48
POLARIZATION AND FORWARD SCATTERING EFFECTS IN LOW-ENERGY
POSITRON INTERACTIONS WITH H₂
Malikov Murod Rasulovich
Associate professor at the Alfraganus University
Email address: malikovmurod55@mail.ru
Orcid ID: 0009-0008-1406-8977
https://doi.org/10.5281/zenodo.14890877
Abstract.
Positron physical-chemistry has been a significant area of scientific
investigation in recent decades. However, low-energy positron scattering by atoms and
molecules still presents several open questions, particularly regarding the effects of low-angle
scattering on measured cross sections and the role of target polarization in the comparison
between theoretical and experimental results. In this study, we examine low-energy positron
collisions with H₂ molecules, focusing on the convergence of the polarization contribution in
the scattering potential.
The interaction between the positron and the molecule is represented by a model
potential, which combines a free-electron gas correlation term with an asymptotic polarization
potential derived from perturbation theory. Specifically, we analyze how polarization effects
beyond second-order perturbation influence scattering observables. Our findings indicate that
a model incorporating up to the quadrupole polarization contribution shows improved
agreement with recent experimental data when corrected for forward scattering effects, as
these measurements were obtained using a transmission beam technique.
We also examine angular distributions by comparing our results with available folded
differential cross-section measurements. A simple correction scheme is proposed for
experimental folded differential cross sections at energies below 1 eV, which we argue aligns
well with the quadrupole polarization model. Finally, a comparison between our phase shifts
and scattering lengths with recent many-div
ab initio
calculations—including virtual
positronium effects—suggests that these effects are inherently accounted for within the
adopted model correlation potential.
Keywords:
molecular hydrogen, positron, elastic scattering, polarization.
Introduction
Positron scattering by atoms and molecules is widely recognized as a challenging
problem. Despite its many applications in fields such as material science, medicine, and
astrophysics, the fundamental nature of positron interactions with atomic and molecular
targets remains a subject of debate. Unlike electron scattering, positron scattering requires
careful treatment of correlation and polarization effects, as the absence of exchange
interactions makes the balance between electrostatic and distortion effects crucial for
determining scattering cross sections. One such correlation effect, virtual positronium
formation, plays a complex role in scattering dynamics that is not yet fully understood.
The case of the hydrogen molecule is no exception. Numerous experimental and
theoretical studies on low-energy positron scattering with H₂ have been conducted. However,
even for elastic scattering, the agreement between theory and experiment remains limited. One
major source of discrepancy is the limited angular resolution of transmission-based
spectrometers, which count positrons elastically scattered below a certain angle as unscattered,
leading to an underestimation of the total cross section (TCS). This issue can be addressed by
49
incorporating reliable differential cross-section data, either from experimental measurements
or theoretical calculations. However, experimental data on this subject remain scarce, and the
choice of the most suitable theoretical approach remains a challenge. While some theoretical
methods offer more convincing results than others, achieving a systematic theoretical
description that aligns with experimental data remains an ongoing effort and a significant open
problem.
Among the various theoretical approaches for positron-molecule scattering, two
predominant methodologies exist: ab initio
many-div calculations and model potential
approaches. Each method has its own advantages. While an ab initio approach aims to compute
a many-div wave function for the positron-molecule system with minimal approximations,
model potential approaches rely on simplified scattering potentials that are easier to interpret
and adjust to match experimental cross sections.
For positron scattering with H₂, there is a wealth of theoretical and experimental data,
though still not sufficient for a complete understanding of the problem. On the theoretical side,
both ab initio
calculations and model potential approximations have been employed. The
agreement between theoretical predictions and experimental results continues to evolve as
new experimental data become available. For instance, when Zecca et al. published their
experimental results for positron energies between 1 eV and approximately 10 eV, they
significantly influenced the theoretical understanding of positron scattering in this energy
range. Prior to this, no data were available for positron energies below 1 eV, and their study
remains the only one providing data below 0.5 eV—an energy region critical for theoretical
comparisons.
The data from Zecca et al. were obtained using a transmission-based positron
spectrometer, which, as mentioned earlier, tends to underestimate cross sections due to the
exclusion of elastically scattered positrons below a certain angle. These cross sections were
later corrected for low-angle scattering effects using a full ab initio
treatment. However, the
experimental data were not enhanced as much as expected, likely due to incomplete treatment
of correlation and polarization effects, as indicated by the comparison of their folded
differential cross sections (FDCS) with available measurements. This discrepancy becomes
even more intriguing when compared with recent many-div calculations that incorporate full
correlation effects, yielding significantly higher cross sections.
Within this context, we propose to study positron scattering by H₂ using a model potential
approach, with the aim of understanding how the inclusion of correlation and polarization
effects influences the calculated cross sections and the forward scattering corrections for
available experimental data. Following our previous work, we will apply the positron
correlation polarization (PCOP) approach
,
modifying the polarization term to include not only
second-order perturbation theory contributions but also relevant hyperpolarizabilities. This
study will assess how these additional effects impact integral and differential cross sections.
Theoretical Approach
The scattering model used in this study is based on the static plus correlation polarization
(SCP) approximation, defined as:
Accurately describing the positron-electron interaction requires a reasonable
representation of the occupied molecular orbitals ϕi(ri)\phi_i(\mathbf{r}_i)ϕi(ri) in the given
molecular state (in this case, the ground state). It is well known that the Hartree-Fock (HF)
50
representation of molecular orbitals provides a sufficiently accurate description for this
purpose, particularly within a model potential approach, as electronic correlation effects can
be incorporated through an external potential.
For the correlation and polarization term, we adopt the approach proposed by Jain and
This matching scheme was originally introduced in the context of electron scattering and
was justified based on its effectiveness in reproducing accurate phase shifts. While it lacks
direct physical interpretation, it has been observed that the cut radius correlates with the
spatial extent of the electronic cloud. This suggests that correlation effects in the outer
molecular region, such as virtual positronium formation, predominantly occur near this point.
Numerical Details
The polarization potential defined in Equation (7) depends on the molecular
polarizabilities. In this study, we utilized the CISD/D6Z values determined by Miliordos and
Hunt. Since these polarizabilities act as parameters in our calculations, we selected values that
provide the most accurate long-range polarization function. Additionally, for convenience, we
chose tabulated data corresponding to equilibrium geometry. It appears that the quadrupole
polarizability value used by Frighetto et al. contains an error by a factor of 2. Equation (7) aligns
with Dalgarno’s definition; however, the relationship between Dalgarno and Buckingham
quadrupole polarizabilities is given by αq=2C0\alpha_q = 2C_0αq=2C0, as noted by
Buckingham, Thakkar and Lupinetti, and Maroulis and Thakkar. This discrepancy is significant
and should be carefully considered when comparing results and analyzing potentials.
All scattering calculations were performed using the MCF method, employing 800 radial
grid points spanning from 0.002 to 125.35 a.u. Special attention was given to the molecular
region when distributing these points, and we determined that 295 grid points within 10.0 a.u.
are sufficient to ensure that the electronic wave function normalizes within
1.0000±1×10−41.0000 \pm 1 \times 10^{-4}1.0000±1×10−4. Additionally, this grid setup
adequately supports the electrostatic potential expansion described in Equation (3), as the
obtained SCF quadrupole moment compares well with reference values.
Since the differential cross sections calculated in this study cover the full angular range
0
∘
0^\circ0
∘
to 180
∘
180^\circ180
∘
for all considered energies, we report folded differential
cross sections (FDCS) for direct comparison with experimental data from Sullivan et al. and
Machacek
et al. These experimental DCS measurements are mirrored and summed around
Experimental total cross-section (TCS) data are typically uncorrected for forward-angle
scattering, as is the case for the TCS measurements by Zecca et al. We compare these results
with our present calculations, along with the data from Machacek et al., both of which were
corrected for forward-scattering effects using theoretical data from Zammit et al. However,
since the measurements from Zecca et al. extend to very low energies (as low as 0.1 eV), and
our theoretical approach differs fundamentally from that of Zammit et al., we opted to use our
own calculated cross sections to correct this dataset. Given that the Zecca et al. cross-section
data extends to such low energies, these corrections may provide valuable insights into the
positron-H₂ scattering system, particularly in analyzing properties such as the scattering
length, as previously explored by Zhang et al. The conclusions drawn by Zhang were later
questioned by Brunger et al., so further discussion of these data may help clarify aspects of this
ongoing debate.
Conclusion
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In this study, we examined the scattering of low-energy positrons by H₂ molecules using
a polarization correlation model (PCOP) and analyzed the effects of second, third, and fourth-
order perturbation theory polarization. An evaluation of the scattering potential indicates that
including polarization up to fourth order is essential for achieving convergence. However, the
calculated cross sections are larger than the most recent many-div ab initio calculations,
which explicitly incorporate virtual positronium effects. The agreement between our PQ model
and the results reported by Rawlins et al.
suggests that the overcorrelation introduced by the
PCOP functional near the van der Waals radius effectively models virtual positronium
formation.
Particular emphasis was placed on correcting for forward scattering effects
in the
total
cross-section (TCS) measurements
of
Zecca et al.
,
as well as on comparing our results with the
folded differential cross-section (FDCS) measurements of Sullivan et al. and Machacek et al.
Since the FDCS measurement technique is influenced by the angular resolution of the
spectrometer, we proposed a correction to the magnitude of the experimental FDCS at 0.5 eV,
based on the direct correction of the TCS data from Machacek et al. for forward scattering
effects, as well as an estimated correction for the TCS data from Zecca et al. Despite the
simplicity of the correction methodology, the adjusted FDCS aligns quantitatively with both the
PQ model and the modified effective range theory (MERT) model, which was extracted from
forward-corrected data from Machacek et al. This consistency suggests that the correction is
both necessary and reliable
.
At other energies, comparison of our results with the measured FDCS from Machacek et
al. indicates that including anisotropic terms in the polarization interaction is essential for the
calculated integral and differential cross sections to reach the correct magnitude
.
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