Tight determination of the positron–molecular-nitrogen scattering length and Bethe moment for a decisive sum-rule test of positron–N2 scattering theory

When the total cross sections for positron scattering from molecular nitrogen (measured from about 20 eV to 3 keV by Dutton, Evans and Mansour in 1987) are tested against the first Born and Bethe–Born descriptions and against the forward-dispersion ('scattering-length') sum rule, the test can only be a consistency check, because the two constants it involves are poorly known: the Bethe moment $M_{\mathrm{tot}}^2$ (extracted from the high-energy Bethe plot with a slope error of nearly 50%) and the zero-energy scattering length $A_0$ (obtained as $-(2.65\pm0.33)\,a_0$ from the sum rule, with an error budget that excludes the uncertainty of the zero-angle Born amplitude $f_{\mathrm{B}}(0)$ and rests on a first-Born extrapolation beyond the measured range). The problem is to determine these constants independently and tightly enough to make the Bethe–Born and sum-rule tests of positron scattering theory in molecular nitrogen decisive. The $M_{\mathrm{tot}}^2$ side is now largely supplied by independent photoabsorption oscillator-strength data, but the scattering-length side remains open; modern determinations of the positron–N2 scattering length still range from about $-2.7$ to $-9.3\,a_0$, and no decisive re-test of the sum rule has been performed.

Background

A positron ($e^+$) is the antiparticle of the electron, and low-energy positron scattering by molecules is a benchmark problem of atomic collision physics: the projectile is distinguishable from the target electrons (no exchange), but the long-range induced polarization attraction and the virtual positronium (the bound $e^+e^-$ state) channel make the low-energy cross sections sensitive to correlation effects that are hard to compute. The experimentally accessible quantity in a transmission ('beam attenuation') experiment is the total scattering cross section $Q_T(E)$, the sum of all elastic and inelastic partial cross sections at incident energy $E$.

For fast charged projectiles, Bethe theory predicts that the total inelastic cross section obeys the asymptotic (Bethe–Born) form $Q_{TI}=\frac{4\pi a_0^2 Z^2}{E/R}M_{\mathrm{tot}}^2\ln\left(\frac{4c_{\mathrm{tot}}E}{R}\right)$, where $a_0$ is the Bohr radius, $R$ the Rydberg energy, $Z$ the projectile charge number, and $M_{\mathrm{tot}}^2$ a property of the target alone: the square of the total dipole moment, i.e. the $S(-1)$-type moment of the target's dipole oscillator strength distribution. On a Bethe plot of $Q_{TI}E/(4\pi a_0^2 Z^2 R)$ against $\ln(4E/R)$ the data should therefore approach a straight line whose slope is $M_{\mathrm{tot}}^2$. Inokuti, Kim and Platzman (1967) established the sum-rule foundations of this Bethe cross-section description.

At zero projectile energy, elastic scattering by a short-range-plus-polarization potential is characterized by the scattering length $A_0$ (the zero-energy limit of the elastic amplitude; the total cross section tends to $4\pi A_0^2$ for s-wave scattering). For positron scattering from molecular nitrogen, a forward dispersion relation at zero energy provides an exact 'sum rule' linking $A_0$, the zero-angle first-Born direct scattering amplitude $f_{\mathrm{B}}(0)$, and the integral of the total cross section over the projectile momentum $k$:

$$-A_0-f_{\mathrm{B}}(0)=\int_0^{\infty}\frac{1}{2\pi}Q_T(k)\,dk .$$

In 1987, absolute total cross sections for positrons in N2 at energies up to 3 keV were measured (Dutton, Evans and Mansour 1987), and a companion analysis (Evans and Mansour 1987) combined these data with the lower-energy (0.5–30 eV) data of Hoffman et al. (1982) to test both descriptions. Subtracting a Thomas–Fermi first-Born elastic estimate from the measured totals, they found the inelastic data above 1.75 keV to follow the Bethe-plot straight line with slope 6.56, interpreted as $M_{\mathrm{tot}}^2$, but with a standard error of nearly 50%; and, evaluating the sum-rule integral (10.96 $a_0$, of which the unmeasured high-energy tail above 1.75 keV, taken from the first-Born asymptotic form, contributes about 30%) with $f_{\mathrm{B}}(0)=-8.31\,a_0$ derived from the Mott–Massey atomic zero-angle intensity, they obtained $A_0=-(2.65\pm0.33)\,a_0$. Both results were only consistency checks: the slope error was dominated by counting statistics in the 1.75–3 keV region and by the elastic subtraction, the $A_0$ error budget excluded the uncertainty of $f_{\mathrm{B}}(0)$ itself, and the previously published values they compared against were widely spread — $M_{\mathrm{tot}}^2$ from 4.74 (oscillator-strength integration by Zeiss et al.) to 5.38–5.59 (magnetic susceptibility), and $A_0$ from $-5.46\,a_0$ (Darewych and Baille 1974) and $-2.82\,a_0$ (Gillespie and Thompson 1975) to $-1.26\,a_0$ (Darewych 1982) and about $-2\,a_0$ from positron lifetime data. Their stated conclusion was that tighter independent determinations of these constants would still be needed for a more decisive test of positron scattering theory in molecular nitrogen.

Problem Statement

Determine, independently and with controlled uncertainties, the constants that enter the decisive test of positron scattering theory in molecular nitrogen: (i) the zero-energy scattering length $A_0$ of the positron–N2 system (equivalently, the sub-eV limit of its total cross section) together with the zero-angle first-Born amplitude $f_{\mathrm{B}}(0)$ that fixes the sum-rule constant, using (ii) the Bethe moment $M_{\mathrm{tot}}^2$ of N2 as independently determined from photoabsorption oscillator-strength data (Zeiss et al. 1977; Chan et al. 1993). Then establish whether the measured positron–N2 total cross sections follow the Bethe–Born high-energy behavior against the independent spectroscopic $M_{\mathrm{tot}}^2$ at the quoted uncertainties, and whether they satisfy, beyond the consistency-level agreement achieved in 1987, the forward-dispersion (scattering-length) sum rule $-A_0-f_{\mathrm{B}}(0)=\int_0^{\infty}\frac{1}{2\pi}Q_T(k)\,dk$. A finding of non-discrimination in that Bethe-behavior comparison is an acceptable outcome. On the sum-rule side, $A_0$ and $f_{\mathrm{B}}(0)$ must be determined with controlled, quantified uncertainties — the sum-rule error budget must include the uncertainty of $f_{\mathrm{B}}(0)$ — sufficient for the test to be decisive rather than consistency-level (i.e. the comparison must decide, at the quoted uncertainties, whether the sum rule holds, rather than leave its residual unresolved within those uncertainties).

References

1. J. Dutton, C. J. Evans and H. L. Mansour, 'The experimental determination of the total scattering cross section of molecular nitrogen for positrons at energies up to 3 keV', Journal of Physics B: Atomic and Molecular Physics 20, 2607–2618 (1987), DOI 10.1088/0022-3700/20/11/028, https://doi.org/10.1088/0022-3700/20/11/028
2. C. J. Evans and H. L. Mansour, 'The total scattering cross section of molecular nitrogen for positrons at energies up to 3 keV: comparison with theory', Journal of Physics B: Atomic and Molecular Physics 20, 5925–5930 (1987), DOI 10.1088/0022-3700/20/21/034, https://doi.org/10.1088/0022-3700/20/21/034
3. K. R. Hoffman, M. S. Dababneh, Y.-F. Hsieh, W. E. Kauppila, V. Pol, J. H. Smart and T. S. Stein, 'Total-cross-section measurements for positrons and electrons colliding with H2, N2, and CO2', Physical Review A 25, 1393–1403 (1982), DOI 10.1103/physreva.25.1393, https://doi.org/10.1103/physreva.25.1393
4. Mitio Inokuti, Yong-Ki Kim and Robert L. Platzman, 'Total Cross Sections for Inelastic Scattering of Charged Particles by Atoms and Molecules. I. A Sum Rule for the Bethe Cross Sections and Its Application to the Helium Atom', Physical Review 164, 55–61 (1967), DOI 10.1103/PhysRev.164.55, https://doi.org/10.1103/PhysRev.164.55
5. J. W. Darewych and P. Baille, 'Interaction of low-energy positrons with molecular nitrogen', Journal of Physics B: Atomic and Molecular Physics 7, L1–L4 (1974), DOI 10.1088/0022-3700/7/1/001, https://doi.org/10.1088/0022-3700/7/1/001
6. E. S. Gillespie and D. G. Thompson, 'Positron scattering by molecular nitrogen', Journal of Physics B: Atomic and Molecular Physics 8, 2858–2868 (1975), DOI 10.1088/0022-3700/8/17/018, https://doi.org/10.1088/0022-3700/8/17/018
7. J. W. Darewych, 'Elastic scattering and annihilation of low-energy positrons by molecular nitrogen', Journal of Physics B: Atomic and Molecular Physics 15, L415–L419 (1982), DOI 10.1088/0022-3700/15/12/003, https://doi.org/10.1088/0022-3700/15/12/003
8. G. D. Zeiss, W. J. Meath, J. C. F. MacDonald and D. J. Dawson, 'Dipole oscillator strength distributions, sums, and some related properties for Li, N, O, H2, N2, O2, NH3, H2O, NO, and N2O', Canadian Journal of Physics 55, 2080–2100 (1977), DOI 10.1139/p77-252, https://doi.org/10.1139/p77-252
9. W. F. Chan, G. Cooper, R. N. S. Sodhi and C. E. Brion, 'Absolute optical oscillator strengths for discrete and continuum photoabsorption of molecular nitrogen (11–200 eV)', Chemical Physics 170, 81–97 (1993), DOI 10.1016/0301-0104(93)80095-q, https://doi.org/10.1016/0301-0104(93)80095-q
10. G. P. Karwasz, D. Pliszka and R. S. Brusa, 'Total cross sections for positron scattering in argon, nitrogen and hydrogen below 20 eV', Nuclear Instruments and Methods in Physics Research Section B 247, 68–74 (2006), DOI 10.1016/j.nimb.2006.01.065, https://doi.org/10.1016/j.nimb.2006.01.065
11. Zbigniew Idziaszek and Grzegorz Karwasz, 'Applicability of modified effective-range theory to positron-atom and positron-molecule scattering', Physical Review A 73, 064701 (2006), DOI 10.1103/PhysRevA.73.064701, https://doi.org/10.1103/PhysRevA.73.064701
12. A. Zecca, L. Chiari, A. Sarkar and M. J. Brunger, 'Positron scattering from the isoelectronic molecules N2, CO and C2H2', New Journal of Physics 13, 115001 (2011), DOI 10.1088/1367-2630/13/11/115001, https://doi.org/10.1088/1367-2630/13/11/115001
13. Kamil Fedus, 'A rigid sphere approach to positron elastic scattering by noble gases, molecular hydrogen, nitrogen and methane', The European Physical Journal D 70, 261 (2016), DOI 10.1140/EPJD/E2016-70452-6, https://doi.org/10.1140/EPJD/E2016-70452-6
14. Luis A. Poveda, Denise Assafrão, Jenifer G. Pinheiro and José R. Mohallem, 'Close-coupling scattering cross sections and a model for positron cooling in a buffer gas of molecular nitrogen', Physical Review A 100, 062706 (2019), DOI 10.1103/PhysRevA.100.062706, https://doi.org/10.1103/PhysRevA.100.062706
15. Lilian Ellis-Gibbings, Francisco Blanco and Gustavo García, 'Positron interactions with nitrogen and oxygen molecules: elastic, inelastic and total cross sections', The European Physical Journal D 73 (2019), DOI 10.1140/epjd/e2019-100459-x, https://doi.org/10.1140/epjd/e2019-100459-x
16. Mahmudul H. Khandker, Nazifa T. Arony, A. K. F. Haque, M. Maaza, M. Masum Billah and M. Alfaz Uddin, 'Scattering of e± from N2 in the energy range 1 eV–10 keV', Molecular Physics 118, e1699183 (2019), DOI 10.1080/00268976.2019.1699183, https://doi.org/10.1080/00268976.2019.1699183
17. Fabio Carelli, Kamil Fedus and Grzegorz Karwasz, 'Total Cross Sections for Electron and Positron Scattering on Molecules: In Search of the Dispersion Relation', Atoms 9, 97 (2021), DOI 10.3390/atoms9040097, https://doi.org/10.3390/atoms9040097
18. A. Zecca, L. Chiari, E. Trainotti, D. V. Fursa, I. Bray, A. Sarkar, S. Chattopadhyay, K. Ratnavelu and M. J. Brunger, 'Positron scattering from argon: total cross sections and the scattering length', Journal of Physics B: Atomic, Molecular and Optical Physics 45, 015203 (2012), DOI 10.1088/0953-4075/45/1/015203, https://doi.org/10.1088/0953-4075/45/1/015203
19. Antonio Zecca, Luca Chiari, Emanuele Trainotti and Michael J. Brunger, 'Very low-energy total cross sections and the experimental scattering length for the positron–xenon system', Journal of Physics B: Atomic, Molecular and Optical Physics 45, 085203 (2012), DOI 10.1088/0953-4075/45/8/085203, https://doi.org/10.1088/0953-4075/45/8/085203
20. Kamil Fedus and Grzegorz Karwasz, 'Virtual and bound states in low-energy positron scattering by atoms and molecules via modified effective range theory', Journal of Physics Communications 8, 045005 (2024), DOI 10.1088/2399-6528/ad4001, https://doi.org/10.1088/2399-6528/ad4001