certain time intervals in the presence of
the pulsed laser field. As a result the pulsed laser field can slow down backward motion of ions in 6 times. |
| format |
Article |
| author |
Starodub, S.S. Roshchupkin, S.P. |
| author_facet |
Starodub, S.S. Roshchupkin, S.P. |
| author_sort |
Starodub, S.S. |
| title |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| title_short |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| title_full |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| title_fullStr |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| title_full_unstemmed |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| title_sort |
non-linear effects at ionization of hydrogen atoms in the strong pulsed light field |
| publisher |
Institute of Applied Physics NAS of Ukraine |
| publishDate |
2012 |
| topic_facet |
Section C. Theory of Elementary Particles. Cosmology |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/107037 |
| citation_txt |
Non-linear effects at ionization of hydrogen atoms in the strong pulsed light field / S.S. Starodub, S.P. Roshchupkin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 153-156. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT starodubss nonlineareffectsationizationofhydrogenatomsinthestrongpulsedlightfield AT roshchupkinsp nonlineareffectsationizationofhydrogenatomsinthestrongpulsedlightfield |
| first_indexed |
2025-07-07T19:24:52Z |
| last_indexed |
2025-07-07T19:24:52Z |
| _version_ |
1837017378503786496 |
| fulltext |
NON-LINEAR EFFECTS AT IONIZATION OF HYDROGEN
ATOMS IN THE STRONG PULSED LIGHT FIELD
S.S. Starodub ∗and S.P. Roshchupkin
Institute of Applied Physics, National Academy of Sciences of Ukraine, 40030, Sumy, Ukraine
(Received October 28, 2011)
It is a model theoretical work of the applied character in which: outside the framework of the dipole approximation
(with an accuracy of about v/c) the effective interaction force between two atoms of hydrogen at their ionization in a
pulsed field of two opposite laser wave is theoretically studied. It is shown that the effective interaction force between
hydrogen ions (atoms after ionization), can become an attractive force on certain time intervals in the presence of
the pulsed laser field. As a result the pulsed laser field can slow down backward motion of ions in 6 times.
PACS: 41.20.Jb, 41.75.Fr
1. INTRODUCTION
There are many works devoted to study of the in-
teraction of similarly charged particles in the pres-
ence of electromagnetic field (see for example [1-9]).
The classical electron interaction in plane electromag-
netic wave was studied in [2, 3]. The possibility of
the electron attraction in the presence of plane elec-
tromagnetic wave was firstly showed by Oleinik [1].
However, the theoretical proof of mentioned process
was given by Kazantsev and Sokolov during investi-
gation of classical relativistic electron interaction in
the field of plane wave [2]. We also note paper [3].
It is very important to point out, that the classical
nonrelativistic electron attraction in the field of plane
monochromatic electromagnetic wave is impossible.
The possibility of the nonrelativistic electrons (ions)
attraction in the pulsed laser field was found out by
authors in [4-7]. Moreover, the electrons (ions) in-
teraction in the field of one pulsed laser wave was
studied in [4, 5]. The nonrelativistic electrons inter-
action in the field of two pulsed laser waves propagat-
ing in opposite directions and normally to the initial
direction of electrons motion was considered in [6].
And the attraction possibility of hydrogen ions mov-
ing as paraxial beam in a pulsed field of two opposite
laser waves in parallel direction to beam was stud-
ied in [7]. Also the heavy nuclei interaction moving
towards to each other in a pulsed field of two oppo-
site laser waves which perpendicularly to the initial
direction of nuclei motion was studied in [9]. In con-
trast to the papers, mentioned above, essential slow
down backward motion of hydrogen ions (in initial
they was atoms) moving towards to each other in a
pulsed field of two opposite laser waves which per-
pendicularly to the initial direction of atoms motion
is investigated in this article. The presence of the
second laser wave has allowed to considerably obtain
the Coulomb repulsion compensation of particles.
2. THE EFFECTIVE INTERACTION
FORCE
Let’s investigate interaction of the two classical non-
relativistic hydrogen atoms (q1, q2) moving towards
to each other along the axis x in the pulsed field of
two opposite laser waves, extending along the axis z.
It is known [8], in the dipole approximation in the
center-of-mass system the external field does not in-
fluence on the relative motion of particles, therefore
we will study interaction of the hydrogen atoms out
of the dipole approximation taking into account the
relativistic correction υ/c � 1 (υ is the relative trans-
verse velocity of atoms, c is the velocity of light in free
space). We assume that the electric and magnetic
strengths of the pulsed field of two contradirectional
laser waves can be written as:
E1 (t, z) = E01 · exp
(−t2/t21
)
cos (ω1ξ−) · ex, (1)
E2 (t, z) = E02 · exp
(−t2/t22
)
cos (ω2ξ+) ex, (2)
ξ± = t ± z
c
, (3)
H1 (t) = H01 · exp
(−t2/t21
)
cos (ω1t) · ey, (4)
H2 (t) = −H02 · exp
(−t2/t22
)
cos (ω2t) · ey. (5)
Here E0j and H0j (j = 1, 2) are the electric and mag-
netic field strengths at the laser pulse peak, respec-
tively; ex and ey are the unit vectors directed along
the x and y axes, respectively; t1,2 and ω1,2 are the
laser pulse durations and the frequencies of the first
and second waves, respectively. Note, that the mag-
netic field was chosen in dipole approximation, as the
correction υ/c � 1 is in the expression of Lorenz force
(see, (6), (7)).
∗Corresponding author E-mail address: starodubss@mail.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N1.
Series: Nuclear Physics Investigations (57), p.153-156.
153
Newton equations for hydrogen atoms motion in
the presence of pulsed laser field of two contradirec-
tional waves (1) – (5) are the following:
mr̈1 = q
[
E (t, z1) +
1
c
ṙ1 × H (t)
]
− q2
|r2 − r1|3
(r2 − r1) ,
(6)
mr̈2 = q
[
E (t, z2) +
1
c
ṙ2 × H (t)
]
+
q2
|r2 − r1|3
(r2 − r1) .
(7)
Here
E (t, z) = E1 (t, zj) + E2 (t, zj) ,
H (t) = H1 (t) + H2 (t) . (8)
In (6), (7) q = qj , m = mj (j = 1, 2) are the charge
and the mass of hydrogen atoms (the charge of atoms
q = 0, but after ionization the charge of ions q = +e);
r1 and r2 are the radius-vectors of the first and the
second atoms, respectively. Let us turn to the center-
of-mass system:
r = r2 − r1, R =
1
2
(r2 + r1) . (9)
In this case, the equations for relative motion of par-
ticles do not depend on a motion of center-of-mass
and are given by following expressions:
d2ξx
dτ2
= η1f1
(
C1 − ξ̇za1
)
− η2f2
(
C2 + ξ̇za2
)
+ β
ξx
ξ3
,
(10)
d2ξy
dτ2
= β
ξy
ξ3
, (11)
d2ξz
dτ2
= ξ̇x (η1f1a1 − η2f2a2) + β
ξz
ξ3
. (12)
Here ξ = (ξx, ξy, ξz) = r/
√
λ̄1λ̄2 (λ̄j = c/ωj, j = 1, 2)
is the radius-vector of the relative distance between
atoms in units of wave-length; τ =
√
ω1ω2t; para-
meter β, the pulse envelopes fj and the velocities of
oscillation of hydrogen atoms ηj (in units of velocity
of light) can be written as:
β =
q2
μc2
√
λ̄1λ̄2
� 1, (13)
f1 = exp
(
−τ2
τ2
1
)
, f2 = exp
(
−τ2
τ2
2
)
, (14)
ηj =
υj
c
=
qE0j
√
λ̄1λ̄2
μc2
� 1, τj =
√
ω1ω2tj , (15)
β � ηj , j = 1, 2. (16)
Here μ = m/2 is the reduced mass of atoms.
Coefficients Cj , aj are given by following expres-
sions: ⎧⎨
⎩
C1 = sin
(
ξz
2
√
φ
)
sin
(
τ√
φ
)
,
C2 = sin
(
ξz
√
φ
2
)
sin
(
τ√
φ
)
;
(17)
{
a1 = cos(τ
/√
φ),
a2 = cos(τ
√
φ); (18)
φ = ω2/ω1. (19)
Note, that in the dipole approximation we must
assume ξz = ξ̇z = ξ̇x = 0 then the influence of an ex-
ternal field on the relative motion of hydrogen atoms,
as one would expect, vanishes.
The effective force of atoms interaction is deter-
mined by the right parts of (10)–(12), and its pro-
jection to a direction of the relative particles motion
is:
Fξ = F · eξ =
1
ξ
(η1f1a− − η2f2a+) + β
1
ξ2
, (20)
where
a− = ξx(C1 − ξ̇za1) + ξza1ξ̇x, (21)
a− = ξx(C2 + ξ̇za2) + ξza2ξ̇x. (22)
We assume that the characteristic oscillation time
(∼ ω−1
j ) is significantly less than the laser pulse du-
ration, so that the following condition is satisfied:
τj � 1, j = 1, 2. (23)
Thereby, we would be averaged the projection of the
effective force (20) and the relative distance between
atoms with respect to the period of fast oscillations:
F̄ξ =
1
2π
∫ 2π
0
Fξ · dτ, ξ̄ =
1
2π
∫ 2π
0
ξ · dτ. (24)
Let’s name the quantity F̄ξ as the average effec-
tive force of the hydrogen atoms (ions) interaction.
One can see that condition F̄ξ > 0 determines the
particles repulsion and F̄ξ < 0 attraction of the par-
ticles. We will set the initial relative coordinates and
velocities of atoms as:
ξx0 = 200, ξy0 = 0, ξz0 = 0;
ξ̇x0 = −10−3, ξ̇y0 = 0, ξ̇z0 = 0.
(25)
The system of (10) – (12), (24) with initial con-
ditions (25) was solved numerically. Thus, fre-
quencies of waves were set by the equal: ω1 =
ω2 = ω = 3 · 1019 s−1 (h̄ω = 19.7 keV), the laser
pulse durations correspond to femtoseconds lasers:
t1 = t2 = 1.5 · 10−16 s (τ1 = τ2 = 5000).
3. THE EFFECTS AT IONIZATION OF
HYDROGEN ATOMS
The oscillation velocity of particles in the first or
second laser wave ηj , j = 1, 2 (15) is the main pa-
rameter. Strength of an external field and possible
character of particles interaction depends on its quan-
tity. The oscillation velocity varied in the range from
10−3 to 10−2 magnitudes of velocity of light in free
space. It corresponds to fields strength from 1011 to
1012 V/cm. Calculations have shown that in all range
154
of the oscillation velocities we can observe the effect
of an attraction of atoms after they ionization. How-
ever, it is valid provided that quantities η1 and η2 will
differ from each other on quantity of the initial rela-
tive atoms velocity. We can see that at increase one
of them the effect of an attraction weakens (Fig. 1).
Fig. 1. The average relative distance ξ̄ in units λ̄
vs. time τ . The dashed line - 0 in figure corre-
sponds to interaction without external field influence
(η1 = η2 = 0). Full lines 1, 2, 3 correspond to oscil-
lation velocities, respectively η1 = 10−3, η2 = 2·10−3;
η1 = 3·10−3, η2 = 7·10−3; η1 = 8·10−3, η2 = 9·10−3
0
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
�
�
2 4 6
Fig. 2. The average relative velocity ¯̇ξ vs. average
relative distance ξ̄. The full line with dots on fig-
ure corresponds to interaction without external field
influence (η1 = η2 = 0). Full line correspond to os-
cillation velocities η1 = 3 · 10−3, η2 = 7 · 10−3. The
dashed line – η1 = 10−3, η2 = 2 · 10−3
One can see that till the moment of the closest ap-
proach of particles, influence of an external field on a
its motion very little (see the left part of Fig. 1). It is
caused by a select of initial conditions (25) at which
the external field does not resist to closest approach
of atoms on distances, comparable with “Coulomb”,
i.e. such on which particles can approach without an
external field. After approach to the several atoms
distance the external field is on and we have ioniza-
tion of atoms. After actually a stopping of ions, the
force of Coulomb repulsion decreases much rather,
than the force caused by an external laser field. There
is a magnetic current generated by strength of a mag-
netic field of the first and second wave. It compensate
the Coulomb repulsion and does not allow hydrogen
ions promptly to go away from each other (see the
right part of Fig. 1). It is visible that in the presence
of a pulsed field the ions leave from each other much
more slowly. Deceleration can reach six times.
From the figure of average relative velocity
(Fig. 2) it is visible that velocity changes a sign on
far distances, i.e. just there where the magnetic cur-
rent of particles is shown. It leads to an attraction
(repulsion) of particles. The given effect confirms also
a view of averaged effective force of hydrogen ions in-
teraction (Figs. 3 and 4).
0 4000 8000 12000 16000
-0.8
-0.4
0
0.4
0.8
�
6
10F�
Fig. 3. The average effective force of the ions inter-
action F̄ξ in units μcω vs. time τ . The dashed line
corresponds to interaction without external field in-
fluence. Full line correspond to oscillation velocities
η1 = 10−3, η2 = 2 · 10−3
0 4000 8000 12000 16000
-4
-2
0
2
4
�
6
10F�
Fig. 4. The average effective force of the ions inter-
action F̄ξ in units μcω vs. time τ . The dashed line
corresponds to interaction without external field in-
fluence. Full line correspond to oscillation velocities
η1 = 3 · 10−3, η2 = 7 · 10−3
155
From Figures 3 and 4 follows that averaged effec-
tive force of ions interaction on certain intervals of
time has the negative value, i.e. ions start to effec-
tively attract to each other. Let’s underline that the
greatest effect of ions confinement is observed when
initial relative velocity and oscillation velocities have
one order of magnitude. At a deviation from this re-
quirement the confinement effect weakens (see lines
1-3 in Fig. 1).
The average effective force of particles interaction
is more often has reverse sign at essential difference
of the oscillation velocities η1 and η2 (see Fig. 4). It
result to decrease of a magnetic current on ions and
the particles go away from each other more promptly.
4. CONCLUSIONS
The research made in the present work allows to ex-
tract the following results:
• outside the framework of the dipole approxima-
tion (with an accuracy of about v/c) the effec-
tive interaction force between hydrogen atoms
(ions after ionization), moving towards to each
other in a pulsed field of two opposite laser
waves which perpendicularly to the initial di-
rection of atoms motion was studied theoreti-
cally;
• possibility of the attraction force amplification
was found out theoretically after atoms ioniza-
tion for the oscillation velocities which differ
from each other on quantity of the initial rel-
ative atoms velocity. The effect is observed at
identical frequencies of waves and is stable to
minor change of the laser parameters.
References
1. V.P. Oleinik. Resonant effects in the field of an
intensive laser beam // Zh. Eksp. Teor. Fiz. 1967,
v. 52, p. 1049-1067 (in Russian).
2. A.P. Kazantsev, V.P. Sokolov. Interaction of elec-
trons in a light field // Zh. Eksp. Teor. Fiz. 1984,
v. 86, p. 896-905 (in Russian).
3. S.T. Zavtrak. Radiative interaction of charges //
Letter in Zh. Eks. Teor. Fiz. 1989, v. 15, p. 14-16
(in Russian).
4. S.S. Starodub and S.P. Roshchupkin. Interaction
of the nonrelativistic electrons in the presence of
a strong pulsed laser field // Laser Phys. 2003,
v. 13, p. 1422-1425.
5. S.S. Starodub and S.P. Roshchupkin. The
coulomb repulsion compensation between the
ions of the beam in the presence of a strong
pulsed laser field // Laser Phys. Lett. 2005, v. 2,
p. 407-411.
6. S.S. Starodub and S.P. Roshchupkin. Interaction
of the nonrelativistic electrons in the pulsed field
of two laser waves // Eur. Phys. J. D. 2007, v.
44, p. 401-405.
7. S.S. Starodub and S.P. Roshchupkin. The hydro-
gen ions attraction effect in the pulsed field of two
laser waves propagating in the opposite directions
// Laser Phys. Lett. 2008, v. 5, p. 691-695.
8. S.P. Roshchupkin and A.I. Voroshilo. Resonant
and Coherent Effects of Quantum Electrodynam-
ics in the Light Field, Kiev: “Naukova Dumka”,
2008, 340 p. (in Russian).
9. S.S. Starodub and S.P. Roshchupkin. Heavy nu-
clei confinement effect in a pulsed light field //
Laser Phys. 2011, v. 21, p. 769-773.
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