Taking the Coulomb effects into account in the reactions of one-electron charge exchange
Within the framework of a single mathematical approach based on the first iteration of the Dodd–Greider equations, direct and two-step mechanisms of electron capture have been described, and their correlation with angular distributions of reaction products has been ascertained. The purpose of this m...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Zitieren: | Taking the Coulomb effects into account in the reactions of one-electron charge exchange / V.Yu. Lazur, V.V. Aleksiy, М.І. Karbovanets, M.V. Khoma, S.І. Myhalyna // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2019. — Т. 22, № 2. — С. 171-181. — Бібліогр.: 24 назв. — англ. |
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| author | Lazur, V.Yu. Aleksiy, V.V. Karbovanets, М.І. Khoma, M.V. Myhalyna, S.І. |
| author_facet | Lazur, V.Yu. Aleksiy, V.V. Karbovanets, М.І. Khoma, M.V. Myhalyna, S.І. |
| citation_txt | Taking the Coulomb effects into account in the reactions of one-electron charge exchange / V.Yu. Lazur, V.V. Aleksiy, М.І. Karbovanets, M.V. Khoma, S.І. Myhalyna // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2019. — Т. 22, № 2. — С. 171-181. — Бібліогр.: 24 назв. — англ. |
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| description | Within the framework of a single mathematical approach based on the first iteration of the Dodd–Greider equations, direct and two-step mechanisms of electron capture have been described, and their correlation with angular distributions of reaction products has been ascertained. The purpose of this modification of the Dodd–Greider integral equations for the quantum mechanical operator of three-particle scattering with rearrangement is to take into account the Coulomb asymptotic behavior of wave functions in the problem of inelastic scattering with redistribution. On this basis, the theory of the reaction of single-electron charge exchange was constructed when a collision of the hydrogen-like atom with a positively charged ion is performed, taking into account the effects of the multiple Coulomb scattering of the electron by the ion target residue. In particular, the amplitude of the reaction is distinguished as the first iterative term for solving the Dodd–Greider equations for the operator of three bodies, and the short-acting interaction that causes the electron transitions is taken into account in the distorting potential. It has been shown that in the one-fold scattering approximation, this method leads to the so-called first Coulomb–Born approximation, where the asymptotic behavior of particles in the input and output channels of the reaction is described by two-particle Coulomb wave functions. A more detailed study of the reaction of the resonance charge transfer between a proton and a hydrogen atom showed that without a correct inclusion of the Coulomb interaction into the wave function of the final state, recreating Thomas’ peak in the angular distributions of the products of this reaction cannot be. The proposed method provides a good agreement with the experimental data of both complete and differential cross-sections due to the advantages of this method, in particular, the full consideration of the interaction after the collision and rapid convergence of the series of the Dodd–Greider perturbation theory.
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ISSN 1560-8034, 1605-6582 (On-line), SPQEO, 2019. V. 22, N 2. P. 171-181.
© 2019, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
171
Semiconductor physics
Taking the Сoulomb effects into account
in the reactions of one-electron charge exchange
V.Yu. Lazur*, V.V. Aleksiy, М.І. Karbovanets, M.V. Khoma, S.І. Myhalyna
Uzhhorod National University,
54, Voloshina str., 88000 Uzhgorod, Ukraine
*E-mail: volodymyr.lazur@uzhnu.edu.ua
Abstract. Within the framework of a single mathematical approach based on the first
iteration of the Dodd–Greider equations, direct and two-step mechanisms of electron
capture have been described, and their correlation with angular distributions of reaction
products have been ascertained. The purpose of this modification of the Dodd–Greider
integral equations for the quantum mechanical operator of three-particle scattering with
rearrangement is taking into account the Coulomb asymptotic behavior of wave functions
in the problem of inelastic scattering with redistribution. On this basis, the theory of the
reaction of single-electron charge exchange was constructed when collision of the
hydrogen-like atom with a positively charged ion is performed with taking into account the
effects of the multiple Coulomb scattering of electron by ion target residue. In particular,
the amplitude of the reaction is distinguished as the first iterative term for solving the
Dodd–Greider equations for the operator of three bodies, and the short-acting interaction
that causes the electron transitions is taken into account in the distorting potential. It has
shown that in the one-fold scattering approximation, this method leads to the so-called first
Coulomb–Born approximation, where asymptotic behavior of particles in the input and
output channels of the reaction is described by two-particle Coulomb wave functions. A
more detailed study of the reaction of the resonance charge transfer between proton and
hydrogen atom showed that without a correct inclusion of the Coulomb interaction into the
wave function of the final state, to recreate Thomas’ peak in the angular distributions of the
products of this reaction cannot be. The proposed method provides a good agreement with
the experimental data of both complete and differential cross-sections due to advantages of
this method, in particular, rather full consideration of the interaction after the collision and
rapid convergence of the series of the Dodd–Greider perturbation theory.
Keywords: Coulomb interaction, operator of three-particle scattering with rearrangement,
Thomas’ mechanism of charge exchange.
https://doi.org/10.15407/spqeo22.02.171
PACS 03.65.-w, 34.50.-s, 34.70.+e, 34.80.Dp
Manuscript received 09.04.19; revised version received 23.04.19; accepted for publication
19.06.19; published online 27.06.19.
1. Introduction
The details of the elementary processes in atomic
collisions are necessary for solving many problems of
nuclear physics and astrophysics, physics and chemistry
of plasma and controlled thermonuclear synthesis, upper
atmosphere physics, quantum electronics, and so on.
Appearance of modern powerful ion accelerators [1] in
many laboratories in the world allowed to get unique
experimental material, especially on high-charged ions
and multielectron ion-atomic processes, which in totality
did stimulating influence on the theory of atomic
collisions.
Experimental studies of charge exchange between
protons and hydrogen or helium atoms [2, 3] confirmed
the important role of the two-step electron capture
mechanism, which was first considered on the basis of
the classical mechanics by Thomas [4] and is called the
Thomas mechanism of charge exchange. According to
the model [4], the electron capture takes place as if in
two stages: first, the flying particle is scattered by
electron of target atom at the angle determined by
kinematics of collision of two free particles which is
called Thomas’ angle. In so doing, this flying particle
causes the ionization of the target with the flight of
electron at the angle 60° to the direction of the initial
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
172
beam, and then the emitted electron rescatters by ion-
residue in the direction of motion of the fast particle and
is captured by it in a bound state. Quantum-mechanical
analogue of this mechanism of charge exchange is
electron transitions through a continuous spectrum from
a target atom into the states related with the fast particle.
With sufficiently large energies of particles, the
two-step mechanism of electron capture is manifested in
differential cross-sections at Thomas’ angle in the form
of maximum – Thomas’ peak that is experimentally
detected and is theoretically reproducible when the inter-
action is included in the final state, which is equivalent to
taking into account many-time electron charge exchange
by the residual ion. If, however, the charge exchange
cross-sections are calculated without taking into account
the interaction in the final state or in the one-time
scattering approximation [5], the Thomas peak does not
appear, but between theoretical and experimental cross-
sections the qualitative differences arise.
The problem of describing the angular and energy
dependences of the cross-sections of single-electron
charge exchange between
+
αZ
A ions and B atoms:
+− +→+
+
α
+
α BABA
ZZ )1(
(1)
became recently the object of not only experimental [2,
3], but also theoretical [5, 6, 7, 8] study. According to the
multiparticle scattering theory, we consider a system
consisting of a flying particle, active electrons, and
residual ion. Since the interaction of particles that take
part in the reactions is the Coulomb one, the basis for the
theoretical description can be taken as the modified
integral equations by Fadeev–Yakubovsky for systems of
several charged particles [9]. However, the practical
realization of the theoretical apparatus of integral
equations is associated with considerable computational
difficulties. In the transition to the systems with a large
number of particles, the theoretical apparatus is sharply
complicated and, accordingly, the ability to carry out a
rigorous quantitative calculation of such systems reduces.
The possibility of analytical solution of the system of
integral equations with potentials close to interatomic
interactions is rather an exception, but not rule.
Well-developed asymptotic (by large interatomic
distances) methods of the theory of ion-atom collisions in
our case do not work, because here, on the contrary,
small interatomic distances are important [10, 11].
Along with the rigorous formulations of the
problem of three bodies in the literature on the theory of
scattering, there are some examples of approximate
dynamical equations that are suitable for a number of
cases and do not need the sophisticated technique to their
solving necessary for finding solutions of exact
equations. In the role of such equations, we give
preference to the Dodd–Greider equations [12] for the
scattering operator with the rearrangement in the system
of three particles. The known difficulties of the
nonrelativistic quantum-mechanical problem of three-
particle scattering with rearrangement (mathematical
fundamentals of the multiparticle scattering theory [9])
are solved in the Dodd–Greider theory by introducing
into consideration two complementary three-particle
potentials that exclude the appearance of disconnected
diagrams in the nucleus of the obtaining equation for the
transition operator. Therefore, the iterative series
obtained on the basis of this equation are manifested as
the rapidly convergence ones in this problem, which
allows us to carry out not only evaluation, but even exact
direct calculations.
2. Application of the Dodd–Greider integral equations
The complex problem of the interaction of atom and ion
in the reaction (1) considered here is an idealized
problem of nonrelativistic interaction of three spinless
particles: α (projectile
αZ
A ), γ (active electron −e ) and
β (target ion +
B ) with the masses αm , γm and βm ,
respectively. The motion of the center of mass is
assumed to be separated. According to the possibility of
splitting the three-particle system into the fragments
α+γβ ),( , β+γα ),( , γ+βα ),( , we introduce the
channel Hamiltonians jj VHH += 0 ( )γβα= ,,j along
with the full Hamiltonian VHH += 0 , where H0 is the
operator of the kinetic energy of the system of three
particles in the system of their center of mass,
∑ γβα=
=
,,j jVV being the full interaction. The lower
index j in Vj defines a particle that does not take part in
this interaction (for example, αV is the operator of the
pair interaction of the particles β and γ ). We shall also
define the channel “interaction”
jυ . Let’s assume that it
can be represented in the form of a sum of the Coulomb
and rapidly descending short-acting parts. Coordinates
used to describe the relative position of particles are
related by the following relationships (the above masses
are denoted by a and b):
( ) αγ −= rxmas
rrr
, ( ) βγ += rsmbx
rrr
, sxR
rrr
−= . (2)
In the terms of the corresponding Jacobian
coordinates of the input and output channels of the
reaction (1), the operator of the kinetic energy H0 can be
represented in two equivalent forms:
ba
H srxr
2222
0
rrrr ∆
−
µ
∆
−=
∆
−
µ
∆
−=
βα
βα , (3)
where
α
∆r
r , x
r∆ ,
β
∆r
r , s
r∆ are the Laplace operators for
the variables αr
r
, x
r
, βr
r
, and s
r
, respectively. The values
αµ and βµ denote the reduced masses of the
corresponding groups of particles:
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
173
γβα
γβα
α
++
+
=µ
mmm
mmm )(
,
γαβ
γαβ
β
++
+
=µ
mmm
mmm )(
. (4)
Let us separate the channel potentials jυ ),( βα=j into
two parts:
jjjj WUVV +=−=υ , (5)
one of which Wj (it is usually called the “distorting”
potential) reveals small by the magnitude of the far-
acting Coulomb background that defines the asymptotic
behavior of wave functions of the scattering problem at
long distances, and another − Uj − gives the remainder
generated by a purely short-acting part of the potential υj
that causes transitions of electron and is considered as
perturbation.
From the definition of channel Hamiltonian
( )βα HH , it follows that it describes the asymptotic
situation, when the particle ( )βα does not interact with
anything, and the other two particles are in the bound
state in the potential ( )βα VV . Thus, the proper states
( )βα ΦΦ fi of the Hamiltonian ( )βα HH have the form
of the products:
( ) ( )αα
α ϕ=Φ rkixii
rrr
exp ,
( ) ( )ββ
β ϕ=Φ rkisff
rrr
exp , (6)
where ( )fi ϕϕ is the wave function of the bound state of
the pair ( )γβ, ( )( )γα, , ( )ααrki
rr
exp ( )( )ββrki
rr
exp is the plane
wave describing the relative motion of free particles
( )βα and ( )γβ, ( )( )γα, in the initial (final) state with the
relative momentum ( )βα kk
rr
. Strictly speaking, in the case
of charged particles in (6) plane waves in the initial and
final states should be distorted by phase factors,
logarithmically dependent on the distance between
particles [9]. This distortion is caused by the physical fact
that the asymptotic motion of particles in the Coulomb
field is never free, and the particles weakly interact at
infinitely large distances between them. It follows that in
the case of long-range action, the above definitions of
channel Hamiltonians require modification.
With taking into account the comments made
above, let us introduce for consideration the modified
channel asymptotic states +αΦ i and −βΦ f , that, in
distinct from αΦi and βΦ f , correctly describe effects
of the far-acting Coulomb field in the processes of charge
exchange. Let’s describe their structure. Let
αααα −=ξ rkr
rr̂
−=ξ ββββ rkr
rr̂
are the parabolic
coordinates of the particle ( )βα before (after) collision;
jk
r̂
( )βα= ,j are the unit vector in the direction of the
vector 1ˆ
: −= jjjj kkkk
rrr
. The functions ( )−β+α ΦΦ fi are the
products of the wave functions by the bound state of pair
( )γβ, ( )( )γα, and distorted plane wave ( )−
β
+
α ff with the
unit amplitude:
( ) ( ) ( ) ( )αααα
+
α
+α σ+ϕ≡ϕ=Φ irkixrfx iii
rrrrr
exp , (7)
( ) ( ) ( ) ( )ββββ
−
β
−β σ−ϕ≡ϕ=Φ irkisrfs fff
rrrrr
exp . (8)
The Coulomb phases ασ and βσ , distorting the plane
waves, are defined by the equations:
( )αααα ξν=σ kln , vnαα =ν , αµ= αv k
rr
,
( )ββββ ξν=σ kln , vnββ =ν , ββ µ= k
rr
v . (9)
The parameter ( )βα nn that characterizes the value of the
effective Coulomb interaction is equal to the product of
the total charge of the pair ( )γβ, ( )( )γα, on the charge of
the third particle ( )βα .
We will realize the further construction on the basis
of separating the distorting potentials αW and βW into
two parts:
dWwW ααα += , dWwW βββ += , (10)
where αw and βw are arbitrary short-acting potentials,
which depend on the relative coordinates αr
r
and βr
r
,
respectively; it is assumed that these potentials descend
enough rapidly at ∞→jr . We also assume that, for
sufficiently large jr , the potentials jdW coincide with
the purely Coulomb ones:
α
α
α→∞α ≡ →
α r
n
WW
c
drd ,
β
β
β∞→β ≡ →
β r
n
WW c
drd , (11)
where ( )c
d
c
d WW βα is the effective Coulomb potential
acting between the particle ( )βα and the center of mass
of the system ( )γβ, ( )( )γα, . Denote with ( )dd HH βα the
modulated channel Hamiltonian generated by the
potential ( )dd WW βα :
dd WHH ααα += , dd WHH βββ += (12)
and will construct ( )dd WW βα
in such calculation in order
to satisfy the Schrödinger equations:
( ) 0=Φ− +α
α id EH , αα µ+= 22
kEE i , (13)
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
174
( ) 0=Φ− −β
β fd EH , ββ µ+= 22
kEE f . (14)
Here, ( )fi EE is the energy of the bound state of the pair
( )γβ, ( )( )γα, , E – total energy of the three-particle
system. Introduction of the Hamiltonian ( )dd HH βα has
deep physical reasons. Electron at any point in the space
undergoes the influence of the Coulomb field of each
center – a fact well known from the general quantum-
mechanical problem of scattering by the Coulomb
potential that distorts the phase of the scattered particle
over the whole area of motion. So, perturbations
dWα , dWβ , approximating the potential of a distant
Coulomb center, must be taken into account in the
channel (i.e., zero) Hamiltonian [13].
Let us now define the full Green function
(resolvent) of the system of three particles:
( ) ( ) 1−± ε±−= iHEEG . (15)
Let’s denote by ( )−
β
+
α dd GG Green’s function of the model
channel Hamiltonian ( )dd HH βα :
( ) 1−
α
+
α ε±−= iHEG dd ,
( ) 1−
β
−
β ε±−= iHEG dd , (16)
where ε is the infinitely small positive number. Let us
introduce in the consideration the wave operator by
Möller ( )−
β
+
α ωω [14], which transforms the channel
eigenfunction ( )−β+α ΦΦ fi into a distorted wave ( )−β+α χχ fi
in the input (output) reaction channel (1):
+α+
α
+α Φω=χ ii , −β−
β
−β Φω=χ ff . (17)
Now we introduce the ±
αβU [15], which possess
such a property that their matrix elements between the
Coulomb asymptotic states +αΦ i and −βΦ f on the
mass surface are the physical amplitudes of the transition
±
αβT from the channel α to the channel β in the “post”
and “prior” formalisms according to:
+α±
αβ
−β±
αβ ΦΦ= if UT . (18)
For the transition operators ±
αβU , we may write the
integral equations obtained and considered for the first
time by Dodd and Greider [15]. Taking into account the
further qualitative analysis, for an illustration we will
write the equation for −
αβU :
+ω−υω= +
ααα
−
β
−
αβ )(* WU
.)(* −
αβ
−
βββ
−
β −υω+ UGW d (19)
In the prior-formalism of this theory, the potential
βW is arbitrary, and the potential αW should not lead to
rearrangement in the channel β . The first term in the
right side of the equation (19) leads to an amplitude in
the Born approximation with distorted waves
≡Φω−υωΦ= +α+
ααα
−
β
−β−
αβ if WDWBT )()(
*
+α
αα
−β χ−υχ≡ if W )( . (20)
Although formally the equation (19) is accurate, its
solution cannot be obtained as based on the approach
associated with the use of standard methods of finding
solutions of integral equations. The fact is that the core of
the integral equation (19) contains disconnected diagrams
that correspond to processes in which one of the particles
does not interact with two other ones. Therefore, the
arguments given in [15] raise doubts concerning the
convergence of the Born series of the method of distorted
waves, that is, iterative decomposition of the equation
(19). This circumstance dictates the necessity of a certain
rearrangement of the equation (19), which is similar to
that performed when the equations of the multiple
scattering theory and Faddeev’s equations are derived
[16]. The integral equations obtained as a result of the
rearrangement, in contrast to (19), do not contain
disconnected diagrams in their nuclei and can be solved
by the standard methods. We will not describe here the
bulky constructions that correspond to such a
rearrangement of the equation (19), because they were
considered in detail in the paper [12]. Let’s bring only
the final result. For this, we introduce the auxiliary
potential ℵυ that corresponds to the virtual intermediate
channel “ℵ ”, as well as the corresponding to it Green’s
operator ( ) 1−
ℵ
+
ℵ ε+υ+−= iHEg . In these notations, the
modified (with taking account the long-range nature of
the Coulomb interactions) Dodd–Greider equation for
quantum mechanical operator −
αβU of a three-particle
scattering with rearrangement results in the final form:
−
αβ
−
αβ += KUIU , (21)
where
( )[ ] ( ) +
ααα
+
ℵββ
−
β ω−υ−υ+ω= WgWI 1* ,
( ) +
βℵ
+
ℵββ
−
β υ−υω= dGgWK * . (22)
The main advantage of Eq. (21) before Eq. (19) is
that the arbitrariness in the choice of the potentials vκ and
Wβ can be used in order to obtain the equations with
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
175
predefined properties. Using (21), the amplitude of the
transition −
αβT (18) can be represented as follows:
+ΦΦ=ΦΦ= +α−β+α−
αβ
−β−
αβ ifif IUT
MSIKU ifif +ΦΦ=ΦΦ+ +α−β+α−
αβ
−β , (23)
where MS are the terms that take into account the
multiple rescattering. If assuming that the processes with
multiple scattering do not affect the shape of the angular
distribution, then the second term in (23) can be omitted.
In this case, the amplitude of the reaction (1) in the prior-
formalism is given by the expression:
( )[ ] ×−υ+ωΦ= ββ
+
ℵ
−
β
−β−
αβ WgT f 1*
( ) +=Φω−υ× −
αβ
+α+
ααα )(DWBTW i
( )[ ] ( ) +α+
αααββ
+
ℵ
−
β
−β Φω−υ−υωΦ+ if WWg
* . (24)
The comparison of the equations (20) and (24) shows
that the first term −
αβT in the right side of (24) indicates
the amplitude of the direct one-step charge exchange
mechanism within the Born approach with the distorted
waves. The second term in (24) directly describes the
two-step mechanism of electron capture through an
intermediate state that is located in discrete or continuous
spectrum. An analogous result takes place also for the
amplitude of the transition +
αβT in the post-formalism:
( ) ×−υωΦ=
ββ
−
β
−β+
αβ WT f
*
( )[ ] .1 +α+
ααα
+
ℵ Φω−υ+× iWg (25)
In conclusion, let us consider again the fundamental
properties of the equation (21). From the formal point of
view, it is difficult to be solved like to the Faddeev-type
equations [9]. However, the equation (21) need not be
precisely solved. The essence of this method is that there
is only an iterative approximation for the operator that
describes its system rearrangement. The Dodd–Greider
theory [12] gives good results in the study of single- and
double-electron processes with redistribution of particles
[6, 8], since the second and higher orders of the series of
perturbation theory, which are obtained when iterating
the integral equation (21) for the transition operator −
αβU ,
do not contain in disconnected diagrams their nuclei, in
contrast to the usual series of perturbation theory. Thus,
transformation of the equation (19) to (21) of the type of
the distorted waves method allows to obtain the iterative
series (they are usually called quasi-Born or Coulomb–
Born series) for the transition operator that, as shown in
[6, 17, 18], converge rapidly, that is, the first iterations of
the corresponding integral equations allows one to obtain
a result that practically coincides with the exact solution.
3. Amplitude of the charge exchange
Let us transform the initial expression (24) for the
amplitude of the reaction −
αβT . For this purpose, we
introduce into consideration the scattering state vector
−βΨ f in such a manner:
( )[ ] −β
ββ
+
ℵ
−β χ−υ+=Ψ ff Wg *1 . (26)
We substitute (17) and (26) into (24), as a result we
obtain the following representation for the amplitude of
the reaction −
αβT with taking into account the direct and
two-step mechanisms:
( ) +α
αα
−β−
αβ χ−υΨ= if WT . (27)
To derive a differential equation for the wave
function −βΨ f , we multiply both parts of the equation
(26) left by ( )ε−υ+− ℵ iHE
*
and go to the boundary
0+→ε . As a result, we obtain the equation:
( ) −β
ℵ
−β
ℵ χυ=Ψυ+− ffHE ** . (28)
Since the search for solutions of the
nonhomogeneous equation (28) with the real local
potential ℵυ is related with great mathematical
difficulties, then it is worthwhile to try replacing this
potential with the operator. It is also necessary that the
solution of the corresponding homogeneous equation
permits the representation in the form:
( ) −
β
−β ϕ=Ψ hsff
r
. (29)
To separate the only solution from the set of
solutions of the differential equation (28), we must
supplement this equation with the boundary conditions:
( ) ( ) =ϕ=Φ →Ψ β
−
β
−β
∞→
−β
β
rfsffrf
rr
( ) ( )[ ]βββββ ξ−ϕ= kivrkisf lnexp
rrr
. (30)
Substituting the function (29) into the equation (28), we
obtain the equation with respect to −
βh :
( )( ) +υ−−−ϕ −
ββ hHEEs ff 0
r
( ) ( ) ( )( ) 0/1
* =ϕυ+∇ϕ∇+ −
βℵ
−
β hshsb fsfs
rr
rr . (31)
To eliminate the disconnected diagrams from the nucleus
K (22), the operator ℵυ must be chosen so that it acts
only on a variable s
r
, which is related to a pair subsystem
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
176
( )γα, . This operator ℵυ is derived, for example, from
the formula:
( ) ( ) ( )( )ssb fsfs
rr
rr (/1 ϕΨ∇ϕ∇−=Ψυℵ (32)
on the set of elements Η∈Ψ , where H is a subspace of
states that corresponds to a continuous spectrum of the
energy operator H [6]. Consequently, with the choice of
the operator ℵυ in the form (32), the nucleus K of the
integral equation (21) is determined only by the terms to
which fully disconnected diagrams correspond. This
means that the iterative series of the equation (21) must
converge faster in the broader energy region than the
initial Born series in the three-body problem.
With taking into account the explicit form (32) of
the operator ℵυ and expression (3) for the operator of
the kinetic energy H0, the equation (31) takes the form:
0
R22
=
−+
∆
+
µ
∆
+− −
β
βαβ
α
α h
ZZ
x
Z
a
EE xr
f
rr
. (33)
According to the formulas (29) and (30), the asymptotic
behavior of the function −
βh for ∞→βr has the form of
a distorted plane wave with the unit amplitude:
= → β
−
β→∞
−
β
β
)(rfh
r
r
( ) ( )
+
υ
−
−= ββββ
αβ
ββ rkrk
ZiZ
rki
rrrr
ln
'
1
exp . (34)
In the case of the change α→ rR , the variables in the
equation (33) are separated, and the corresponding wave
functions are explicitly expressed in two-part terms:
( ) ( )α
−−−−
β ℑℜ= rxCh
rr )()()( . (35)
The two-particle Coulomb wave functions of scattering
( )x
r)(−ℜ and ( )α
−ℑ r
r)(
are determined through a confluent
hypergeometric function by equalities:
×
π
+=ℜ
ββ−
q
aZ
q
iaZ
x
2
exp1Г)(
)( r
( ) ( )xqiiqxqiaZFxqi
rrrr
−−× β ,1,exp , (36)
×
πµ
−
µ
−=ℑ
βαα
α
βαα−
q
ZZ
q
ZZi
x
2
exp1Г)(
)( r
( ) ( )ααααβαααα −µ−× xqixiqqZZiFxqi
rrrr
,1,exp . (37)
The coefficient )(−C and variables q
r
and αq
r
,
which appear when separating the variables, can be
determined by stitching −
βh with the eikonal asymptotic
limit (34) at .∞→βr
Summing up, let us write the wave function of the
finite state −βΨ f that describes the scattering the charged
particle β by a hydrogen-like system ( )γα, in our
problem:
( ) ( ) ( ) ( )α
−−
βββ
−β ℑℜϕµ=Ψ rxrkisf
iv
f
t
rrrrr )()(
exp
'
, (38)
where
( ) ( ) ( )xixiviFvNx tt
rrr
υ′−υ′−′−′=ℜ +− ,1,)()( ,
υ′=′ βZvt , ββ µ= k
rr
v ,
( ) ( ) ( )αβαβ
−
α
− −−′′=ℑ rkirikviFvNr
rrr
,1,)()( ,
( ) ( ) ( )2exp1Г
)(
ttt vvvN ′π′+=′+ ,
( ) ( ) ( )2exp1Г)(
vvvN ′π′−=′− .
The function −βΨ f takes into account interaction of
the bound electron γ with the residual target ion β and
interaction of heavy particles α and β between
themselves. The wave function of the initial state
+αχi is
determined from [19]. Using these expressions for wave
functions and transition operator [14], as well as the
relations (27) and (38), we can obtain the following
representation for the amplitude of the charge-transfer
reaction with account of Coulomb interaction in the final
state:
( ) ( ) ( )×ϕ−υ= ββαααα
−
αβ ∫∫ srkirkixdrdvNT f
rrrrrrr *
exp,
[ ] ( ) ( )×υ′+υ′′ϕ−× ααα xixiviFxsZrZ ti
rrr
,1,
( )×−−× ααααα rkirikivF
rr
,1,
( )αβαβ −′−× rkirikviF
rr
,1, , (39)
where
( ) ( ) ( )( )×′+′−+µ=υ α
′−
βα viviivvN t
vt 11Г1Г,
( )[ ]2exp tvvv ′−′+π−× α . (40)
Calculation of the amplitude (39) in the general case,
when 1≠βZ , is complicated by the presence of three
confluent hypergeometric functions under the integral
sign. However, there is an important special case when
calculating the matrix element in (39) can be reduced to
one-dimensional numerical integration. The above is
related to the charge exchange reaction at the collision of
proton (or some other charged particle: positron, nucleus
etc.) with hydrogen atom )1( =βZ . In this case, the
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
177
Coulomb parameter 0=αv and the confluent
hypergeometric function ( )ααααα −− rkirikivF
rr
,1, in the
formula (39) is equal to unity. We note that in our
consideration the effects of multiple Coulomb
rescattering of captured electron by the ion-residue of the
target are already approximately summed up in the
distorting factor ( )x
r)(−ℜ .
For further calculations, we will use the expressions
for wave functions of bound states [6, 14] and the
integral representation for the confluent hypergeometric
function [20]:
( )
( )
×
−
=
acaB
zcaF
,
1
,,
× ( )∫
−−− −
1
0
1)1( )exp(1 ztttdt
aca , (41)
where ),( yxB is the Euler beta-function [20] that simply
expresses through the Г -function
( ))(Г)(Г)(Г),( yxyxyxB += , and the integration must
be carried out in the complex plane z, choosing the
correct contour (dependent on a ) that bypasses the
points 0 and 1. Changing the order of integration for
integrals included in (40) (the integral function possesses
properties sufficient for such a transposition of integrals),
we obtain the representation for −
αβT :
( )
( ) ( )
×
′+′−′−′
=υ
= α−
αβ
viviBviviB
NvN
T
tt 1,1,
0, 2
( )
( ) ( )
,
,1
1
1
0
1
0
1
2
2122
1
1
11∫ ∫ +′
′
′
−′
−
−
×
vi
vi
vi
vi
t
ttItdt
t
tdt
t
t
(42)
( ) ,lim,
22
021 JZttI
ε∂λ∂
∂
−
λ∂λ∂
∂
=
ββα
α→ε (43)
( ) ( )
×
λ−
−= α
ββα∫∫ s
s
rkrdxdJ
exp
exp
rrrr
( ) ( ) ( ) ( )xit
x
x
r
r
tkirki
rrrrr
vexp
expexp
exp 1
21
2
δ−δ−
+×
α
α
βαα (44)
with 21 tikβ−ε=δ , 12 tvi ′−λ=δ β , αα =λ Z , ββ =λ Z .
Using the results of our previous works [6, 14] for
the matrix element J, we obtain:
∫
∞
γ
++
π
=
0 2212
2
11
2
2
2
16
cxcxc
dx
a
m
J , (45)
where
( )2
21
2
2111 ρ+ρ+−= qqc
rr
,
( )[ ] ( )[ ]2
2
2
2
2
1
2
122 αα λ+ρ+λ+ρ+= qqc ,
( )[ ] +ρ+ρ+−λ= α
2
21
2
2112 qqc
rr
[ ] [ ]2
2
2
1
2
1
2
2
2
1
2
2 qqq ++λ=ρ+ρ++λρ+ αα .
Let us transform the formula (45) by separating the
dependence of t1 and t2 in the explicit form in the
denominator of the subintegral expression. After
completing this transformation, the obtained expressions
can be combined in the following representation for the
amplitude of charge exchange
( ) ×=υ
π
= αα
−
αβ ZNvN
a
T 22
2
0,
16
∫
∞
ββα
→ε ∏
ε∂λ∂
∂
−
λ∂λ∂
∂
×
0
22
0 )(lim dxx , (46)
where
( ) ( )( ) ×′+′−′−′=∏ −1
1,1,)( viviBviviBx tt
( )
( )
( )∫ ∫
+++
−
−
×
+
′
′
−′1
0
1
0 2121
1
2
22
1
1
1
1 1
1 tCtDtBtAt
dtt
t
dtt
iv
vi
vi
vi
t
t
. (47)
Explicit expressions for the coefficients A, B, C, and D
are given in Appendix. After integrating by t1 and t2 [20],
the final expression for )(x∏ has the form:
( ) ( ) ×++=∏
′′−− tvivi
ABADAx 11)( 1
( )( )
++
−
−′×
BADA
ACBD
ivviF t ;1,, . (48)
We compare the proposed approach with the
method of continuum distorted wave (CDW)
approximation. For the first time, the CDW approach was
used by Cheshire [21] for calculations of the cross-
sections of resonance charge exchange of fast protons on
hydrogen atoms. Later in his work [22] Gayet showed
that the amplitude of the transition in the CDW approach
can be obtained as the first quasi-Born term of series of
the perturbation theory by Dodd–Greider [15] for the
operator of three-particle scattering with rearrangement.
It is worthwhile to emphasize that, in the standard CDW
approximation, only the interaction before and after
collisions of active electron with far removed core is
taken into account. At large scattering angles, as it
follows from the calculations of differential cross-
sections of charge exchange in the eikonal
approximation, an important role is played by interaction
of the heavy particles α and β between themselves,
which, however, is not taken into account in the CDW
approximation.
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Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
178
4. Results of calculations
We consider the application of the foregoing formalism
to the calculation of angular and energy dependences of
the cross-sections of the reaction of charge exchange of
proton on hydrogen atom:
++ +→+ H)1(H)1(HH ss . (49)
This reaction represents a special interest and serves as a
standard for checking different theories of processes with
rearrangement, since in this case potentials of interaction
in the channels and the wave functions of bound states
are precisely known.
First, before proceeding to discuss the results of the
investigating the reaction (49), let us note the following
things. For the process (1), the scattering amplitude −
αβT
has a distinct maximum (in fact, there are two ones)
within the region of small angles 1<<µ≤Θ γm ,
( )βαβα +=µ mmmm . It is this area of the angles of
scattering that is considered below. In the case of the
reaction (49), when the particles α and β are protons,
the exchange part of the scattering amplitude is
negligibly small.
Results of calculations of total cross-sections with
amplitudes from [19] (dashed curve) and equation (46)
(solid curve) in comparison with the results of the CDW
method (dashed-point curve) and the smoothed results of
experiments [17] (dots) are presented in Fig. 1. With
decreasing the velocity of colliding particles, the CDW
approximation becomes incorrect [17] and, as seen from
Fig. 1, leads to overestimated values of the cross-
sections, but the proposed in this work model of
approximation of distorted waves with accounting the
Coulomb interaction in the final state is better agreed
(Fig. 1, solid curve) with the experimental data.
Fig. 1. Total cross-sections of the charge exchange process at
the collision of proton with hydrogen atom.
The demands for the improvement of theoretical
representations have led to the fact that main emphasis of
theoretical and experimental works shifted from the
study of integral values, which characterize different
processes, to the study of the differential values that give
more detailed information about the role of different
mechanisms of the reaction and is more rigid test for
theoretical models. In differential cross-sections of
charge exchange, the two-step mechanism of electron
capture is manifested in the form of a characteristic sharp
maximum at the angle of scattering by Thomas TΘ –
Thomas’ peak. Experimentally, the Thomas peak was
first observed in the angular distributions of hydrogen
atoms formed during the charge exchange of protons on
helium [23] at an energy of several Mega-electron-Volts
at the angle 0.5 milliradian.
The results of calculating the differential cross-
sections of electron capture by protons in hydrogen with
the amplitudes from the work [13] and equation (46) for
two energy values are compared with the experimental
Fig. 2. Differential cross-sections of charge exchange of
protons on hydrogen in dependence on the scattering angle in
the coordinates of the center of mass system. The energy of
protons is 125 keV (a) and 500 keV (b).
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
179
data [24], the results of the CDW method [17] and
Oppenheimer–Brinkman–Kramer (OBK) approximation
in Fig. 2. The greatest interest for methodical
comparisons is calculations of angular distributions with
the amplitude of simple one-step charge exchange
mechanism [19] and calculations by formula (46), when
into the analysis of the reaction (49), except for the one-
step one, the two-step (Thomas’) electron capture
mechanism is also included. It is seen that taking into
account two-step effects leads to appearance of the
pronounced maximum (Thomas’ peak) at the site of the
Jackson–Schiff “laydown”, which is obtained within the
framework of the simple one-step mechanism.
At large scattering angles, the interaction of heavy
particles plays an important role, which, however, is not
accounted in the CDW approximation. In our
consideration, accounting of this interaction in the wave
function of the finite state
−βΨ f
(multiplier ( )α
−ℑ r
r)(
in
the formula (39)) leads to a more smooth decrease of the
cross-sections with the growth of the scattering angle,
which corresponds to the observed experimental behavior
of the cross-sections.
In conclusion, it should be emphasized that, when
using the Coulomb–Born approximation, we neglect the
effects of rescattering, that is, we do not take into account
the possible multi-step mechanisms of the reaction. With
increasing incident particles energy, we observe increase
of the role of two-step transitions through the
intermediate state that is located in a discrete or
continuous spectrum. Quantitative description of these
transitions becomes possible only with the total inclusion
of the interaction after collision into the wave function of
the final state, which is equivalent to accounting the
effects of the multiple rescattering of electron by the ion-
residue of target.
5. Conclusion
As can be seen from the above discussion, the universal
mathematical basis for construction of approximated
charge exchange theory can be based on equations of the
quantum scattering theory in systems of several particles,
and the iterations of these equations form representation
for the amplitudes in the form of the series (23), and the
number of terms taken into account defines the order of
rescattering.
Summing up the results of the theoretical studies of
Thomas’ peak in differential cross-sections, first of all, it
should be noted that the Coulomb rescattering of electron
by the ion-residue of the target in the final state affects
stronger than all the others on the form of the angular
distributions, which is equivalent to rather total
accounting the interaction after the collision. If so charge
exchange cross-sections are calculated without
accounting the Coulomb interaction in the final state or
in the one-step approximation, then the Thomas peak
does not occur in the angular distributions, and there are
qualitative differences between the theoretical and
experimental cross-sections. Fig. 2 shows the example of
such a “direct” analysis of the experimental cross-section
for the reaction (49) on the base of the formulae of the
OBK approximation, which leads to increasing the cross-
sections at small scattering angles and very fast their
descending at large scattering angles.
In general, the obtained correlations of theoretical
and experimental data allows one to conclude about the
adequacy of the method of calculating differential cross-
sections of charge exchange in the wide area of energies
and proton scattering angles, which is based on the use in
calculations of the amplitude of the first quasi-Born term
of the iterative series the Dodd–Greider equation,
modified for the Coulomb interaction.
Appendix
Here are the expressions for the coefficients A, B, C, and D from (47):
( ) −
−−−−= αγγ
γ
γ
γ
22
2
224
kmbam
m
b
a
xm
m
bx
A ( ) ( ) +
λ+λ− βαβγα
γ
kkam
m
b rr22
( )( ) ++λ+ε++ αγαα xkmbk
222222 ( ) ( )( ) +λ++ελ+ε+ ββγβγα
222222 2 xkamamk
( ) ( )( )2222
αβγα λ+ε+++ kmbk ( ) ( )( )222
amkammb βγαβγγ λ+λ+− .
( )[ ( )( −+−= βγαγ
2222
(2 kmbkamabB ( ) ( ) ) ( ) ) ( ) ×+′−λ+ε+
γ
βγαγβα
a
m
kxammkkb v2 222 rrrr
( )( ( ) ) ( ) ( ) ( ) ( )( )( ) ( ) ]υ′λ+λλ+ε++−−υ′λ+ε−′× βγαααγβαγββγα makbmbkkmkabixamik
222222 2v
rrrr
.
( )([ ( ) ( )( )) ( ) ( )( ×λ+λ+−υ′−λ+ε−−+= βγαβαγββαβαγγγ amkxkmbkkikkammbxamC 222 222 rrrr
( ) ( )( )( )) ]υ′−+λ+ε× γββαβα mbkkkik 2
rr
.
( ) ( )( )( )[ ( ) ( ) ( )( ×λ+ε+λ+ε+ε−λ+λ+−++= αβγβαβγαβγγ
22222 222 xamxikkamkammbxxD
rr
( ) ) ( )( ) ( )( ) ]423222
βγγγβγγαββγα −−−λ+ε−λ+λ× kammbmbkammbikam .
SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
180
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SPQEO, 2019. V. 22, N 2. P. 171-181.
Lazur V.Yu., Aleksiy V.V. et al. Taking the Сoulomb effects into account in the reactions of one-electron
181
Authors and CV
Volodymyr Yu. Lazur. Doctor of
Physics and Mathematics Sciences,
Leading Researcher, Professor of the
Department of Theoretical Physics at
the Uzhhorod National University,
Ukraine. He is the author of more
than 290 scientific publications. His
main research interests include
theoretical physics, theory of ion-
atom and ion-molecular collisions.
E-mail:
volodymyr.lazur@uzhnu.edu.ua
Vitaliy V. Aleksiy. Head of
Laboratories of the Department of
Theoretical Physics at the Uzhhorod
National University, Ukraine. He is
the author of about 20 scientific
publications. The central research
focus on theory of ion-atom and ion-
molecular collisions.
E-mail: vitaliy.aleksiy@uzhnu.edu.ua
Myroslav I. Karbovanets. Candidate
of Physics and Mathematics Sciences,
Head of the Department of
Theoretical Physics at the Uzhhorod
National University, Ukraine. He is
the author of more than 80 scientific
publications. His main research
interests are the theory of ion-atom
collisions, ion-molecular collisions
and theoretical physics.
E-mail:
myroslav.karbovanets@uzhnu.edu.ua
Authors and CV
Mykhaylo V. Khoma. Candidate of
Physics and Mathematics Sciences,
Docent of the Department of
Theoretical Physics at the Uzhhorod
National University, Ukraine. He is
the author of about 30 scientific
publications. The area of his scientific
interests includes theoretical physics,
theory of ion-atom collisions, non-
adiabatic processes in ion-atom
collisions.
E-mail:
mykhaylo.khoma@uzhnu.edu.ua
Svitlana I. Myhalyna. Senior
Lecturer of the Department of
Computer Systems and Networks at
the Uzhhorod National University,
Ukraine. She is the author of about 30
scientific publications. Her research
areas are the theoretical physics and
theory of ion-atom collisions.
E-mail:
svitlana.mihalina@uzhnu.edu.ua
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| id | nasplib_isofts_kiev_ua-123456789-215468 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-23T18:52:46Z |
| publishDate | 2019 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Lazur, V.Yu. Aleksiy, V.V. Karbovanets, М.І. Khoma, M.V. Myhalyna, S.І. 2026-03-18T11:39:56Z 2019 Taking the Coulomb effects into account in the reactions of one-electron charge exchange / V.Yu. Lazur, V.V. Aleksiy, М.І. Karbovanets, M.V. Khoma, S.І. Myhalyna // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2019. — Т. 22, № 2. — С. 171-181. — Бібліогр.: 24 назв. — англ. 1560-8034 PACS: 03.65.-w, 34.50.-s, 34.70.+e, 34.80.Dp https://nasplib.isofts.kiev.ua/handle/123456789/215468 https://doi.org/10.15407/spqeo22.02.171 Within the framework of a single mathematical approach based on the first iteration of the Dodd–Greider equations, direct and two-step mechanisms of electron capture have been described, and their correlation with angular distributions of reaction products has been ascertained. The purpose of this modification of the Dodd–Greider integral equations for the quantum mechanical operator of three-particle scattering with rearrangement is to take into account the Coulomb asymptotic behavior of wave functions in the problem of inelastic scattering with redistribution. On this basis, the theory of the reaction of single-electron charge exchange was constructed when a collision of the hydrogen-like atom with a positively charged ion is performed, taking into account the effects of the multiple Coulomb scattering of the electron by the ion target residue. In particular, the amplitude of the reaction is distinguished as the first iterative term for solving the Dodd–Greider equations for the operator of three bodies, and the short-acting interaction that causes the electron transitions is taken into account in the distorting potential. It has been shown that in the one-fold scattering approximation, this method leads to the so-called first Coulomb–Born approximation, where the asymptotic behavior of particles in the input and output channels of the reaction is described by two-particle Coulomb wave functions. A more detailed study of the reaction of the resonance charge transfer between a proton and a hydrogen atom showed that without a correct inclusion of the Coulomb interaction into the wave function of the final state, recreating Thomas’ peak in the angular distributions of the products of this reaction cannot be. The proposed method provides a good agreement with the experimental data of both complete and differential cross-sections due to the advantages of this method, in particular, the full consideration of the interaction after the collision and rapid convergence of the series of the Dodd–Greider perturbation theory. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Semiconductor physics Taking the Coulomb effects into account in the reactions of one-electron charge exchange Article published earlier |
| spellingShingle | Taking the Coulomb effects into account in the reactions of one-electron charge exchange Lazur, V.Yu. Aleksiy, V.V. Karbovanets, М.І. Khoma, M.V. Myhalyna, S.І. Semiconductor physics |
| title | Taking the Coulomb effects into account in the reactions of one-electron charge exchange |
| title_full | Taking the Coulomb effects into account in the reactions of one-electron charge exchange |
| title_fullStr | Taking the Coulomb effects into account in the reactions of one-electron charge exchange |
| title_full_unstemmed | Taking the Coulomb effects into account in the reactions of one-electron charge exchange |
| title_short | Taking the Coulomb effects into account in the reactions of one-electron charge exchange |
| title_sort | taking the coulomb effects into account in the reactions of one-electron charge exchange |
| topic | Semiconductor physics |
| topic_facet | Semiconductor physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/215468 |
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