Eur. Phys. J. D (2011)
DOI: 10.1140/epjd/e2011-20099-x
THE EUROPEAN
PHYSICAL JOURNAL D
Regular Article
Electron and positron scattering from atomic potassium
K. Ratnavelua and W.E. Ong
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia
Received 9 February 2011 / Received in final form 7 May 2011
c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2011
Published online 6 October 2011 – Abstract. The present coupled-channels-optical method (CCO) provides a comprehensive theoretical calculation of electron-potassium atom and positron-potassium atom scattering at intermediate energies. The
CCO calculations for various physical observables are compared with the convergent close-coupling method
(CCC) and other theoretical results, as well as experimental data where available. The CCO provides good
to fair agreement with results from the convergent close coupling calculations, which are considered to be
the state of the art for alkali atoms in the electron channel.
1 Introduction
While electron scattering from atoms and molecules has
remained the main focus of atomic collision physics for
most of the last century, there has now been an increased
focus on positron scattering from these systems. Early theoretical studies into positron-atom scattering have seen
a tremendous boost with the exceptional computational
power that has been available in the last 2 decades. During
the same period, experimental data on positron-atom scattering began to grow, with the availability of monoenergetic positron beams that were first developed by Costello
et al. [1]. Reviews by Charlton and Laricchia [2], Laricchia
and Charlton [3] and Surko et al. [4] give an historical insight of the various theoretical and experimental developments in positron-atom physics over the last 30 years.
With the advent of the convergent close-coupling
(CCC) method and relativistic convergent close coupling
(RCCC), the theoretical study of electron scattering for
hydrogenic atoms has to some extent has been addressed
quite convincingly. In a series of calculations, Bray [5,6]
had reported on the various electron-alkali atom scattering systems. Their CCC and RCCC results on a number of physical observables provide good agreement with
the available experimental data. In the case of electronpotassium (K) scattering, calculations of the total ionization cross section up to an incident energy of 100 eV using
the CCC method, were first reported in Bray [6]. Further
CCC calculations generated the reduced Stokes parameters and the derived charge-cloud parameters for the 4p
state of K at 10 eV [7], 54.4 eV [8] and 4 and 80 eV [9]. For
10 eV, 56 states were used in the CCC to obtain convergence with minor discrepancies (less than 15%) compared
to the experimental measurements. At 54.4 eV, the calculations were performed using 48 states in order to oba
e-mail: [email protected]
tain convergence to better than 10% with the experiments.
Generally, the CCC theory accurately predicted the components of the equivalent Stokes parameters. Nevertheless,
the agreement between the experiment and the CCC theory in terms of differential cross section (DCS) is only
satisfactory [10–13], although this most likely is a reflection of the difficulty of these experiments. More recently,
the CCC has been extended for the calculation of electronimpact ionization of potassium [14]. Their calculation has
also explained the unusual nature of the spin asymmetries
(as measured in [15]) for this system. Further, the CCC
shows qualitative agreement with the fully differential (e,
2e) cross sections [16]. However, the CCC also continues to
show substantial differences with the available ionization
cross sections for e− -K [17,18].
Williams and Trajmar [12] measured the e− -K absolute DCSs for elastic and excitation of the 4p and (5s+3d)
states at 6.7 eV, 16 eV and 60 eV. We note that their DCS
were qualitatively different from those of Slevin et al. [13].
Other elastic and inelastic DCS measurements were reported by [11] at various intermediate energies. Vuskovic
and Srivastava [10] made a comprehensive comparison
with the available experiments as well as the theoretical data, in differential, integral, ionization and total
cross sections. This comparison indicated agreement was
marginal at best (see later).
Other discrete excitation measurements were made by
Goldstein et al. [19–21]. Phelps et al. [22] also determined
the direct excitation functions for 14 states (4s, 6s, 7s, 8s,
4p, 5p, 6p, 7p, 3d, 5d, 6d, 5f , 6f , 7f ) from the measured
optical excitation functions in the energy range 5–40 eV.
In addition, Visconti et al. [23], Kasden et al. [24] and the
Detroit Group [25–27] have reported TCS measurements
for e− -K scattering. Ionization cross sections were also
measured by McFarland and Kinney [28], Zapesochnyi and
Aleksakhin [17] and Korchevoi and Przonski [29], with a
recommended data set being compiled by Zecca [18].
2
The European Physical Journal D
Among the earliest theoretical close-coupling (CC)
calculations for e− -K scattering were those from Karule
and Peterkop [30], Moores [31] and Phelps et al. [22].
McCarthy et al. [32] also presented the elastic and inelastic (4s−4p) differential cross section at 54.4 eV using
a 4-state (4s, 4p, 5s, 5p) CC calculation in momentumspace. Further, Msezane et al. [33] calculated the 7-state
(4s, 4p, 5s, 3d, 5p, 6s, 4d) CC cross sections (7CC) for
elastic scattering and discrete excitation of the lowest six
states in the energy range of 4–200 eV. Configurationinteraction target wave functions that take correlation and
polarization effects into account were used in that work of
Msezane et al. [33]. However, there were a number of discrepancies between their TCS results and the corresponding measurements by Kwan et al. [26]. Besides the CC calculations, there has been an array of Born, Glauber and
Distorted-Wave methods [11,34–40]. We note that Zeman
et al. [39] used the relativistic DWA, while Mitroy [41]
reported results from a unitarised distorted wave born
approximation (UDWBA) using 6-states. His results for
the 4s–4p excitation are in reasonable agreement with the
measured integral cross section data [20,21], but the DCS
shows discrepancies at the reported energies between 54.4
to 200 eV. Mitroy has argued that the incorporation of the
continuum into the theoretical calculations would provide
a better insight into the discrepancies.
The so-called Stokes parameters [42] for electron-atom
scattering are considered critical in determining the quality and strength of the various theories. The most recent measurements for superelastic e− -K scattering were
by the Flinders group and their results can be found
in [7–9]. Alignment and orientation parameters, L⊥ , P̄l ,
γ, as described by Andersen et al., were deduced directly
from the measured Stokes parameters P̄1 , P̄2 , P̄3 and P + .
They claimed that the agreement between the experimental measurements and the CCC calculations was excellent,
suggesting that this system has been benchmarked for the
case of electron scattering.
Up to the present time, all the measurements for
positron-potassium (e+ -K) scattering were performed by
the Detroit group. The first TCS was measured by Stein
et al. [25], for the energy range of 5–49 eV. Though they
claimed that the uncertainty in the determination of the
vapor pressure in the oven was the major potential source
of uncertainty in the measurement, the resulting total
cross section values were only reported to within their
statistical uncertainties. An improved measurement of the
absolute TCS was undertaken by Kwan et al. [26], in the
3–102 eV energy range. They reported a “total” experimental uncertainty in the TCS of 21% at each projectile
energy. The measured TCS is slightly lower than the results of the calculations of Ward et al. [43,44] while the
modified Glauber calculations by Khare and Vijayshri [45]
and the second-Born full model potential calculations by
Gien [46] are in less satisfactory agreement with the measured TCS. Parikh et al. [27] also remeasured this TCS,
now in the energy range from 1–102 eV, citing direct
(QT M ) and relative (QT M ) cross sections. The relative
measurements were taken for several different energies.
Their measurements indicated a maximum near 6 eV and
a significant decrease in magnitude is observed before and
after that maximum. This was the main discrepancy between their measurements and the calculations by Ward
et al. It must be noted however, that none of the measured TCS were corrected for forward angle scattering effects. Such a correction, which is energy dependent, would
increase the magnitude of the measured TCS [47].
The first measurement of Ps formation cross sections
was reported by Zhou et al. [48]. The measured upper and
lower limits were within 15% of each other above 7 eV, but
they tend to diverge significantly at energies below 7 eV.
These differences were attributed to backward scattering
that caused the upper limit to be too high. They also
found agreement (in magnitude as well as shape) between
their lower limit results and the CC calculations of Hewitt
et al. [49] that had coupled the Ps channels in the calculations. It should be noted that the Ps formation cross
section accounted for about 40% of the total cross section
at 6 eV.
For the theoretical positron (e+ ) case, a number of
works were carried out for e+ -alkali metal scattering processes applying the CC method. Ward et al. [43] performed 2-, 4- and 5- state calculations (CC(2,0), CC(4,0),
CC(5,0)) from the set of target states(4s4p5s3d5p) for the
energy range 4–50 eV. In their calculations, they assumed
that the interaction of the valence e− with the core is
weak, thus, it was treated as a frozen-core (FC). It must
be noted that the Ps formation and ionization processes
were ignored in their calculation. Their TCS was in reasonable agreement with the experimental results [25] for
the energy range of 7.9–98.5 eV. They also performed a
separate CC(2,0) calculation, using the numerical frozencore Hartree-Fock (FCHF) wave functions. The calculated
TCS with these wave functions was larger than that calculated with the model potential wave functions by about
16–28%. They also found that the FCHF method overestimated the static dipole polarizability for K, this implied
that valence-core correlation must be included in the target description for a higher accuracy determination. In
Ward et al. [44], the previous calculation that we just discussed was extended down to the energy of 0.5 eV. Here
they found good agreement between their latest CC(5,0)
model potential calculations and the experimental TCS
measurements [25]. Of note was that they explained the
slow convergence they observed at lower energies (<10 eV)
might be due to the neglect of Ps formation which is important at very low energies.
Hewitt et al. [49] were the first to employ the CC
method to investigate the Ps formation in e+ -K scattering. They used two sets of basis states, CC(4,0) – K(4s,
4p, 5s, 5p) and CC(4,3) – K(4s, 4p, 5s, 5p) + Ps(1s,
2s, 2p), in their calculations at impact energies ranging
from 0.5 to 60 eV. Their elastic cross section showed excellent agreement with those of Ward et al. [44] for all
energies >5 eV. However, the inclusion of the Ps formation channel significantly reduced the elastic cross section
for scattering energies less than 7 eV. For the Ps formation cross sections, comparisons were made with the earlier
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
distorted-wave approximation (DWA) and first Born approximation (FBA) results by Guha and Mandal [50] and
Mandal and Guha [51]. They found that the Ps(2p) formation cross section was the dominant contribution to
the rearrangement cross section for energies below 7 eV.
In addition, the corrected TCS, augmented by the n3 correction, was found to be in good agreement with Kwan
et al. [26]. Ray et al. [52] reported another realistic calculation of positron-potassium scattering using static and
coupled-static expansion schemes. Nevertheless, their results were somewhat limited due to the simple approximation they used.
In a further investigation of the e+ -K scattering
system, the first R-matrix calculation by McAlinden
et al. [53] used a K(4s, 4p, 5s, 5p, 3d) + Ps(1s, 2s, 2p,
3s, 3p, 3d) approximation. Elastic and excitation cross
sections for both the atomic levels and Ps were calculated in the energy range of 0.5 to 60 eV, where the
atomic wave functions were generated using the local potential of Stein [54]. Other theoretical results were done
by McAlinden et al. [53,55]. For Ps(1s), Ps(2s) and Ps(2p)
formation, McAlinden et al. [53] and Hewitt et al. [49]
showed relatively good agreement with each other. This
implies that the results are not sensitive to the different
atomic wave functions used in their respective calculations. We note that when the n = 3 Ps channels were
additionally included, the Ps(2s) and Ps(2p) cross sections were reduced at the maximum. Since that result for
Ps(n = 3) formation was first reported in the literature,
no comparison with other models has been made. However, their total Ps formation cross section showed poor
agreement with the measurements of Zhou et al. [48]. That
poor agreement led McAlinden et al. to use the 1/n3 scaling formula to estimate the contributions of Ps(n > 3)
formation to the overall value. On making this correction
it raised the total Ps formation to the lower experimental
bound. This relatively large effect at the lower energies
implied the importance of including the higher Ps channels. Their calculated TCS was in good agreement with
the experimental data by Kwan et al. [26] and Parikh
et al. [27] once the latter were normalized upwards by
10%. Campbell et al. [56], employed a more sophisticated
coupled-pseudo-state calculation, to investigate the effect
of increasing the number of excited-states of Ps in the
approximation. The CCC method was also extended to
study positron-hydrogen and positron-lithium atom scattering [57–59]. No calculations for positron-sodium and
positron-potassium scattering within that formalism were,
however, reported up to the present time.
For e− -K scattering, the CCO method of McCarthy
and Stelbvoics [60], which attempts to allow for the neglected continuum channels via an optical potential, has
however not been extended beyond electron scattering
from atomic sodium. All details of the computation of the
continuum optical potentials are described in their paper.
However, the calculation of ionization asymmetry is still
to be addressed. Notwithstanding that general comment,
we acknowledge that another optical potential version was
used to study electron-potassium scattering [61]. In this
3
work, we extend the CCO method to incorporate the 9
states of atomic K(4s, 4p, 5s, 3d, 5p, 4d, 6s, 6p and 5d),
with the continuum optical potential in the main 4s-4s,
4s-4p and 4p-4p couplings. In the e+ -K case, we will report
the CCO method as implemented within the CC formalism of Mitroy and Ratnavelu [62]. This method has been
used to study a number of positron-hydrogenic atomic systems [63–65]. For e+ -Na and e+ -Li, the CCO had some
success as in the H case. The present work on K will thus
provide another test for the efficacy of the CCO in describing these sorts of systems.
In the next section, we discuss details of our calculations. Thereafter our results, and a discussion of those
results, for both the electron and positron channels are
given. Where possible, comparison of our CCO results are
made to available data and other theoretical results. Finally, some conclusions from the present work are summarised.
2 Details of theoretical calculations
For electron-potassium scattering, the following calculations were performed:
(a) CC6: this coupled-channel calculation includes the
atomic potassium states K(4s), K(4p), K(5s), K(3d),
K(5p), and K(4d).
(b) CCO6: the continuum optical potentials in the 4s-4s,
4s-4p and 4p-4p couplings are incorporated into the
CC6 calculations.
(c) CC9: this coupled-channel calculation includes the 6
potassium states in (a) and also couples the K(6s),
K(6p) and K(5d) states.
(d) CCO9: the continuum optical potentials in the 4s-4s,
4s-4p and 4p-4p couplings are incorporated into the
CC9 calculations.
The atomic wave functions used in the CC6 were taken
from Mitroy [41] who had described the use of HartreeFock (HF) calculation with STO (Slater-type orbitals) basis set to generate the K(4s), K(4p), K(5s), K(3d), K(5p),
and K(4d). We also used the ionic core functions from
Mitroy [41] in our present calculations. For the K(6s),
K(6p) and K(5d) wave functions, we used the HF program
of Mitroy [66] to generate these wavefunctions.
These calculations were done in the various models to
study for convergence and the effects of the optical potential. For conciseness, we report only the most elaborate
CCO9 results except in some selected cases.
All the calculations were performed in the energy region of 4–100 eV. When solving the Lippmann-Schwinger
(LS) equations, the partial-wave cross section of the Tmatrix elements were allowed for 0 < J < JM AX.
JM AX must be less or equal to 180. On the other hand,
the matrix elements that incorporated the continuum optical potentials (CCO) were allowed for 0 < J < JOP T .
JOP T is always less than JM AX since the continuum
effects on the partial-wave cross section at larger J are
negligible. In addition, the number of partial wave used
4
The European Physical Journal D
Table 1. Elastic, excitation of the K(4p) state and total cross sections (in units of πa20 ) at selected energies for the present
calculated cross sections.
Model
Elastic
CCO9
CCO6
K(4s-4p)
CCO9
CCO6
TCS
CCO9
CCO6
Energy (eV)
20
30
54.42
4
10
98.57
89.52
34.80
35.83
21.06
21.16
16.13
15.98
51.84
52.82
50.68
52.47
49.77
50.49
169.28
167.19
121.66
126.17
107.39
96.97
in the calculations strongly depends on the incident energy. Thus, we conclude optimally that, JOP T = 8 at
4 eV, JOP T = 11 at 10 eV, JOP T = 25 at 60 eV and
JOP T = 30 at 100 eV.
Similarly, for positron-potassium scattering, the calculations performed were:
(a) CC(5,6): this close-coupling calculation includes the
atomic potassium states K(4s), K(4p), K(5s), K(3d)
and K(5p) together with the Ps states Ps(1s), Ps(2s),
Ps(2p), Ps(3s), Ps(3p) and Ps(3d).
(b) CCO(5,6): in this calculation, the continuum optical
potentials in the 4s-4s, 4s-4p and 4p-4p couplings are
incorporated into the 11-state calculation in (c).
(c) CC(9,6): this close-coupling calculation includes the
atomic potassium states K(4s), K(4p), K(5s), K(3d),
K(5p), K(4d), K(6s), K(6p) and K(5d) together with
the Ps states Ps(1s), Ps(2s), Ps(2p), Ps(3s), Ps(3p)
and Ps(3d).
(d) CCO(9,6): in this calculation, the continuum optical
potentials in the 4s-4s, 4s-4p and 4p-4p couplings are
incorporated into the 15-state calculation in (e).
As in the electron case, we only report the most elaborate
results except in some selected cases. All the calculations
were performed in the energy region of 5–100 eV, which is
above the ionization threshold. The higher partial waves
for the Ps formation were excluded for J > 16 as it’s
time consuming. The prescription for the optical potential
(JOP T ) and the maximum JM AX is similar as in the
electron case.
Further, the ionisation cross sections were calculated
from the continuum optical model [60,67]. These calculations are denoted by COPM(−) and COPM(+) for the
electron and positron case respectively.
3 Electron-potassium scattering
We begin our discussion by concentrating on our elastic
(integral and differential) cross sections, inelastic (4s-4p)
integral and differential cross sections and total cross sections, as calculated using our 6-state and 9-state formalism. All these data are summarised in Table 1. In each
case comparison, where possible, is also made against the
60
100
11.03
11.04
10.47
10.39
7.60
7.57
44.08
44.92
31.63
32.08
29.46
29.66
18.27
18.83
86.17
85.79
54.70
54.45
51.06
50.80
33.06
33.32
results from other calculations and measurements. Thereafter, the current orientation and alignment parameter
results are presented and again, where possible, are compared to corresponding previous results from the literature. Note that for the integral and total cross sections
the calculations are performed at discrete values of the
electron energy, with a linear interpolation between those
energies then being made for the relevant figures.
3.1 Elastic scattering
In Figure 1, we see that all the qualitative integral cross
section features are reproduced by the present models except for a few cases at certain energies. Other than that,
all the theoretical models are in reasonable agreement
with each other and with the experimental data to within
the experimental uncertainties. The elastic cross section
is found to be the major contributor to the total cross
section at the lower energies. For example, it accounts for
more than 50% of the total cross section at 4 eV, whereas
at 10 eV and 60 eV, it constitutes only 28% and 20%
to the total cross section respectively. Below 10 eV, the
steep increase in the cross section can be attributed to
the long-range dipole polarisability (α) interaction which
is common amongst all the alkali metals.
The DCSs for elastic scattering, calculated using the
present CCO9 model is shown in Figures 2a–2b (54.42 eV
and 100 eV). To avoid cluttering of the Figures, the
CCO6 data are not shown. In general, we have observed
that there is good qualitative and quantitative agreement between the CCO6 and CCO9 results at all energies
shown except at 4 eV. The experimental measurements
of Buckman et al. (BNT) and Vuskovic and Srivastava
(VS), as well as the results from other calculations (Bray
et al. [61] – 17CCO6, Madison et al. [38] – DWB2) are
also shown where available. Considering these figures in
more detail then at first glance, the qualitative features
of these DCS are reminiscent of those for the elastic scattering DCSs of other alkali atoms such as Li and Na. We
reiterate that it is the strong dipole polarisability of K
that underlies the large forward scattering we observe in
Figure 2.
At 54.42 and 100 eV, the CCO model show good qualitative agreement with the measurements of BNT except
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
5
3
100
CCO9
DWB2
17CCO6
VS
BNT
a) 54.42
2
10
1
10
80
2
Cross Sections (π a0 )
10
CCO9
CCO6
17CCO6
UBA6
CC7
VS
0
2
Differential Cross Sections (π a0 )
10
60
40
20
-1
10
-2
10
-3
10
b) 100 eV
2
10
1
10
0
10
0
0
20
40
60
80
100
Energy (eV)
Fig. 1. (Color online) Elastic integral cross section for
electron-potassium scattering: present CCO9 (solid); CCO6
(dashed); 17CCO6 [61] (); UBA6 [41] (∗ ); CC7 [33] ();
VS [10] (•).
-1
10
-2
10
-3
10
0
20
40
60
80
100
120
140
160
180
Angle (Degree)
for some very minor details. At 54.42 eV, the CCO9 model
reproduce the minima and the secondary maximum, at
middle and large-angle scattering, when compared with
the experimental data. These effects are also observed at
higher energies. Thus, it appears that the CCO approach
can reproduce in great detail aspects of electron scattering
even from a relatively large atom such as potassium. From
a quantitative perspective the experimental measurements
are generally lower in magnitude than all the theoretical
calculations in the middle-angle region. Such behaviour
has also been seen in sodium, where it is now understood
to be due to limitations in the measurements due to the
some difficulties involved in making those measurements.
3.2 4s-4p transition
For the resonant transition the present CCO9 shows reasonable agreement at the integral cross section level with
existing theoretical and experimental data, throughout
the energy range studied, as displayed in Figure 3 (also
see Tab. 1 for discussion below). The only possible exception to this is at energies below 10 eV. Unlike the elastic
cross section, this transition dominates at the intermediate and higher energies. For example at 4, 10 and 60 eV,
the contribution of this transition to the total cross section is at about 30%, 41% and 58% respectively for both
the CCO9 and CCO6 calculations. At energies of 5 and
20 eV, the CCO9 is a little smaller in magnitude relative to
that for the 17CCO6 cross sections. Overall, however, our
CCO and the 17CCO6 are in good accord to within 10–
15%. The integral cross sections (ICSs) for this excitation
Fig. 2. (Color online) Elastic differential cross section for
electron-potassium scattering at (a) 54.42 eV and (b) 100 eV –
present CCO9 (solid line); DWB2 [38] (-·-); 17CCO6 [61] (-··-);
VS [10] (•); BNT [11] ().
are also given in Table 1. At 54.42 eV the experimental
data of BNT gives an ICS of 36.8(±6.7) πa20 , whereas the
CCO9 and CCO6 cross sections are about 20% smaller at
31.63 and 32.08 πa20 respectively. Nonetheless the present
theory results are all within the experimental error. Using their optical excitation functions, Chen and Gallagher
(hereafter denoted as CG) and Zapesochnyi et al. (ZPS)
have derived the ICS at this energy to be 34.4(±2.8) and
32.4(±1.9) πa20 , respectively. Mitroy has argued that CG
did not appropriately take cascade effects into consideration, from the higher n states, and thus gives a recommended value (CG*) of 27 πa20 . Further, Mitroy’s UBA6
calculation gives 31.40 πa20 (without cascade correction)
and 35.78 πa20 (with cascade corrections taken into account). These are in excellent agreement with the present
results. At 100 eV, the experimental data of BNT gives
21.4(±3.6) πa20 whereas the CCO9 and CCO6 cross sections are about 15% smaller at 18.27 and 18.83 πa20 respectively. The CG, CG* and ZPS values at this energy are
21.8(±1.3), 19.4 and 21.7(±1.7) πa20 respectively. Further,
the UBA6 calculation gives 20.72 πa20 (without cascade
correction) and 23.11 πa20 (with cascade corrections taken
into account). So once again the accord with our results is
good. The 100 eV 17CCO6 cross section (18.99 πa20 ) and
the present CCO ICS are also in near agreement at this
energy.
6
The European Physical Journal D
identical to the results of the 17CCO6 with only minor
discrepancies at the angles 70◦ and 140◦ . At both these
energies the present CCO9 and the independent 17CCO6
calculations are larger in magnitude, particularly at middle and more backward scattering angles, than the available measurements (VS and BNT). This discrepancy is
again likely to reflect difficulties encountered in the experiments.
CCO9
17CCO6
UBA6
VS
BNT
Phelps et. al. (1979)
70
2
Cross Section (πa0 )
60
50
3.3 Total cross sections (TCSs)
40
30
20
10
0
20
40
60
80
100
Energy(eV)
Fig. 3. (Color online) Excitation K(4s-4p) cross section
for electron-potassium scattering: present CCO9 (solid);
17CCO6 [61] (); UBA6 [41] (∗); VS [10] (•); BNT [11] ();
Phelps et al. [22] (◦).
The DCSs for excitation of the 4p state of potassium
for incident electron energies of 4, 54.42 and 100 eV are
shown in Figures 4a–4c respectively. There are a number
of experimental measurements performed on this transition including those from BNT and VS. One of the most
recent, however, is the joint experimental and theoretical
investigation as reported by Stockman et al. [7–9]. Their
DCSs were derived from a series of superelastic scattering
experiments. Theoretically, they also reported the results
from convergent close-coupling (CCC) calculations.
Let us now consider some of the energies where the
present 4s-4p CCO results can be compared against the
available data and previous computations. At 4 eV (see
Fig. 4a), the present CCO9 reproduces the same qualitative structure as the others at most angles, except at
around 130◦ where a sharp dip is clearly observed. However, there are some differences seen between the CCC and
the CCO’s at the middle and larger scattering angles with
the experimental data supporting the CCC results. This
clearly shows that the CCO has some remaining inadequacies at the lower energies, although overall the agreement
remains fair. At 54.42 eV, the present DCS are however, in
excellent agreement with the CCC and the experimental
measurement of Stockman et al. This can be clearly seen in
Figure 4b. Considering 100 eV, the present DCS are compared with the available theoretical calculations of DWB2
and 17CCO6. We note that at this energy, the present
DCS again agrees very well with the 17CCO6 differential
cross sections. Indeed the present CCO9 model is almost
In Figure 5, the TCSs for electron-potassium scattering
are presented. With the exception of the data of VS,
which is in quite good accord with the theoretical CCO6,
CCO9 and 17CCO6 TCS results, at all mutual energies of
study, all the theories tend to be significantly stronger in
magnitude than the TCS measurements from the Detroit
group [26,27]. However, that group did not attempt to correct their TCS data for forward scattering effects which,
if they had done so, would have increased the magnitude
of their TCS. Note that this effect is energy dependent,
generally being more significant at lower energies than at
higher energies. A recent study [68], albeit for positron
scattering, found that such a correction could be as large
as 60%–70%, depending on the energy. Therefore the level
of agreement in Figure 5, between the present CCO calculations and the Detroit measurements’, is probably quite
a bit better than what first appears to be the case.
Other points of note from Figure 5 include the very
large TCS as we go towards zero energy. This observation
is again consistent with atomic potassium having a significant dipole polarisability. Our CCO6 and CCO9 calculations are found to be in agreement with each other
across the energy range of consideration, to better than
10%. Agreement with the earlier 17CCO6 computation is
also satisfactory in this case.
3.4 Alignment and orientation parameters
The present alignment and orientation parameters for
e− -K scattering are compared with the earlier work of
Stockman et al. [7–9] in Figures 6–8 and also with the
corresponding CCC results reported in Stockman et al.
Experimentally, the three reduced Stokes parameters P̄1 ,
P̄2 and P̄3 were measured at incident electron energies
of 2.4 eV, 8.4 eV and 52.8 eV and 78.4 eV, which corresponded to the superelastically scattered electron energies
of 4 eV, 10 eV, 54.4 eV and 80 eV respectively. The alignment and orientation parameters L⊥ , P̄ and γ, which were
introduced by Andersen et al. [42], can be easily deduced
from the Stokes parameters by:
1
L⊥ = −P̄3 P̄ = P̄12 + P̄22 γ = arg P̄1 + iP̄2 ,
2
while the coherence properties of the collision can be represented by the P + parameter,
P + = P̄12 + P̄22 + P̄32 .
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
CCO9
DW B2
17CCO6
CCC
VS
BNT
a) 4 eV
10
2
10
1
10
0
7
10
-1
10
4
10
3
10
2
10
1
10
0
b) 54.42 eV
2
Differential Cross Sections (πa0 )
Stockm an et al.
10
-1
10
-2
10
-3
10
3
10
1
10
-1
10
-3
10
-5
c) 100 eV
0
20
40
60
80
100
120
140
160
180
Angle (Degree)
Fig. 4. (Color online) Excitation DCS for (4s-4p) electron-potassium scattering at (a) 4 eV (b) 54.42 eV (c) 100 eV: present
CCO9 (solid); 17CCO6 [61] (-··-); VS [10] (•); BNT [11] (); DWB2 [38] (-·-); CCC [8,9] (· · · ·); Stockman et al. [8,9] (♦).
In Figures 6 and 7, the present calculated reduced Stokes
parameters, charge cloud parameters and coherence parameter at 4 and 10 eV, are depicted together with the
available data. The agreement between the CCC calculation and the experiment data is excellent (better than
15%), due to the large number of states that were used
to obtain convergence. Considering initially Figure 6 at
4 eV, all the calculations qualitatively reproduce the gross
features of the experimental alignment and orientation
parameters to some degree. Quantatively, however, the
present CCO6 and CCO9 computations are clearly inferior to these from the CCC, which is not surprising given
what we had previously noted for the 4s-4p DCS at this
energy. By 10 eV, however (see Fig. 7), the CCO6 and
CCO9 are doing a reasonable job of both qualitatively
and quantatively reproducing all the Stokes parameters,
charge cloud parameters and the coherence parameter.
This is very strong evidence for the optical potential providing a physical description of the scattering dynamics
at this energy.
Figures 8 and 9 now compare the same parameters at
the incident electron energy of 54.42 and 80 eV, respectively. At 54.42 eV the CCC calculation, which had incorporated 48 states into their calculation in order to obtain
convergence to better than 10%, when compared to our
CCO9 result, with only 9 states included together with
the optical potential, are in excellent agreement with one
another and the experimental data. The only exceptions
to this general statement, at 54.42 eV, are that our predictions for the P̄2 and P̄3 results are differ with the CCC
by about 40% at the scattering angles 70◦ and 45◦ . The
agreement between all the theories and the experiment is
acceptable for the P̄3 results. At 80 eV, our present CCO9
calculation again agrees well with the experimental Stokes,
alignment and orientation parameters and with the CCC.
It is therefore clear that at 54.42 eV and 80 eV our CCO9
results also provide a physical representation of scattering
dynamics, suggesting that from at least 10 eV–80 eV our
formalism is an accurate and reliable approach for e− -K
scattering.
8
The European Physical Journal D
200
160
180
140
160
120
2
Total Cross Sections (πa0 )
140
120
100
100
80
0
5
10
15
20
80
60
CCO9
CCO6
17CCO6
BNT
VS
Kwan et. al. (1991)
Parikh et. al. (1993)
40
20
0
0
20
40
60
80
100
Energy (eV)
Fig. 5. (Color online) Total cross section for electron-potassium scattering: present CCO9 (solid line); CCO6 (dashed line);
17CCO6 [61] (); VS [10] (•); BNT [11] (); Kwan et al. [26] (•); Parikh et al. [27] ().
4 Positron-potassium scattering
As with our previous discussion on electron-potassium
scattering, we will now in principle report on the elastic
(integral and differential) cross sections, inelastic (4s-4p,
4s-3d) integral and differential cross sections and the total
cross sections, as calculated using our formalism (only the
most elaborate model cross sections will be shown except
for some special cases). Further, we will also present a subset of our relevant integral cross sections for the Ps(1s),
Ps(2s), Ps(2p) and Ps(n = 3) channels. However, the
availability of other data and calculations is rather more
limited in the positron case compared to the electron case.
As a consequence, specific results will in general only be
discussed, where comparison can be made against the results from other calculations and/or measurements. Note
that for the integral and total cross sections, the calculations are performed at discrete values of the positron
energy, with a linear interpolation between those energies
then being made for the relevant figures.
4.1 Elastic K(4s-4s) transition
The elastic integral cross section for positron-potassium
scattering is shown in Figure 10, where all the present
model results and the McAlinden et al. (CC(5,6)M ) calculation are seen to predict similar qualitative features. It
is clear from that figure that the present elastic cross sections are very strongly peaked in magnitude as go to lower
energies. The present elastic cross section is also found
to be the major contributor to the total cross sections at
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
_
P1
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.0
0.8
0.6
0.4
0.2
_
P2
_
PL
_
P3=-L
T
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
90
γ
60
30
0
CCO9
CCO6
CCC
Stockman et al.
-30
-60
-90
0
20
40
60
80
100
120
140
160
180 0
20
40
60
9
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
+ 0.3
0.2
0.1
0.0
160 180
P
80
100
120
140
Scattering Angle (degree)
Scattering Angle (degree)
Fig. 6. (Color online) The reduced Stokes parameters’, charge cloud parameters’ and coherence parameter, for 4s-4p excitation
of potassium at an electron energy of 4 eV: present CCO9 (solid); CCO6 (dashed); CCC [7] (dash-dot); Stockman et al. [7] (◦).
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
90
_
P1
_
P3=-L
T
_
P
60
γ
30
P
0
CCO9
CCO6
CCC
Stockman et al (1998)
-30
-60
-90
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.1
1.0
0.9
0.8
0.7
0.6
0.5
L 0.4
0.3
0.2
0.1
0.0
1.1
1.0
0.9
0.8
+ 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
180
_
P2
0
30
60
90
120
Scattering Angle (degree)
150
180 0
30
60
90
120
150
Scattering Angle (degree)
Fig. 7. (Color online) The reduced Stokes parameters’, charge cloud parameters’ and coherence parameter, for 4s-4p excitation
of potassium at an electron energy of 10 eV: present CCO9 (solid); CCO6 (dashed); CCC [7] (dash-dot); Stockman et al. [7]
(◦).
10
The European Physical Journal D
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
_
P1
0.4
_
P2
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
_
P3=-L
T
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
_
PL
80
γ
60
40
-1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.1
1.0
0.9
+ 0.8
0.7
0.6
0.5
0.4
CCO9
0.3
CCC
0.2
Stockman et al. (1999) 0.1
0.0
80 100 120 140 160 180
P
20
0
-20
-40
-60
-80
0
20
40
60
80
100
120
140
160
180 0
20
40
60
Scattering Angle (degree)
Scattering Angle (degree)
Fig. 8. (Color online) The reduced Stokes parameters’, charge cloud parameters’ and coherence parameter, for 4s-4p excitation
of potassium at an electron energy of 54.42 eV: present CCO9 (solid); CCC [8] (dash-dot); [8] (◦).
1.0
0.8
0.6
0.4
0.2
_
P1
1.0
0.8
0.6
0.4
0.2
_
P2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.0
0.8
0.6
0.4
0.2
_
P3=-L
T
_
PL
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
90
γ
60
+
P
30
0
-30
CCO9
CCC
Stockman et al.
-60
-90
20
40
60
80
100
120
Scattering Angle (degree)
140
20
40
60
80
100
120
140
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Scattering Angle (degree)
Fig. 9. (Color online) The reduced Stokes parameters’, charge cloud parameters’ and coherence parameter, for 4s-4p excitation
of potassium at an electron energy of 80 eV: present CCO9 (solid); CCC [8] (dash-dot); [8] (◦).
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
120
70
110
CCO(9,6)
CCO(5,6)
CC(9,6)
CC(5,6)
M
CC(5,6)
60
100
90
100
65
40
90
60
80
55
2
Cross Sections (πa )
2
Cross Sections (πa0 )
30
70
20
60
10
50
0
40
5
10
15
20
30
20
20
30
40
50
60
70
80
90
100
Energy (eV)
Fig. 10. (Color online) Elastic integral cross section for
positron-potassium scattering: present CCO(9,6) (solid line);
CCO(5,6) (dashed line); CC(9,6) (dash dot); CC(5,6) (dash
double dot); CC(5,6)M [53] ().
5
10
15
20
25
30
35
80
90
40
50
20
10
45
60
40
0
50
70
30
10
CCO(9,6)
CC(9,6)
M
CC(5,6)
70
50
80
0
75
110
0
120
11
10
20
30
40
50
60
70
100
Energy (eV)
Fig. 11. (Color online) Excitation K(4s-4p) integral cross section for positron-potassium scattering: present CCO(9,6) (solid
line); CC(9,6) (dash dot); CC(5,6)M [53] ().
lower energies, just like we saw originally for the electrons.
Similarly, the long-range dipole polarisability continues to
influence the scattering system in this atomic channel, as
evidenced by the steep increase in the cross section magnitude being observed at energies below 10 eV.
The comparison between the present CC(5,6) and the
corresponding Belfast calculation, at 5, 10, 30 and 60 eV,
indicates that the present elastic ICS is larger in magnitude than the CC(5,6)M by about 14%, 4%, 1.6% and
9%, respectively. This reflects the fact that the quality of
the wave functions in both these computations are different, which may be important for positron scattering
especially at lower energies. We should also note that the
CC(9,6) and the CC(5,6) results are in agreement to better than 6%, throughout the entire energy region. This
suggests that the convergence in increasing the number
of atomic states has largely been achieved. However, the
optical potential calculations do show significant effects
on the magnitude of the elastic ICS. This is also observed
in Figure 10, where the CCO(5,6) and CCO(9,6) cross
sections are significantly smaller than their CC counterparts, sometimes by nearly 30%. This is plausible due to
the continuum effects on this transition, whereby the flux
is absorbed into the continuum channels and materializes
in the form of a significant ionization cross section (see
later).
tures of this cross section are also similar to the corresponding electron scattering case with the maximum occurring at about 10–20 eV. In general, the qualitative
shapes of the present models also show good agreement
with the CC(5,6)M with all the calculations converging
above 50 eV. Although not shown in Figure 11, we found
that the present CC(5,6) shows reasonable agreement with
the CC(5,6)M calculation with the differences usually being less than 6.5%. The difference between the CC(5,6)
and the CC(9,6) was also typically less than 4% at all
energies. Note that as the incident energy of the positron
increases, this transition dominates the scattering process.
For example, at 5 eV it accounts for about 31% of the total cross section, which increases to about 65% of the total
cross section at 100 eV. However, the inclusion of the optical potentials to the CC(9,6) calculations has a major
effect on the magnitude of the cross sections. In particular, our CCO(9,6) cross sections are generally smaller in
magnitude than the corresponding CC cross sections. For
example, at 15 eV the CCO(9,6) show a maximum difference with the corresponding CC(9,6) results of about
15%. However, again, all the calculations tend to converge
in magnitude at higher energies.
While not being specifically plotted, the 4s-4p DCS
exhibit characteristics similar to that described earlier for
the electron scattering case. Namely, they exhibit some
angular structure at middle and backward angles and their
respective magnitudes are very strongly forward peaked.
4.2 K(4s-4p) transition
4.3 K(4s-3d) transition
The integral cross section for the resonant excitation 4s4p transition is depicted in Figure 11. The qualitative fea-
The integral cross section for the 4s-3d transition of
the 11-state and 15-state calculations are depicted in
12
The European Physical Journal D
24
50
25
CCO(9,6)
CC(9,6)
M
CC(5,6)
30
45
25
20
20
CCO(9,6)
CC(9,6)
Nan et al (2005)
M
CCO(5,6)
40
20
15
35
15
2
Cross Sections (πa0 )
2
Cross Sections (πa0 )
16
10
12
5
5
10
15
20
30
10
25
20
5
15
0
8
0
5
10
15
10
4
5
0
0
10
20
30
40
50
60
70
80
90
100
Energy (eV)
Fig. 12. (Color online) Excitation K(4s-3d) integral cross section for positron-potassium scattering: present CCO(9,6) (solid
line); CC(9,6) (dash dot); CC(5,6)M [53] ().
Figure 12. We note that this transition is quite significant
in the strength of its excitation at all energies with its contribution to the total cross section being up to 15%. All
the 11-state models (CC(5,6)M and CC(5,6)) are in qualitatively good agreement with each other. A maximum in
the ICS is observed at around 8 eV and each of the calculations tend to converge in value above 15 eV. However,
the CC(9,6) and CCO(9,6) results seem to show differences in qualitative detail, compared to the other models, in the energy region of 6–8 eV (see Fig. 12). The
differences between the CC(9,6) and the CC(5,6) cross
sections are, however, better than 5% throughout the entire energy region. With the inclusion of optical potentials, the magnitude of the cross section at 5 eV for the
CC(9,6) is 12.686 πa20 which is reduced to 9.557 πa20 for
the CCO(9,6).
5 Ps formation cross sections
The Ps-formation cross sections in the Ps(1s), Ps(2s) and
Ps(2p) states are depicted in Figures 13, 14a and 14b respectively, while in Figure 15 the sum of cross section in
the Ps(n = 3) states is also presented. Finally, the present
total Ps-formation cross section, which is Ps(n = 1+2+3),
are shown in Figure 16. All the present CCO(9,6) and
0
0
12
24
36
48
60
72
84
96
Energy (eV)
Fig. 13. (Color online) Ps(1s) formation cross section: present
CCO(9,6) (solid line); CC(9,6) (dash dot); Nan et al. [70] (dotted); CC(5,6)M [53] ().
CC(9,6) results are depicted together with the earlier work
of the Belfast group [CC(5,6)M ]. In addition, we also display the results on Ps formation from Nan et al. [69,70],
who used another variant of the continuum optical potential to incorporate Ps formation but do not take explicitly
into account the channel couplings in the CC expansion
of the total wave function. As for the total Ps-formation
cross section, we have included the only available experimental measurement by Zhou et al. [48] for comparison.
The measured lower and upper bound total Ps-formation
cross sections are abbreviated as Exp-LB and Exp-UB,
respectively.
5.1 Ps(1s) formation
In Figure 13, the Ps(1s) formation cross section are shown.
The present cross sections demonstrate similar qualitative
features with each other and converge in absolute value
above 20 eV. The cross section of Nan et al. generally
lies below the results from all the other models at energies above 3 eV. However, below 3 eV its magnitude
suddenly increases dramatically as the energy further decreases. The structure that is predicted by other models
for energies below 5 eV is also not seen in the calculation
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
12
13
15
CCO(9,6)
CC(9,6)
Nan et al (2005)
M
CCO(5,6)
10
CCO(9,6)
CC(9,6)
Nan et al (2005)
M
CCO(5,6)
15
8
10
6
10
2
Cross Sections (πa0 )
2
Cross Sections (πa0 )
4
2
0
20
5
0
5
5
10
15
20
15
10
5
0
0
0
5
10
15
20
25
30
Energy (eV)
0
10
20
30
40
50
60
70
80
90
100
Energy (eV)
Fig. 14. (Color online) (a) Ps(2s) and (b) Ps(2p) formation
cross section: present CCO(9,6) (solid line); CC(9,6) (dash
dot); Nan et al. [70] (dotted); CC(5,6)M [53] ().
Fig. 15. (Color online) Ps(3s + 3p + 3d) formation cross
section: present CCO(9,6) (solid line); CC(9,6) (dash dot); Nan
et al. [70] (dotted); CC(5,6)M [53] ().
of Nan et al. The difference in the treatment of the optical
potential between the work of Nan et al. and the present
CCO(9,6) does however explain this latter observation.
As expected from our previous descriptions, the CCO
calculations show a reduction in the magnitude of the
Ps(1s) cross sections compared to the CC results, in the
energy region 5–30 eV. As the energy increases, this cross
section diminishes gradually and can soon be neglected.
Therefore, it can be concluded that the effects of the Psformation channel are less important above 30 eV. However, it does accounts for about 8% of the total cross section in the lower energy region.
throughout the entire energy region below 20 eV, except
in the 15-state calculations at 5 eV where the cross section
has increased 1% from 1.689 πa20 to 1.711 πa20 . Overall, all
the models tend to converge in magnitude above 20 eV.
Figure 14b depicts the Ps(2p) formation cross section.
The same conclusions can be drawn here as given above
for the Ps(2s) formation. The cross sections calculated
by Nan et al. [69] are once again, below 10 eV, observed
to be significantly larger than the other calculated cross
sections. Finally, we note that this transition accounts for
about 8% of the total cross section.
7 Ps (n = 3) formation
6 Ps(2s) and Ps(2p) formation
The Ps(2s) and Ps(2p) formation cross sections for the
CC(9,6) and CCO(9,6) models are depicted separately in
Figures 14a and 14b respectively. Cross sections from the
CC(5,6)M and Nan et al. [69] computations are also included for a comparative study. For the Ps(2s) transition,
the qualitative shapes of all the models are in reasonable
agreement with each other except for the results of Nan
et al. [69]. Significant differences are in particular observed
at energies below 10 eV. With the inclusion of the optical
potentials, the Ps(2s) cross sections are reduced further
In Figure 15, the Ps (n = 3) cross sections are presented.
It’s apparent from this figure that the present Ps (n =
3) formation cross section are also shown together with
the result of Nan et al. [70] and with results from the
Belfast group. The cross section from our CC(9,6) and
CCO(9,6) calculations show respective maxima at around
8 eV, while the cross section of Nan et al. [70] increases
sharply and reaches a maximum of 13 πa20 at 4.2 eV. The
Ps (n = 3) formation cross section (see Fig. 15) plays an
important role for the total Ps-formation cross section at
higher energies. For instance at 10 eV it accounts for about
14
The European Physical Journal D
70
70
CCO(9,6)
CCO(5,6)
CC(9,6)
CC(5,6)
Nan et al (2005)
M
CC(5,6)
Exp - LB
EXP - UB
60
60
50
16
14
40
12
2
Cross Sections (πa0 )
30
2
Cross Sections (πa0 )
50
Positron-Potassium
CCO(9,6)
+
COPM
Electron-Potassium
CCO9
COPM
CCC
MKKP
ZA
40
20
30
10
10
8
0
20
0
5
10
15
20
6
4
10
2
0
0
20
40
60
80
100
Energy (eV)
Fig. 16. (Color online) Total Ps formation cross section for
positron-potassium scattering: present CCO(9,6) (solid line);
CCO(5,6) (dashed line); CC(9,6) (dash dot); CC(5,6) (dash
double dot); CC(5,6)M [53] (); Nan et al. [70] (· · · ·); expt
data [48] - LB (♦), UB ().
40% of the total Ps-formation cross section, while at 20 eV
it accounts for about 54% of the total Ps-formation cross
section. The Ps (n = 3) formation cross section can be,
however, largely neglected above 30 eV.
8 Total Ps-formation cross section
The present total Ps-formation cross section is compared
in Figure 16 with the only available experimental data of
Zhou et al. [48]. Other theoretical calculations are also
included in the comparison. The present CC(9,6) and
CCO(9,6) calculations are in fair agreement with each
other, and with the CC(5,6)M , but all are consistently
lower than the experimental lower bound. As expected, as
the energy increases, all the models predict cross sections
that descend monotonically in value until they converge
above 30 eV. Overall, the present total Ps-formation cross
section accounts for about 22% and 17% of the grand total
cross section at 5 and 10 eV, respectively.
9 Direct ionization cross sections
In Figure 17, the direct ionization cross sections for
positron-K scattering, calculated using the CCO(9,6)
0
10
Energy (eV)
100
Fig. 17. Ionisation cross section for electron – and positronpotassium scattering: positron – present CCO(9,6) (♦);
present COPM+ (solid line); electron – present CCO9 (×);
present COPM− (dash-dot); CCC [6] (dash-double dot);
MKKP [28,29] (◦); ZA [17] ().
are displayed. Due to the unavailability of experimental ionisation cross sections in positron-potassium
scattering, we have included for comparison, the corresponding experimental ionisation cross sections for
electron-potassium scattering. Those electron data were
measured by Zapesochnyi and Aleksakhin [17] (hereafter
abbreviated as ZA), McFarland and Kinney [28] and
Korchevoi and Przonski [29] (hereafter abbreviated as
MKKP). Note that the MKKP ionisation cross sections
were recommended in Zecca et al. [18]. Besides that, the
present CCO9 calculations for the electron-potassium case
are also depicted for comparative purposes. For electronpotassium scattering, the current ionisation cross sections
calculated using the CCO9 agree better with those from
the COPM− . The CCO models generally give larger cross
sections than those from the other calculations as well as
the measurements, except at energies below 7 eV. The discrepancy between the present CCO calculations, with the
measurements of ZA is, however, still observable at the
energy region 8–28 eV.
In the positron-potassium case, all the present CCO
calculations predict larger ionisation cross sections than
from the COPM+ model except above 15 eV. Note that
the ionisation cross sections from the CCO(9,6) results is
K. Ratnavelu and W.E. Ong: Electron and positron scattering from atomic potassium
200
180
190
160
170
150
160
140
150
130
140
2
Cross Sections (πa0 )
earlier, these corrections might be significant and would
serve to increase the magnitude of the experimental data,
perhaps significantly.
We must note that the inclusion of optical potentials
in the present CC models has an appreciable effect on the
cross sections (this is not depicted as it clutters Fig. 18).
The CCO cross sections are found to be smaller in absolute value than the CC with the differences being about
1–11%. For CCO(9,6), the cross sections are exceptionally
larger than their corresponding CC cross sections at 5 eV.
The total cross sections tend to increases as the number
of states increases in the calculation. However, all the calculations as well as the measurement converge gradually
above 50 eV.
CCO(9,6)
M
CC(5,6)
Kwan et al (1991)
Parikh et al (1993)
170
180
120
130
110
120
100
110
90
100
80
90
70
80
0
5
10
15
20
70
60
50
40
30
20
15
0
10
20
30
40
50
60
70
80
90
100
Energy (eV)
Fig. 18. Total cross sections for positron-potassium scattering:
positron – present CCO(9,6) (solid line); CC(5,6)M [53] ();
Kwan et al. [26] (); Parikh et al. [27] ().
about 20% larger than the COPM+ , at the maximum of
12.6 πa20 .
As the energy increases, all the present CCO calculations tend to converge to the COPM model, for both the
positron and electron cases at about 10 times the ionization threshold. We believe it is very important that the
direct ionisation cross sections be measured in the near
future.
10 Total cross section
The total cross section for positron-potassium scattering
is shown in Figure 18. The present calculated CCO(9,6)
are compared to the available theoretical calculations that
incorporate the Ps channel. We have also included the
experimental measurements of the Detroit Group [26,27]
for a further comparative study.
As can be observed, the qualitative shapes of all the
theoretical calculations tend to agree well with the measurements of Kwan et al. and Parikh et al. At energies below 10 eV, however, all the theoretical total cross sections
seem to overestimate the measurements by Parikh et al.
However, this is not surprising as none of the Detroit data,
as noted previously, has been corrected for forward angle
scattering effects. Given the strongly forward peaked nature of the elastic differential cross sections, as observed
11 Conclusion
The present work has provided a comprehensive study of
electron as well as positron scattering from atomic potassium by using an optical potential method. For electronpotassium scattering, the CCOM method has been implemented by performing the CCO9 and CCO6 calculations
in the energy region of 4–100 eV. It is ambitious to extent the CCO calculations to low energy region since the
COPM is a high-energy approximation. However, we observed that the general features predicted in that region
are acceptable except for the weaker transitions that are
plagued with many structures. For positron-potassium
scattering, the CCOM method was implemented in the
CCO(9,6) and CCO(5,6) calculations at the energies ranging from 5–100 eV.
For electron-potassium scattering, we have also presented the cross sections for total, elastic and K(4s-4p)
transition. Differential cross section for the elastic and the
K(4s-4p) transition as well as the alignment and orientation parameter are also presented at selected energies.
Generally, good agreement is found with experiment measurements where available. We are also encouraged by the
excellent agreement with the CCC calculations on Stokes
parameters at energy 54.42 eV and above.
In the case of positron-potassium scattering, we have
generally confirmed the works of McAlinden et al. [53] in
the CC(5,6) calculation. Any noticeable differences may
be due to the different numerical treatment used in each
respective calculation. The present largest CCO(9,6) calculations has shown that the use of the optical potentials
in the coupled-channel method is comparable to a certain extent with other theoretical calculations as well as
experimental measurements. For the total cross section,
all the theoretical models (including the present calculations) predicted larger cross section than the experimental
measurement. The effect of increasing the number of state
(in both atomic and Ps channel) in the calculation is observable especially around 5–15 eV. Thus, contributions
of the higher excited states are important and should not
be ignored. More refined experimental measurements of
the total cross section would be most welcome to test the
theories.
16
The European Physical Journal D
The present results support the conclusion of
McAlinden et al. [53] that the Ps formation plays a significant role in the positron-potassium scattering below
20 eV. Cross sections for the present Ps formation in the
Ps(1s), Ps(2s), Ps(2p), Ps(n = 2) and Ps(n = 3) are
shown and in good accord with the work of McAlinden
et al. [53] but differ quite significantly with Nan et al. [70]
at energy below 10 eV. The present total Ps formation
also observed to fall outside the experimental bounds of
Zhou et al. [48]. Since the present work are still not definitive, it would be necessary to include higher Ps states in
calculation.
In the calculation of the total ionization cross section, discrepancies between the CCO calculations and the
COPM models can be observed in the low and intermediate energy region for both electron- and positron- potassium scattering. At energy about 10 times the ionization
threshold, all the CCO calculations tend to converge to
their respective COPM models. Lastly, differential cross
section (DCS) for positron-potassium scattering were also
calculated (but not shown) for the elastic, K(4s-4p),
Ps(1s) and Ps(n = 2) formation at selected energies.
Among those DCSs presented, comparison could be made
only for the Ps(1s) formation where the calculated DCS
by Guha and Mandal [50] was reported at 5 and 10 eV.
However, the present DCS showed significant difference
with the DCS by Guha and Mandal [50] at both energies.
It would be of great interest if the DCS for the positronpotassium scattering are measured in the near future. This
will provide a more discriminating test for the present theory.
KR wishes to thank the University of Malaya Research Grant
(UMRG 089/10AFR) for partial funding of this project. The
initial preparation of this work was done during a research trip
undertaken on the sponsorship of the Australia-Malaysia Institute and Center for Matter-Anti-Matter Studies at Flinders
University (May–June 2010). KR acknowledges the critical
reading of this paper by Prof Mike Brunger of Flinders University and the technical support of Chin Jia Hou of the University
of Malaya.
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