Article pubs.acs.org/JPCA
Cross Sections for Positron Impact with 2,2,4-Trimethylpentane Luca Chiari,† Antonio Zecca,‡ Francisco Blanco,§ Gustavo García,∥ Michael V. Perkins,⊥ Stephen J. Buckman,○,∇ and Michael J. Brunger*,†,∇ †
ARC Centre of Excellence for Antimatter−Matter Studies, School of Chemical and Physical Sciences, Flinders University, Adelaide, SA 5001, Australia ‡ Department of Physics, University of Trento, Via Sommarive 14, I-38123 Povo, Trento, Italy § Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain ∥ Instituto de Física Fundamental, Consejo Superior de Investigationes Científicas (CSIC), Serrano 113-bis, E-28006 Madrid, Spain ⊥ School of Chemical and Physical Sciences, Flinders University, Adelaide, SA 5001, Australia ○ ARC Centre of Excellence for Antimatter−Matter Studies, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia ∇ Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia ABSTRACT: 2,2,4-Trimethylpentane (C8H18), a hydrocarbon produced all over the world on a large scale in the processing of crude oil, has long been known and used in the energy sector. It has also recently attracted the attention of the radiation physics and chemistry community, owing to its applications in medical imaging techniques. Charged-particle interactions with this species unfortunately remain mostly unknown. In this study, we report on measured total cross sections for positron scattering from 2,2,4-trimethylpentane in the energy range from 0.12 to 50 eV. We also present calculations of the total cross sections, elastic integral and differential cross sections, positronium formation cross sections, and inelastic integral cross sections at energies from 1 to 1000 eV using the independent atom model with screening corrected additivity rule. A knowledge of those scattering cross sections might, through simulation models, help to improve the accuracy of current radiation detection devices and hence provide better estimates of the extent of any charged-particle-induced damage in biomolecular systems.
1. INTRODUCTION
referred to as isooctane. It is commonly found in petrol and, therefore, is of great interest in the petrochemical industry, which, in turn, produces large amounts of it every year.1 TMP has been assigned the standard 100 point on the octane rating scale. This high octane rating is a desirable property of a fuel for an internal combustion engine,1 and for TMP, it is related to its highly branched structure compared with the straight chain hydrocarbon. Given the above-mentioned properties, it is not surprising that TMP has attracted the attention of the broader atomic and molecular physics community and, in particular, its antimatter branch. In that respect, we note some studies on positron mobility2 and positron annihilation spectroscopy,3−5 in either pure TMP or solutions, that aimed at investigating the structure and properties of organic fluids. The mechanisms of positronium formation6 and annihilation3,7,8 in liquid hydrocarbons were also examined using TMP as a target. TMP does not offer itself just to fundamental studies, as it has also become very useful in biomedical applications. In fact, more recently it
2,2,4-Trimethylpentane (TMP) is a highly branched hydrocarbon with the chemical formula C8H18 (see Figure 1). It is the most important 8-carbon hydrocarbon and, in fact, is often
Special Issue: Franco Gianturco Festschrift Figure 1. Schematic diagram of the 2,2,4-trimethylpentane molecule. Carbon atoms are indicated in black, while the hydrogen atoms have a white color. © 2014 American Chemical Society
Received: March 17, 2014 Revised: March 31, 2014 Published: March 31, 2014 6466
dx.doi.org/10.1021/jp502632m | J. Phys. Chem. A 2014, 118, 6466−6472
The Journal of Physical Chemistry A
Article
The TCS of interest is determined by performing attenuation measurements and applying the Beer−Lambert law (eq 1 in Zecca et al.15). Specifically, the transmitted intensity of the positron beam is measured both in the presence and in the absence of the target vapor in the scattering chamber at each given incident energy. Measurements of the target pressure and temperature within the cell are also carried out at the same time, while the geometrical length of the scattering region is well-known. The transmittance of the positron beam is maintained at values greater than 0.7 when the target vapor is present in the chamber, so as to minimize multiple scattering events. Throughout the measurements we used a high-purity (99.8%) TMP source purchased from Sigma-Aldrich. Note that, each time we begin an experiment on a new target, we carry out preliminary measurements in order to validate our experimental techniques and procedures. Targets whose positron scattering TCSs are well-known, such as the noble gases,20−22 are employed for this purpose, although molecular nitrogen15 is also often used as a reference gas. The zero of the positron energy scale and the beam energy distribution are determined through a retarding potential analysis (RPA) of the incident beam without the target vapor in the chamber.23 In this case the error on the energy scale amounts to ∼±0.05 eV, whereas the energy resolution is found to be ∼0.25 eV (fwhm). Because of this finite energy resolution, the cross sections we measure are actually convoluted over the beam energy distribution. However, we anticipate that this effect might be largely insignificant at most positron energies and become important only below ∼0.5 eV, as the incident energy is comparable to the beam energy width. The measured data were corrected for some inevitable instrumental effects before being used to determine the TCSs. First, the geometrical length of the interaction region (L = 22.1 ± 0.1 mm) was corrected for the gyration of the positrons in the magnetic field.24 That correction increased L by 5.5% (with B ≈ 11 G) at all incident energies, except at E ≥ 37.5 eV when the magnetic field was reduced to B ≈ 4 G, and hence, the correction was just 2%. Moreover, as an MKS 627B capacitance manometer operating at 45 °C was used to measure the pressure in the scattering chamber, whereas the temperature of the vacuum chamber was ∼64 ± 2 °C, a thermal transpiration correction was applied to the pressure readings. That correction was calculated using the method described by Šetina,25 and it decreased the TCS magnitude by 2.7% at most. Note that we made use of that model, instead of the semiempirical method of Takaishi and Sensui26 that we typically employed in our previous papers, as Daudé et al.27 have recently shown that the latter approach might overestimate the thermal transpiration correction for large molecules, such as, in this case, TMP.28 The forward angle scattering effect has by far the most important impact on the measured cross sections, particularly for polar molecules.29 Its extent depends on the angular discrimination of the apparatus at a given energy and the shape of the elastic DCSs, for the relevant target species, in this forward angular region.29 The angular discrimination of the Trento apparatus varies from 17.5° at 1 eV to 2.4° at 50 eV.15 Elastic DCSs for positron-TMP scattering have not been measured so far, but they can be calculated using our IAMSCAR method (see Figure 2). Hence, by following the approach outlined in one of our earlier studies,24 our measured TCSs (see Table 1) could, at least in principle, be corrected for the forward scattering effect. However, because of the absence of independent experimental or theoretical DCSs for TMP, we
has been employed in the development of new detectors for imaging in radiation therapy.9 When those imaging devices are irradiated by the primary ionizing radiation, TMP acts as a liquid ionization medium between two planes of electrodes, which form a scanning matrix ionization chamber.9 However, in order to accurately characterize the interactions between that primary radiation and the target molecules, a quantitative knowledge of the scattering cross sections is required.10 Unfortunately they are typically unknown for the large variety of ionizing radiation employed in current medical practice. Positrons, in particular, are today amply used in many biomedical technologies, for both therapy and imaging,11 such as positron emission tomography (PET).12 Nonetheless, positron interactions with living matter largely remain quite poorly understood. The growing interest within the radiation physics and chemistry community, over the past decade and a half or so, in improving our comprehension of charged-particleinduced damage in biomolecular systems has led to a large number of studies into electron and positron scattering from the subunits of DNA and their analogues.13 Therefore, a practical need for cross sections of positron-TMP scattering represents an important rationale for conducting the present study. While there were neither measurements nor calculations of the scattering cross sections for positron collisions with TMP prior to the present study, a similar situation with respect to electron scattering from that target is also found. To the best of our knowledge there is only one investigation into electronimpact ionization of TMP.14 In order to, at least in part, fill this gap in the literature, in this article we present experimental total cross sections (TCSs) for low-energy positron impact with TMP. Those measurements were undertaken using the positron spectrometer at the University of Trento and span the incident energy range of 0.12−50 eV. The present study thus extends and concludes our previous experimental investigations of positron scattering from hydrocarbons.15−18 We also present theoretical TCSs, elastic integral cross sections (ICSs), and differential cross sections (DCSs), as well as inelastic ICSs for positronium formation and the sum of the electronic excitations and direct ionization, calculated using the independent atom model within the screening-corrected additivity rule formalism (IAM-SCAR) for positron energies between 1 and 1000 eV. The article is structured as follows. Section 2 gives a short overview of the experimental techniques and data analysis. We then briefly provide the details of our theoretical framework in section 3, followed by a presentation and discussion of the results in section 4. Finally, some concluding remarks in relation to the present investigation are given in section 5.
2. EXPERIMENTAL METHODS The present TCS measurements were conducted using the positron apparatus at the University of Trento, that was developed by Zecca and collaborators.15 As that spectrometer and the underlying techniques have been described in detail elsewhere,15 we only give a brief summary here. A low-energy positron beam is generated by a radioactive 22Na isotope (activity ∼1.4 mCi) in combination with a tungsten moderator of thickness 1 μm.19 Some electrostatic optics and a weak axial magnetic field (B ≈ 4−11 G) guide and focus the beam into the scattering cell where the target molecules may be present. The positrons are finally detected by a channel electron multiplier. 6467
dx.doi.org/10.1021/jp502632m | J. Phys. Chem. A 2014, 118, 6466−6472
The Journal of Physical Chemistry A
Article
Figure 2. Elastic differential cross sections for positron-2,2,4trimethylpentane scattering, computed with the present IAM-SCAR approach, at selected impact energies between 1 and 50 eV.
Figure 3. Present experimental total cross sections for positron collisions with 2,2,4-trimethylpentane are compared to our IAM-SCAR calculations for the total cross section, elastic integral cross section, positronium formation cross section, and the integral cross section for electronic excitation plus direct ionization. Also shown are the measured TCSs corrected for the forward angle scattering effect (see text) at selected incident energies. The arrows denoted by “Ps” and “IP” indicate the threshold energies for positronium formation and first ionization, respectively.
Table 1. Present Experimental Total Cross Sections for Positron Scattering from 2,2,4-Trimethylpentane; the Errors Represent the Statistical Uncertainties (1σ) on the Measurements energy (eV)
TCS (10‑20 m2)
TCS error (10‑20 m2)
energy (eV)
TCS (10‑20 m2)
TCS error (10‑20 m2)
0.12 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.80 1.00 1.25 1.50 1.75 2.00 2.50 3.00 4.00
230.87 220.89 207.66 189.33 182.62 159.73 147.63 127.82 112.67 102.00 90.59 83.43 76.13 67.22 63.25 57.13 50.15
17.13 9.22 13.75 11.74 14.63 8.02 9.86 3.91 1.33 2.54 1.59 2.81 0.77 1.50 0.42 0.40 1.06
5.00 6.00 7.00 8.00 9.00 10.00 12.50 15.00 20.00 25.00 30.00 35.00 37.50 40.00 45.00 50.00
47.95 46.77 45.30 44.38 43.73 43.11 41.78 39.91 38.37 36.08 34.98 34.74 34.92 35.49 36.05 35.04
0.60 0.53 2.86 0.75 1.76 0.64 0.66 1.71 0.70 0.38 0.85 0.84 0.79 0.71 0.35 1.02
positron24,30,35−39 scattering cross sections for a large variety of molecular targets and over an extensive energy range, typically from 1 to 5000 eV. Therefore, we only briefly summarize the main features of that formalism here. The first subjects of the computations are the individual atoms that form the target molecule, i.e., carbon (C) and hydrogen (H) in this case. The atomic optical model is based on a potential scattering approach, where the local complex potential V(r) is given by V (r ) = Vs(r ) + Vp(r ) + iVa(r )
(1)
The real part of eq 1 drives the elastic scattering dynamics and embraces the electrostatic (Vs(r)) and polarization (Vp(r)) interactions. The imaginary part (Va(r)) describes all inelastic processes that are considered as absorptions from the incident positron beam. Owing to this last term in eq 1, the optical model potential method yields a complex phase shift δl = λl + iμl. This allows for the calculation of the DCSs and ICSs for elastic and inelastic scattering, as well as the TCS as the sum of those ICSs. The static potential was obtained from the charge density derived from Hartree−Fock atomic wave functions, using a procedure analogous to that of Reid and Wadehra.40−42 The dipole plus quadrupole polarization potential was developed from that reported by McEachran et al.43 for Ne, but scaled by a constant in order to match the known dipole and quadrupole polarizabilities of the C and H atoms.44−46 The absorption potential accounts for the electronic excitations, positronium formation, and direct ionization. However, owing to the challenging nature of representing the Ps formation channel, the definition of the threshold energy for the absorption potential can be critical. Here we adopt a phenomenological approach where the absorption threshold is considered as an energy-dependent parameter:
have, in general, not corrected our measured TCSs. The values reported in Table 1 are, therefore, a lower bound on the “true” TCSs. Nevertheless, in order to estimate the extent of that effect, we corrected some of our measured TCSs using the IAM-SCAR DCSs of Figure 2, at a few selected energies between 1 and 50 eV (see Figure 3): we found that this correction varies between 2.7% and 5.7% in that energy range. The present TCS measurements span the energy range between 0.12 and 50 eV. The statistical uncertainties are typically smaller than 8% and, on average, amount to ∼3%. The overall errors on the TCSs are estimated to lie within the ∼5− 9% range. They originate from the statistical uncertainties, the uncertainty in the pressure and temperature measurements (
Cross Sections for Positron Impact with 2,2,4-Trimethylpentane Luca Chiari,† Antonio Zecca,‡ Francisco Blanco,§ Gustavo García,∥ Michael V. Perkins,⊥ Stephen J. Buckman,○,∇ and Michael J. Brunger*,†,∇ †
ARC Centre of Excellence for Antimatter−Matter Studies, School of Chemical and Physical Sciences, Flinders University, Adelaide, SA 5001, Australia ‡ Department of Physics, University of Trento, Via Sommarive 14, I-38123 Povo, Trento, Italy § Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain ∥ Instituto de Física Fundamental, Consejo Superior de Investigationes Científicas (CSIC), Serrano 113-bis, E-28006 Madrid, Spain ⊥ School of Chemical and Physical Sciences, Flinders University, Adelaide, SA 5001, Australia ○ ARC Centre of Excellence for Antimatter−Matter Studies, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia ∇ Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia ABSTRACT: 2,2,4-Trimethylpentane (C8H18), a hydrocarbon produced all over the world on a large scale in the processing of crude oil, has long been known and used in the energy sector. It has also recently attracted the attention of the radiation physics and chemistry community, owing to its applications in medical imaging techniques. Charged-particle interactions with this species unfortunately remain mostly unknown. In this study, we report on measured total cross sections for positron scattering from 2,2,4-trimethylpentane in the energy range from 0.12 to 50 eV. We also present calculations of the total cross sections, elastic integral and differential cross sections, positronium formation cross sections, and inelastic integral cross sections at energies from 1 to 1000 eV using the independent atom model with screening corrected additivity rule. A knowledge of those scattering cross sections might, through simulation models, help to improve the accuracy of current radiation detection devices and hence provide better estimates of the extent of any charged-particle-induced damage in biomolecular systems.
1. INTRODUCTION
referred to as isooctane. It is commonly found in petrol and, therefore, is of great interest in the petrochemical industry, which, in turn, produces large amounts of it every year.1 TMP has been assigned the standard 100 point on the octane rating scale. This high octane rating is a desirable property of a fuel for an internal combustion engine,1 and for TMP, it is related to its highly branched structure compared with the straight chain hydrocarbon. Given the above-mentioned properties, it is not surprising that TMP has attracted the attention of the broader atomic and molecular physics community and, in particular, its antimatter branch. In that respect, we note some studies on positron mobility2 and positron annihilation spectroscopy,3−5 in either pure TMP or solutions, that aimed at investigating the structure and properties of organic fluids. The mechanisms of positronium formation6 and annihilation3,7,8 in liquid hydrocarbons were also examined using TMP as a target. TMP does not offer itself just to fundamental studies, as it has also become very useful in biomedical applications. In fact, more recently it
2,2,4-Trimethylpentane (TMP) is a highly branched hydrocarbon with the chemical formula C8H18 (see Figure 1). It is the most important 8-carbon hydrocarbon and, in fact, is often
Special Issue: Franco Gianturco Festschrift Figure 1. Schematic diagram of the 2,2,4-trimethylpentane molecule. Carbon atoms are indicated in black, while the hydrogen atoms have a white color. © 2014 American Chemical Society
Received: March 17, 2014 Revised: March 31, 2014 Published: March 31, 2014 6466
dx.doi.org/10.1021/jp502632m | J. Phys. Chem. A 2014, 118, 6466−6472
The Journal of Physical Chemistry A
Article
The TCS of interest is determined by performing attenuation measurements and applying the Beer−Lambert law (eq 1 in Zecca et al.15). Specifically, the transmitted intensity of the positron beam is measured both in the presence and in the absence of the target vapor in the scattering chamber at each given incident energy. Measurements of the target pressure and temperature within the cell are also carried out at the same time, while the geometrical length of the scattering region is well-known. The transmittance of the positron beam is maintained at values greater than 0.7 when the target vapor is present in the chamber, so as to minimize multiple scattering events. Throughout the measurements we used a high-purity (99.8%) TMP source purchased from Sigma-Aldrich. Note that, each time we begin an experiment on a new target, we carry out preliminary measurements in order to validate our experimental techniques and procedures. Targets whose positron scattering TCSs are well-known, such as the noble gases,20−22 are employed for this purpose, although molecular nitrogen15 is also often used as a reference gas. The zero of the positron energy scale and the beam energy distribution are determined through a retarding potential analysis (RPA) of the incident beam without the target vapor in the chamber.23 In this case the error on the energy scale amounts to ∼±0.05 eV, whereas the energy resolution is found to be ∼0.25 eV (fwhm). Because of this finite energy resolution, the cross sections we measure are actually convoluted over the beam energy distribution. However, we anticipate that this effect might be largely insignificant at most positron energies and become important only below ∼0.5 eV, as the incident energy is comparable to the beam energy width. The measured data were corrected for some inevitable instrumental effects before being used to determine the TCSs. First, the geometrical length of the interaction region (L = 22.1 ± 0.1 mm) was corrected for the gyration of the positrons in the magnetic field.24 That correction increased L by 5.5% (with B ≈ 11 G) at all incident energies, except at E ≥ 37.5 eV when the magnetic field was reduced to B ≈ 4 G, and hence, the correction was just 2%. Moreover, as an MKS 627B capacitance manometer operating at 45 °C was used to measure the pressure in the scattering chamber, whereas the temperature of the vacuum chamber was ∼64 ± 2 °C, a thermal transpiration correction was applied to the pressure readings. That correction was calculated using the method described by Šetina,25 and it decreased the TCS magnitude by 2.7% at most. Note that we made use of that model, instead of the semiempirical method of Takaishi and Sensui26 that we typically employed in our previous papers, as Daudé et al.27 have recently shown that the latter approach might overestimate the thermal transpiration correction for large molecules, such as, in this case, TMP.28 The forward angle scattering effect has by far the most important impact on the measured cross sections, particularly for polar molecules.29 Its extent depends on the angular discrimination of the apparatus at a given energy and the shape of the elastic DCSs, for the relevant target species, in this forward angular region.29 The angular discrimination of the Trento apparatus varies from 17.5° at 1 eV to 2.4° at 50 eV.15 Elastic DCSs for positron-TMP scattering have not been measured so far, but they can be calculated using our IAMSCAR method (see Figure 2). Hence, by following the approach outlined in one of our earlier studies,24 our measured TCSs (see Table 1) could, at least in principle, be corrected for the forward scattering effect. However, because of the absence of independent experimental or theoretical DCSs for TMP, we
has been employed in the development of new detectors for imaging in radiation therapy.9 When those imaging devices are irradiated by the primary ionizing radiation, TMP acts as a liquid ionization medium between two planes of electrodes, which form a scanning matrix ionization chamber.9 However, in order to accurately characterize the interactions between that primary radiation and the target molecules, a quantitative knowledge of the scattering cross sections is required.10 Unfortunately they are typically unknown for the large variety of ionizing radiation employed in current medical practice. Positrons, in particular, are today amply used in many biomedical technologies, for both therapy and imaging,11 such as positron emission tomography (PET).12 Nonetheless, positron interactions with living matter largely remain quite poorly understood. The growing interest within the radiation physics and chemistry community, over the past decade and a half or so, in improving our comprehension of charged-particleinduced damage in biomolecular systems has led to a large number of studies into electron and positron scattering from the subunits of DNA and their analogues.13 Therefore, a practical need for cross sections of positron-TMP scattering represents an important rationale for conducting the present study. While there were neither measurements nor calculations of the scattering cross sections for positron collisions with TMP prior to the present study, a similar situation with respect to electron scattering from that target is also found. To the best of our knowledge there is only one investigation into electronimpact ionization of TMP.14 In order to, at least in part, fill this gap in the literature, in this article we present experimental total cross sections (TCSs) for low-energy positron impact with TMP. Those measurements were undertaken using the positron spectrometer at the University of Trento and span the incident energy range of 0.12−50 eV. The present study thus extends and concludes our previous experimental investigations of positron scattering from hydrocarbons.15−18 We also present theoretical TCSs, elastic integral cross sections (ICSs), and differential cross sections (DCSs), as well as inelastic ICSs for positronium formation and the sum of the electronic excitations and direct ionization, calculated using the independent atom model within the screening-corrected additivity rule formalism (IAM-SCAR) for positron energies between 1 and 1000 eV. The article is structured as follows. Section 2 gives a short overview of the experimental techniques and data analysis. We then briefly provide the details of our theoretical framework in section 3, followed by a presentation and discussion of the results in section 4. Finally, some concluding remarks in relation to the present investigation are given in section 5.
2. EXPERIMENTAL METHODS The present TCS measurements were conducted using the positron apparatus at the University of Trento, that was developed by Zecca and collaborators.15 As that spectrometer and the underlying techniques have been described in detail elsewhere,15 we only give a brief summary here. A low-energy positron beam is generated by a radioactive 22Na isotope (activity ∼1.4 mCi) in combination with a tungsten moderator of thickness 1 μm.19 Some electrostatic optics and a weak axial magnetic field (B ≈ 4−11 G) guide and focus the beam into the scattering cell where the target molecules may be present. The positrons are finally detected by a channel electron multiplier. 6467
dx.doi.org/10.1021/jp502632m | J. Phys. Chem. A 2014, 118, 6466−6472
The Journal of Physical Chemistry A
Article
Figure 2. Elastic differential cross sections for positron-2,2,4trimethylpentane scattering, computed with the present IAM-SCAR approach, at selected impact energies between 1 and 50 eV.
Figure 3. Present experimental total cross sections for positron collisions with 2,2,4-trimethylpentane are compared to our IAM-SCAR calculations for the total cross section, elastic integral cross section, positronium formation cross section, and the integral cross section for electronic excitation plus direct ionization. Also shown are the measured TCSs corrected for the forward angle scattering effect (see text) at selected incident energies. The arrows denoted by “Ps” and “IP” indicate the threshold energies for positronium formation and first ionization, respectively.
Table 1. Present Experimental Total Cross Sections for Positron Scattering from 2,2,4-Trimethylpentane; the Errors Represent the Statistical Uncertainties (1σ) on the Measurements energy (eV)
TCS (10‑20 m2)
TCS error (10‑20 m2)
energy (eV)
TCS (10‑20 m2)
TCS error (10‑20 m2)
0.12 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.80 1.00 1.25 1.50 1.75 2.00 2.50 3.00 4.00
230.87 220.89 207.66 189.33 182.62 159.73 147.63 127.82 112.67 102.00 90.59 83.43 76.13 67.22 63.25 57.13 50.15
17.13 9.22 13.75 11.74 14.63 8.02 9.86 3.91 1.33 2.54 1.59 2.81 0.77 1.50 0.42 0.40 1.06
5.00 6.00 7.00 8.00 9.00 10.00 12.50 15.00 20.00 25.00 30.00 35.00 37.50 40.00 45.00 50.00
47.95 46.77 45.30 44.38 43.73 43.11 41.78 39.91 38.37 36.08 34.98 34.74 34.92 35.49 36.05 35.04
0.60 0.53 2.86 0.75 1.76 0.64 0.66 1.71 0.70 0.38 0.85 0.84 0.79 0.71 0.35 1.02
positron24,30,35−39 scattering cross sections for a large variety of molecular targets and over an extensive energy range, typically from 1 to 5000 eV. Therefore, we only briefly summarize the main features of that formalism here. The first subjects of the computations are the individual atoms that form the target molecule, i.e., carbon (C) and hydrogen (H) in this case. The atomic optical model is based on a potential scattering approach, where the local complex potential V(r) is given by V (r ) = Vs(r ) + Vp(r ) + iVa(r )
(1)
The real part of eq 1 drives the elastic scattering dynamics and embraces the electrostatic (Vs(r)) and polarization (Vp(r)) interactions. The imaginary part (Va(r)) describes all inelastic processes that are considered as absorptions from the incident positron beam. Owing to this last term in eq 1, the optical model potential method yields a complex phase shift δl = λl + iμl. This allows for the calculation of the DCSs and ICSs for elastic and inelastic scattering, as well as the TCS as the sum of those ICSs. The static potential was obtained from the charge density derived from Hartree−Fock atomic wave functions, using a procedure analogous to that of Reid and Wadehra.40−42 The dipole plus quadrupole polarization potential was developed from that reported by McEachran et al.43 for Ne, but scaled by a constant in order to match the known dipole and quadrupole polarizabilities of the C and H atoms.44−46 The absorption potential accounts for the electronic excitations, positronium formation, and direct ionization. However, owing to the challenging nature of representing the Ps formation channel, the definition of the threshold energy for the absorption potential can be critical. Here we adopt a phenomenological approach where the absorption threshold is considered as an energy-dependent parameter:
have, in general, not corrected our measured TCSs. The values reported in Table 1 are, therefore, a lower bound on the “true” TCSs. Nevertheless, in order to estimate the extent of that effect, we corrected some of our measured TCSs using the IAM-SCAR DCSs of Figure 2, at a few selected energies between 1 and 50 eV (see Figure 3): we found that this correction varies between 2.7% and 5.7% in that energy range. The present TCS measurements span the energy range between 0.12 and 50 eV. The statistical uncertainties are typically smaller than 8% and, on average, amount to ∼3%. The overall errors on the TCSs are estimated to lie within the ∼5− 9% range. They originate from the statistical uncertainties, the uncertainty in the pressure and temperature measurements (
