Cent. Eur. J. Phys. • 11(4) • 2013 • 423-430
DOI: 10.2478/s11534-013-0191-7
Central European Journal of Physics
K-shell single-electron capture in collision of fast
protons with multi-electron atoms
Research Article
Ebrahim Ghanbari-Adivi∗ , Azimeh N. Velayati
Department of Physics and Isfahan Quantum Optics Group (IQOG), Faculty of Science,
University of Isfahan,
Isfahan 81746-73441, Iran
Received 31 October 2012; accepted 04 February 2013
Abstract:
Single-electron capture from the K shell of atomic targets by impact of protons at moderate and high energies has been studied using a first-order three-body Coulomb-Born continuum distorted wave approximation. The applied formalism satisfies the correct Coulomb boundary conditions. Single-zeta RoothaanHartree-Fock wave functions are used to describe the initial electronic bound state of the exchanged electron. Both differential and integral capture cross sections are calculated for impact of protons on carbon,
nitrogen, oxygen, neon and argon atoms. The results are compared with the available measurements and
other theories. The agreement between the calculations and experimental data is remarkable.
PACS (2008): 34.70.+e
Keywords:
distorted-wave approximation • K-shell electron capture • coulomb boundary conditions • differential and
total cross sections
© Versita sp. z o.o.
1.
Introduction
Inner-shell electron capture in collisions of ions with
atomic and molecular targets plays an important role in
understanding the electronic structure of the targets, the
correlation between electrons, collision mechanisms and
the dynamics of various evolution processes in complex
targets. Inner shell vacancy production is necessary for
inner shell spectroscopic studies of atomic and molecular
structures. Accordingly, the transfer of bound electrons
from K and other inner shells in targets to fast ion pro∗
E-mail: [email protected]
jectiles is an important process for inner shell vacancy
production during ion-atom collisions. Thus a great deal
of the experimental data and theoretical work on fast ionatom collisions has concentrated on inner-shell ionization
and electron capture processes [1–22].
In a recent paper [23], we applied a first-order three-body
Coulomb-Born continuum distorted-wave model (CBDW3B), originally proposed by Datta et al. [24], to positronium formation from K shells of multi-electron atomic targets. For positronium formation from helium atoms at intermediate and high impact energies, the formalism gave
reliable results for integrated cross sections which are
both comparable with the corresponding measurements
and compatible with other complicated formalisms. In
a second paper [25], the model was applied to single423
Unauthenticated
Download Date | 6/15/17 2:01 PM
K-shell single-electron capture in collision of fast protons with multi-electron atoms
Theory
A typical collision system for which there is no Coulomb
residual interaction in the final channel has been the subject of many investigations. For single-electron capture in
a such system, a detailed derivation of the post and prior
forms of the CBDW-3B transition amplitude was presented in Refs. [23–25]. However, in order to keep the article self-contained, a summary of the method is given for
the three-body rearrangement process, H + +A → H +A+ ,
in which H is an hydrogen atom and A is a neutral atomic
target. The CBDW-3B post and prior forms of the transition amplitude are given as
post
Tif
prior
Tif
= hχf |Uf | χi i ,
= hχf |Ui | χi i .
(1)
xT
H)
xP
(c.
m.
of
A)
2.
e
. of
(c.m
electron capture in proton-helium collisions. For both differential and integral cross sections, at some of the considered impact energies, the agreement between the results
and experimental data was better than that obtained from
other complicated three- and four-body formalisms.
In this paper, the CBDW-3B model is applied to evaluate the transition amplitude for one-electron capture from
the K shell of atomic targets in collision with energetic
protons. The single zeta wave functions are used to describe the initial bound state of the active electron and
the differential and integral cross sections are calculated
for a range of impact energies for which the approach is
valid. The results are compared with available experimental data as well as other theories. Atomic units are used
unless otherwise stated.
The plan of the paper is as follows: the next section
presents a summary of the CBDW-3B theory. The obtained results are reported and discussed in the third
section and the concluding remarks are given in the last
section.
XT
XP
A+
H+
Figure 1.
Two sets of Jacobi coordinate vectors for relative motion of
active electron e and heavy particles H + and A+ in initial
and final channels.
Γ(x) is Euler’s gamma function. Two vector sets (xT , XT )
and (xP , XP ) are the proper Jacobi coordinates in the initial and final channels to describe the kinematic of the
whole system in the C.M. frame. Fig. 1 presents these sets
of coordinates. qi and qf are the momenta of the incident
and scattered particles, ψi (xT ) and ψf (xP ) are bound-state
wave functions of the active electron in the entrance and
exit channels, ν = (ZT − 1)/v is the initial Sommerfeld
parameter, v is the impact velocity, and X is the distance
between two heavy positive cores participating in the collision process.
Using Eqs. (2) and (3), the post and prior forms of the
transition amplitude can be written in their explicit integral forms in coordinate representation as;
post
Tif
= Z T e−
πν
2
Z
Γ(1 + iν)
1
1
−
X
xT
dxP dXP
× ψf∗ (xP )ψi (xT )ei(qi ·XT −qf ·XP )
The modified interaction potentials and the initial and final distorted wave functions are given in coordinate space
as
1
1
1
1
− ; Uf = ZT
−
,
(2)
Ui =
X
xP
X
xT
(4)
× 1 F1 (−iν; 1; +iqi XT − iqi · XT ),
prior
Tif
= e−
πν
2
Z
Γ(1 + iν)
dxP dXP
1
1
−
X
xP
× ψf∗ (xP )ψi (xT )ei(qi ·XT −qf ·XP )
and
χi (xT , XT ) = e−
πν
2
× 1 F1 (−iν; 1; +iqi XT − iqi · XT ).
Γ(1 + iν)ψi (xT )eiqi ·XT
× 1 F1 (−iν; 1; +iqi XT − iqi · XT ),
(5)
(3)
χf (xP , XP ) = ψf (xP )eiqf ·XP .
ZT is the effective nuclear charge of the target ion,
1 F1 (a; b; x) is the confluent hypergeometric function, and
Now, we assume that the initial K-shell bound state
of the active electron is a linear combination of
the single-zeta Slater basis orbitals as ψi (xT ) =
C1 φ1 (xT ) + C2 φ2 (xT ) + C3 φ3 (xT ), in which C1 , C2 and
C3 are the combination coefficients and the normalized single-zeta s-type orbitals are given as φ1 (x) =
424
Unauthenticated
Download Date | 6/15/17 2:01 PM
Ebrahim Ghanbari-Adivi, Azimeh N. Velayati
√
√
ζ13/2 exp(−ζ
φ2 (x) = √
ζ25/2 x exp(−ζ2 x)/ 3π and
√1 x)/7/2 π,
φ3 (x) = 2ζ3 x 2 exp(−ζ3 x)/3 5π. The expansion coefficients Ci and exponent parameters ζi are both given in
Ref. [26]. For transition to the 1s state of the produced hydrogen atom, inserting the initial and final wave functions
in Eq. (5) leads to;
post
Tif
ZT − πν
=
e 2 Γ(1 + iν)
π
× C1 ζ13/2 [F1 (ζ1 ) − F2 (ζ1 )]
∂
− C2 √
[F1 (ζ2 ) − F2 (ζ2 )]
∂ζ
3 2
√ 7/2 2
2ζ
∂
[F1 (ζ3 ) − F2 (ζ3 )]} ,
+ C3 √3
3 5 ∂ζ32
ζ25/2
(6)
Z
dxP dXP
1
X
× e−ζxT +iqi ·XT e−ζP xP −iqf ·XP
(7)
× 1 F1 (−iν; 1; +iqi XT − iqi · XT ).
The explicit integral forms of F2 (ζ) and F3 (ζ) are the
same as that of F1 (ζ) but X should be replaced by xT and
xP , respectively. Using the method explained in detail in
Refs. [23–25], the appearring six-dimensional integrals in
F1 (ζ), F2 (ζ) and F3 (ζ) can be reduced to one dimensional
integrals as;
2
Z 1
1 d
1
F1 (ζ) = 8π 2 ζ dκκ(1 − κ)
K(ρ, p) , (8)
ρ dρ
ρ
0
2
Z 1
1 d
F2 (ζ) = 8π 2
dκ(1 − κ)
K(ρ, p),
(9)
ρ
dρ
0
2
Z 1
1 d
dκκ
K(ρ, p),
(10)
F3 (ζ) = 8π 2 ζ
ρ dρ
0
in which
p = −q − κv;
ρ2 = (1 − κ)(1 + κv2 ) + κζ 2
K(ρ, p) =
[(ρ − iqi )2 + (p − qi )2 ]iν
.
[ρ2 + p2 ]1+iν
Here, the wave vector q is the momentum transfer experienced by the projectile during the collision process. The
differential cross section for single K-shell electron capture in the laboratory frame is given by;
σ (θ) =
dσ
dΩ
lab
=2
MP
2π
2
|Tif |2 ,
Z
σtotal =
(11)
σ (θ)dΩ,
(12)
which should be multiplied by 1.202 to account for the
transition to the final excited states of the projectile according to the well-known Oppenheimer scaling rule.
3.
A similar form can be derived for the prior form of the
amplitude by changes ZT → 1 and F2 (ζ) → F3 (ζ).
In Eq. (6), F2 (ζ) has the following six-dimensional integral
form;
F1 (ζ) =
where MP is the mass of the proton and factor 2 is for two
equivalent K-shell electrons in the initial channel. Tif
denotes the post or prior form of the transition amplitude.
The total capture cross section is given by;
Results
In this section the results for electron capture from the K
shell of carbon, nitrogen, oxygen, neon and argon atoms by
impacting protons are presented. The numerical values of
cross sections calculated from post and prior amplitudes
show that although the post values depend more sensitively on the reasonable choices of the effective nuclear
charge of ZT , the prior cross sections are not very sensitive to such choices. In a three-body model, the three
inter-related quantities, i.e. the effective potential, the
one-electron orbital and the orbital energy of the active
electron should be known in order to give a full description of the electron in its initial and final states [6]. Although, there is no way to derive the unique values of
these three inter-related quantities, we have used a standard independent-electron model, in which the electron
is assumed to move in an averaged Coulomb potential
field with an effective nuclear charge ZT . Also, a combination of the single-zeta Slater basis wave functions
along with its corresponding binding energy are used in
our procedure to describe the initial bound state of the
active electron. Let us consider εi as the binding energy
of the active electron in its initial state. One way to relate
the one-electron orbital and the orbital energy to ZT is
to use the relation between the εi obtained from the selfconsistent Roothaan-Hartree-Fock method and the ZT for
hydrogen-like atoms. Such a choice of these inter-related
quantities is reasonable, since the K-shell electrons of the
many-electron atoms behave nearly like a K-shell electron
in a hydrogen-like atom, and also our findings show that
the CBDW-3B formalism with this value of nuclear charge
gives a better description of both the differential and total
cross sections. In following, our discussion continues only
√
with ZT = −2εi .
Figure 2 shows the calculated differential cross sections (DCS) for the single-electron transition from the K
shell of carbon atoms to the ground state of produced hydrogen at incident energies of 200, 300, 400 and 600 keV.
425
Unauthenticated
Download Date | 6/15/17 2:01 PM
K-shell single-electron capture in collision of fast protons with multi-electron atoms
4
4
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [14]
10
3
10
CBDW-3B (post)
CBDW-3B (prior)
CDW [11]
Exp. Data [14]
3
10
2
10
2
1
10
2
(dσ / dΩ)lab (units of a0 /Sr)
2
(dσ / dΩ)lab (units of a0 /Sr)
10
1
10
p-C
E=200 keV
0
10
4
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [14]
3
10
0
p-C
E=400 keV
10
-1
10
4
10
CBDW-3B (post)
CBDW-3B (prior)
CDW-3B [11]
Exp. Data [14]
3
10
2
10
2
10
1
10
1
10
0
10
p-C
E=300 keV
0
10
-1
-2
10
10
0
1
2
3
4
5
6
7
8
9
Scattering Angle (mrad)
Figure 2.
p-C
E=600 keV
-1
10
10
0
1
2
3
4
5
6
7
8
9
10
Scattering Angle (mrad)
(Color online) Angular distribution of differential cross sections for K-shell single-electron capture in collision of proton with carbon at
energies of 200, 300, 400 and 600 keV.
As is seen from the graphs, both post and prior cross sections have a kinematical peak at the extreme forward scattering angles. Prior CBDW-3B DCS for all considered
impact energies show a sharp minimum after the kinematical peak. In contrast to the theoretical prediction of
such a sharp dip in the first-order perturbative approaches
for different charge-exchange reactions [6, 23, 25, 27, 28],
the dip is called “unphysical" in the literature since it is
not observed in the experimental measurements [6, 29–32].
For lower energies the post curves fall almost monotonically, hence this minimum is not present in these curves.
But with increase in energy this minimum appears gradually in post curves too. For an incident energy of 200 keV,
this minimum occurs in the prior curve approximately at an
angle of 1 mrad. With increasing energy it shifts smoothly
toward smaller angles. The occurrence of this minimum is
due to the destructive interference of the partial amplitudes from the attractive and repulsive parts of the interaction potential. Although the mentioned minimum comes
from the destructive interference of the partial amplitudes,
both its occurrence and its location depend on the integrand functions in Eqs. (4) and (5). The initial wave function is a function of xT while the final wave function is
a function of xP . For post cross sections, the sharp dip
occurs if the complex functions F1 (ζ) and F2 (ζ) have the
same arguments and the same order of magnitude, while
for the prior cross sections this dip is observed if a similar situation occurs for F1 (ζ) and F3 (ζ). Below a scattering angle of 1.5 mrad, the prior results are always less
than the results of post amplitude. Although, at the incident energy of 200 keV the post results differ significantly
from prior amplitude results at this angular region, with
increase in energy the difference decreases. For the angular region beyond 1.5 mrad, the difference between post
and prior results is very marginal and the corresponding
related curves nearly coincide with each other at most of
the scattering angles. As is expected at very high energies
the post and prior results tend towards each other.
The experimental data for K-shell single-capture DCS
measured by Horsdal-Pedersen et al. [14] are also shown
in Fig. 2. As is seen from the graphs, the agreement between the present calculations and the depicted measurements is good especially for moderate impact energies of
200, 300 and 400 keV. As the impact energy increases, the
second- and higher-order terms in perturbation methods
find more significant contribution in the transition amplitude and a first-order approximation is not appropriate
for the presentation of all the features exhibited in the
experimental values.
426
Unauthenticated
Download Date | 6/15/17 2:01 PM
Ebrahim Ghanbari-Adivi, Azimeh N. Velayati
For incident energies of 400 and 600 keV, the three-body
continuum distorted wave (CDW-3B) calculations [11] are
compared with the present results in Fig. 2. The CDW3B calculations show a kinematical peak, a dip due to
the occurrence of the cancelation between partial amplitudes and a secondary maximum. The dip in the CDW3B results occurs nearly at the same angle in which the
sharp minimum of the prior CBDW-3B DCS appears. For
400 keV protons, the agreement between the present results and experimental measurement is better than CDW3B agreement with such measurments, while the situation
reverses for 600 keV protons.
Similar calculations are carried out for the DCS for Kshell single-electron capture from nitrogen atoms at 2.5
and oxygen atoms at 3.4 MeV. As will be seen, for the
considered impact energies the post and prior total cross
sections of these reactions are nearly equal. However, the
calculations show some marginal differences between their
post and prior differential cross sections. It is reasonable
because the differential cross sections give more detailed
information on collision dynamics.
clearly compatible with CDW-3B calculations. The integrated cross sections for K-shell single electron capture
in collisions of protons with carbon atoms are displayed
in Fig. 5. The results are also compared with other theoretical results and available experimental findings [16].
As can be seen from the figure, the prior CBDW-3B results show good agreement with the observed data, especially at higher energies. The agreement of prior values
with measurements is obviously compatible with distorted
strong-potential Born (DSPB) calculations [7] and CDW3B findings [17]. However, post results overestimate other
calculations and are in poor agreement with the experiment.
The predictions included in Fig. 5 show that for protoncarbon collisions, prior calculations give rather good
agreement with measurements over the whole considered energy region, except for energies below 400 keV
for which calculations overestimate experiment. In general, for this reaction, the prior CBDW-3B approximation
is found to result in better agreement with experimental
cross-sections compared to the DSPB [7] formalism for
energies over 400 keV and to the CDW-3B model [17] for
energies lower than 1 MeV.
In Fig. 3, the angular distributions of the K-shell singlecapture DCS for collisions of fast protons with neon atoms
at incident energies 500, 700, 1000 and 1500 keV are
shown and compared with the corresponding measurements from Horsdal-Pedersen et al. [14]. For an impact
energy of 1500 keV the CDW-3B results [11] are also
depicted. Although most of the features which were discussed for Fig. 1 appear in these graphs, the difference
between post and prior results at larger scattering angles
is considerable for lower energies. For lower energies the
prior CBDW-3B formalism gives a better description of
collision dynamics with respect to post calculations. For
larger scattering angles and higher impact energies the
calculated cross-sections of both the post and prior approximations agree very well with experimental data. It
is obvious that the prior approximation is in reasonable
accord with the existing experimental data beyond the unphysical minimum. In contrast to 400 keV proton-carbon
collisions, the figure shows that for 1.5 MeV proton-neon
collisions, the CDW-3B results are a little better than the
present results.
The results from the continuum distorted wave-eikonal initial state (CDW-EIS) [4] and DSPB [7] approximations are
depicted in Figs. 8 and 9. The prior CBDW-3B results are
in general larger than CDW-EIS and DSPB predictions
and the present formalism is less satisfactory in comparison to that given by these theories.
Differential cross-sections for single-electron capture by
6 MeV protons from the K-shell of argon atoms are shown
in Fig. 4 and compared with experimental values [20] and
with the CDW-3B theory [11]. The largest differences
between the post and prior curves occur between 0 and
1 mrad. At 1 mrad, the curves cross and then gradually diverge with an increase in the scattering angle.
However, for this angular region the agreement between
both the post and prior calculations with measurements is
very good. Further, in this region the present results are
An overview of the figures presented in this section and
their related discussions raises questions. The questions we are interested in are: for the specified chargeexchange reactions, which one of the post and prior versions of the CBDW-3B formalism better describes the collision dynamics? Can the post approximation be improved
and if so, how? It is well-known that the post-prior discrepancy for rearrangement collisions originates from the
fact that the Hamiltonians defining initial and final states
are different. Then, if approximate wave functions are used
The present calculated integral cross sections for 1s-1s
single electron capture by fast protons from nitrogen, oxygen, neon and argon atoms are displayed respectively in
figures 6-9 and compared with their experimental values
and other theoretical findings.
As can be seen from
the figures, for all specified collision systems, except for
the proton-nitrogen collision, prior results generally agree
well with experiment while post results generally overestimate the cross sections, with increasing deviation due
to decreasing energy. For proton-nitrogen collisions the
post and prior CBDW-3B total cross sections are in close
agreement with each other and are in excellent agreement
with experimental measurements.
427
Unauthenticated
Download Date | 6/15/17 2:01 PM
K-shell single-electron capture in collision of fast protons with multi-electron atoms
3
3
10
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [14]
2
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [14]
2
10
1
10
1
10
0
(dσ / dΩ)lab (units of a0 /Sr)
10
p-Ne
500 keV
-1
10
-1
10
2
2
(dσ / dΩ)lab (units of a0 /Sr)
10
0
3
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [14]
2
10
1
10
p-Ne
1000 keV
-2
10
CBDW-3B (post)
CBDW-3B (prior)
CDW -3B [11]
Exp. Data [14]
2
10
1
10
0
10
0
10
-1
10
-1
10
-2
10
p-Ne
700 keV
0
1
2
3
4
5
6
7
8
9
10
p-Ne
1500 keV
0
Scattering Angle (mrad)
Figure 3.
1
2
3
4
5
6
7
8
9
10
Scattering Angle (mrad)
(Color online) Same as Fig. 2 but for collision of proton with neon at energies of 500, 700, 1000 and 1500 keV.
1
10
2
Total Cross Sections (units of a0 )
CBDW-3B (post)
CBDW-3B (prior)
CDW-3B [11]
Exp. Data [20]
0
2
(dσ / dΩ)lab (units of a0 /Sr)
10
-1
10
-2
10
p-Ar
E = 6 MeV
-3
10
CBDW-3B(post)
CBDW-3B(prior)
DSPB [7]
CDW-3B [17]
Exp. Data [16]
-2
10
-3
10
p-C
-4
10
-4
10
0
2
0.5
4
Figure 4.
(Color online) Same as Fig. 2 but for collision of proton
with argon at impact energy of 6 MeV.
to describe the initial and final states of the system, the
post-prior equality does not hold. For many-electron targets or dressed projectiles the post-prior discrepancy depends on the procedure applied to describe atomic states.
One way to avoid this inequality is the use of the arithmetic average of the post and prior forms of the transition
amplitude [33]. The second way to preserve the post-prior
equivalence, is to adopt the parameterization by Green,
Sellin, and Zachor which is useful to determine atomic
1
1.5
2
2.5
Energy (MeV)
Scattering Angle (mrad)
Figure 5.
(Color online) Total cross sections for K-shell singleelectron capture in collision of proton with carbon.
quantities with a good approximation and which has been
successful in various collision calculations [34–36]. However, as is generally accepted and our results show, for
the asymmetric three-body rearrangement collisions with
ZT > ZP , the prior form of the amplitude would be preferred since the expansions in terms of the weaker perturbation Ui might be expected to converge more rapidly [37].
Although, it is found that the prior CBDW-3B approximation is remarkably successful in describing some of the
428
Unauthenticated
Download Date | 6/15/17 2:01 PM
Ebrahim Ghanbari-Adivi, Azimeh N. Velayati
-2
10
p-N
-4
10
-5
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [27]
Exp. Data [28]
Exp. Data [29]
Exp. Data [30]
-6
10
-7
10
CBDW-3B (post)
CBDW-3B (prior)
Exp. Data [19]
2
Total Cross Section (units of a0 )
2
Total Cross Section (units of a0 )
-3
10
-3
10
-4
10
-5
10
p-O
-8
10
-6
10
1
10
1
2
Energy (MeV)
aspects of the existing experimental (differential and integral) cross-sections for the considered asymmetric collision systems, it is not expected that a three-body model
would exhibit all the features of these complex collision
systems. Accordingly, for some cases (for example for
proton-neon collisions), the post results are a little better
than their corresponding prior values. This exceptional
behavior possibly originates from the outer shell electron
configurations. However, for such cases, the difference
between the post and prior calculations is marginal and
using the average amplitude is proposed to avoid the postprior discrepancy. Also, as is mentioned above, the post
CBDW-3B formalism is sensitive to the effective nuclear
charge of the target. On the other hand, there exist two
equivalent electrons in the K shell of the atomic target.
Therefore, it is expected that the post results would be improved if the theory is extended to a four-body formalism.
We may also remark that the static inter-electron correlation effects which originate from the pure Coulombic
interactions between the active electrons in multi-electron
atomic targets occasionally become important. Static correlation refers to the difference between an exact manybody structure calculation and a Hartree-Fock (HF) evaluation. This means , that a wave function more complicated
than the scaled hydrogenic wave functions is required to
describe more of the aspects of such systems adequately.
Hence, considering more accurate wave functions for the
initial state of the active electron improves the post results and also decreases the post-prior discrepancy of the
specified charge-exchange reactions.
4.
Summary and conclusions
In summary, it has been shown that a CBDW-3B approximation to the transition operator for K-shell single-
Figure 7.
2
(Color online) Same as Fig. 5 but for proton-nitrogen collision.
Total Cross Sections (units of a0 )
Figure 6.
3
4
5
Energy (MeV)
(Color online) Same as Fig. 5 but for proton-oxygen collision.
CBDW-3B (post)
CBDW-3B (prior)
CDW-EIS [4]
DSPB [7]
Exp. Dtta [16]
Exp. Data [19]
-3
10
-4
10
p-Ne
-5
10
0.4
1
6.5
Energy (MeV)
Figure 8.
(Color online) Same as Fig. 5 but for proton-neon collision.
electron capture from multi-electron atomic targets leads
to differential and total cross sections which are in good
agreement with experimental measurements at moderate
and higher impact energies. Both post and prior amplitudes were derived as one dimensional integrals which
have quick and straightforward numerical computations.
Although the prior amplitudes are not so sensitive to the
target nuclear effective charge, the post results are very
sensitive to different values of this parameter. In comparison with the post results, the prior results in general give
a better description of both differential and total cross
sections. For total cross sections, the proton-nitrogen
collision is an exception for which the post and prior results are compatible and even the post results are in better agreement with measurements. For such cases, since
the difference between the post and prior cross sections
is not considerable, using the average transition amplitude is proposed. The post results can be improved if
the CBDW-3B theory is extended to its four-body version
and/or more accurate wave functions are used to describe
the initial bound-state of the involved electrons. The ob429
Unauthenticated
Download Date | 6/15/17 2:01 PM
K-shell single-electron capture in collision of fast protons with multi-electron atoms
-4
CBDW-3B (post)
CBDW-3B (prior)
CDW-EIS [4]
DSPB [7]
Exp. Data [12]
Exp. Data [21]
2
Total Cross Sections (units of a0 )
10
-5
10
-6
10
p-Ar
-7
10
1.5
2
3
4
5
6
7
8
9 10 11 12
Energy (MeV)
Figure 9.
(Color online) Same as Fig. 5 but for proton-argon collision.
tained differential cross sections are compatible with the
results obtained from the CDW-3B formalism. Although,
for proton-carbon collision the obtained prior CBDW-3B
total cross sections are in better agreement with experiment in comparison with CDW and DSPB formalisms, the
situation is reversed for the impact of proton on neon and
argon atoms.
References
[1] X. Y. Ma, X. Li, S. Y. Sun, X. F. Jia, Europhys. Lett.
98, 53001 (2012)
[2] M. Zapukhlyak et al., J. Phys. B-At. Mol. Opt. 38,
2353 (2005)
[3] N. Abufager, P. D. Fainstein, A. E. Martínez, R. D.
Rivarola, J. Phys. B-At. Mol. Opt. 38, 11 (2005)
[4] N. Abufager, A. E. Martínez, R. D. Rivarola, P. D.
Fainstein, J. Phys. B-At. Mol. Opt. 37, 817 (2004)
[5] E. Ghanbari Adivi, M. A. Bolorizadeh, J. Phys. B-At.
Mol. Opt. 37, 3321 (2004)
[6] D. P. Dewangan, J. Eichler, Phys. Rep. 247, 59 (1994)
[7] S. Alston, Phys. Rev. A 41, 1701 (1990)
[8] K. M. Dunseath, D. S. F. Crothers, T. Ishihara, J. Phys.
B-At. Mol. Opt. 21, L461 (1988)
[9] M. Ghosh, C. R. Mandal, S. C. Mukherjee, Phys. Rev
A 35, 2815 (1987)
[10] D. P. Dewangan, J. Eichler, J. Phys. B-At. Mol. Opt.
19, 2939 (1986)
[11] R. D. Rivarola, A. Salin, J. Phys. B-At. Mol. Opt. 17,
659 (1984)
[12] E. Horsdal-Pedersen, C. L. Cocke, J. L. Rasmussen, S.
L. Varghese, W. Waggoner, J. Phys. B-At. Mol. Opt.
16, 1799 (1983)
[13] J. Macek, S. Alston, Phys. Rev. A 26, 250 (1982)
[14] E. Horsdal-Pedersen, F. Folkmann, N. H. Pedersen,
J. Phys. B-At. Mol. Opt. 15, 739 (1982)
[15] R. D. Rivarola, R. D. Piacentini, A. Salin, Dž. Belkić,
J. Phys. B-At. Mol. Opt. 13, 2601 (1980)
[16] M. Rødbro, E. Horsdal-Pedersen, C. L. Cocke, J. R.
MacDonald, Phys. Rev. A 19, 1936 (1979)
[17] Dž. Belkić, R. Gayet, A. Salin, Phys. Rep. 56, 279
(1979)
[18] Dž. Belkić, A. Salin, J. Phys. B-At. Mol. Opt. 11, 3905
(1978)
[19] C. L. Cocke, R. K. Gardner, B. Curnutte, T. Bratton, T.
K. Saylor, Phys. Rev. A 16, 2248 (1977)
[20] C. L. Cocke, J. R. MacDonald, B. Curnutte, S. L.
Varghese, R. Randall, Phys. Rev. Lett. 36, 782 (1976)
[21] J. R. MacDonald, C. L. Cocke, W. W. Eidson, Phys.
Rev. Lett. 32, 648 (1974)
[22] J. S. Briggs, J. Macek, J. Phys. B-At. Mol. Opt. Phys.
5, 579 (1972)
[23] E. Ghanbari-Adivi, Eur. Phys. J. D 62, 389 (2011)
[24] S. K. Datta, D. S. F. Crothers, R. McCarroll, J. Phys.
B-At. Mol. Opt. 23, 479 (1990)
[25] E. Ghanbari-Adivi, J. Phys. B-At. Mol. Opt. 44,
165204 (2011)
[26] E. Clementi, C. Roetti, Atom. Data Nucl. Data 14, 177
(1974)
[27] E. Ghanbari-Adivi, H. Ghavaminia, J. Phys. B-At. Mol.
Opt. 45, 235202 (2012)
[28] E. Ghanbari-Adivi, H. Ghavaminia, Eur. Phys. J. D 66,
318 (2012)
[29] P. J. Martin et al., Phys. Rev. A 23, 3357 (1981)
[30] T. R. Bratton, C. L. Cocke, J. R. Macdonald J. Phys.
B-At. Mol. Opt. 10, L517 (1977)
[31] V. Mergel et al., Phys. Rev. Lett. 86, 2257 (2001)
[32] M. S. Schöffler et al., Phys. Rev. A 79 064701 (2009)
[33] D. Belkić, Phys. Scripta 40, 160 (1989)
[34] F. Decker, J. Eichler, Phys. Rev. A 39, 1530 (1989)
[35] E. Ghanbari-Adivi, Braz. J. Phys. 42, 172 (2012)
[36] J. M. Monti, R. D. Rivarola, P. D. Fainstein, J. Phys.
B-At. Mol. Opt. 44, 195206 (2011)
[37] J. Eichler, Lectures on ion-atom collisions from nonrelativistic to relativistic velocities (Elsevier Science,
Amsterdam, 2005)
430
Unauthenticated
Download Date | 6/15/17 2:01 PM