Calculations of Positron Scattering from Boron, BH, BF, BF₂, and BF₃

Abstract

The single-center convergent close-coupling (CCC) method is applied to calculate positron scattering from boron. A model potential approach is utilized to extract the positronium formation, direct ionization, and values between the positronium formation and ionization thresholds. We present results for total, electron loss, elastic, momentum transfer, total bound state excitation, positronium formation, direct ionization, stopping power, and mean excitation energy from ${10}^{-5}$ eV to 5000 eV. For boron, there is only one other set of theoretical positron calculations for elastic and momentum transfer above 500 eV, which is in excellent agreement with the current CCC results. Using the current results for boron atoms and previous CCC calculations for hydrogen and fluorine atoms, positron scattering from BF, ${\mathrm{BF}}_2$, ${\mathrm{BF}}_3$, and BH molecules is calculated for energies between 0.1 eV and 5000 eV with a modified independent atom approach.

1. Introduction

Interest in scattering from boron (B) largely extends from fusion research. Along with carbon, these atoms form viable materials for the shielding and walls in fusion reactors, such as ITER [1,2]. Boron, in particular, is often used to coat the first wall of these reactors through boronization, which reduces impurities in the plasma [3]. Furthermore, recent work has also highlighted the potential of boron to be utilized for aneutronic fusion through the proton–boron fusion reaction [4,5]. Within fusion reactors, positrons are produced in large quantities due to pair production [6,7]. Only 0.1% of the produced positrons are expected to annihilate within the plasma, with the rest escaping the plasma and colliding with the reactor walls [8]. Accurate scattering data is, therefore, required to fully understand the impact of these collisions on the degradation of the reactor materials.

Research for positron boron scattering is scarce. To date, only a single set of calculations for elastic scattering is available in the literature [9]. Theoretical work for electron scattering on boron is not as limited, with the B-spline R-matrix (BSR) [10], binary encounter Bethe (BEB) [11], R-matrix with pseudostates (RMPS) [12,13], optical potential [14], R-matrix [15], and time-dependent close-coupling (TDCC) [16] calculations for elastic, excitation, and ionization cross sections. These calculations, however, are mostly constrained to incident energies below 150 eV.

In fusion research, there is considerable interest in molecules, such as boron hydride (BH), which will form as impurities at the plasma’s edge due to interactions between the plasma and reactor materials [17]. Results for positron scattering on boron hydride are not available within the literature, with theoretical work focused solely on electron scattering at low energies [18].

Other molecules of interest are BF, ${\mathrm{BF}}_2$, and ${\mathrm{BF}}_3$, which are important due to the use of ${\mathrm{BF}}_3$ plasmas in ion implantation [19]. Beyond this, ${\mathrm{BF}}_3$ is also commonly used in neutron detectors [20] and as a feed gas in plasmas [21]. As with BH, there are no positron scattering studies within the literature for these targets. There exist, however, numerous theoretical electron scattering studies for ${\mathrm{BF}}_3$ [21,22,23,24,25,26]. For BF and ${\mathrm{BF}}_2$, calculations for electron scattering have only been conducted by Gupta et al. [27] and Kim and Irikura [28]. As these molecules are highly reactive, corrosive, and toxic, they are difficult to perform measurements upon. To date, the only experimental results are the total ionization [29] and attachment [30] cross sections for electron scattering from ${\mathrm{BF}}_3$. For positron and electron scattering, cross sections for each projectile are equal at high incident energies. Therefore, the current results are expected to be useful for future research on the scattering of both projectiles from the considered targets.

We have conducted single-center convergent close-coupling (CCC) calculations for positron scattering from atomic boron. This technique is based on finding an accurate multiconfigurational description of the target structure through utilization of the codes developed by Zatsarinny [31] and Fischer [32] together with the scattering formulation of the CCC method [33]. The CCC formalism has been previously used to study positron scattering from various atoms and molecules [34,35,36]. Cross sections for total, elastic, momentum transfer, electron loss, positronium formation, direct ionization, excitation, stopping power, and mean excitation energy have been calculated for incident energies between ${10}^{-5}$ eV and 5000 eV. Through application of the independent atom model screening-corrected additivity rule (IAM-SCAR) approach [37] combined with previous CCC calculations for atomic fluorine and hydrogen [38], results for total, electron loss, direct ionization, positronium formation, elastic, and total electronic excitation cross sections for BH, BF, ${\mathrm{BF}}_2$, and ${\mathrm{BF}}_3$ are also produced. We use atomic units throughout unless otherwise specified.

2. Method

The convergent close-coupling method is extensively documented within the literature for single and two-center calculations of positron scattering for both atomic and molecular targets [36,38,39]. The use of a complex model potential to supplement single-center CCC calculations is also well documented [36,40,41]. Therefore, only a brief overview of the current method is provided here.

2.1. Single-Center CCC Formalism

The first step in treating the boron atom is to perform a self-consistent field Hartree–Fock (SCHF) calculation to obtain the 1s, 2s and 2p orbitals. We then obtain one-electron orbitals by diagonalizing the ${\mathrm{B}}^{2+}$ one-electron frozen-core Hamiltonian in the non-orthogonal Laguerre basis,

$$\begin{array}{cc}\hfill {\phi }_{k\ell }\left(r\right)& =\sqrt{\frac{{\alpha }_{\ell }(k-1)!}{(k+\ell )(k+2\ell )!}}{\left(2{\alpha }_{\ell }r\right)}^{\ell +1}\hfill \end{array}$$

$$\begin{array}{c}\hfill \times e^{-{\alpha }_{\ell }r}{L}_{k-1}^{2\ell +1}\left(2{\alpha }_{\ell }r\right),k=1,\dots ,N_{\ell }.\end{array}$$

(1)

Here, ℓ is the angular orbital momentum, $L_{k-1}^{2\ell +1}$ are the associated Laguerre polynomials, ${\alpha }_{\ell }$ are exponential fall-off parameters, and $N_{\ell }$ is the number of functions for each ℓ. The resulting one-electron orbitals are still not optimized for describing the Boron structure. The 2s and the 2p orbitals are replaced with those obtained from the SCHF calculation. Then, the $n=3$ and $n=4$ orbitals are obtained from a multiconfigurational Hartree–Fock (MCHF) calculation for Boron. The collective one-electron orbitals, suitably orthogonalized, are then used to form N three-electron configurations to obtain the ground and excited states (${\Phi }_n^N$) of the Boron atom via diagonalizing the Boron Hamiltonian,

$$H_{\mathrm{T}}=\sum _{i=1}^{N_{\mathrm{e}}}\left(-\frac{1}{2}{\nabla }_i^2-\frac{Z}{r_i}\right)+\sum _{i>j=1}^{N_{\mathrm{e}}}\frac{1}{|{\mathit{r}}_i-{\mathit{r}}_j|}.$$

(2)

All configurations assume $1s^2$ for the frozen core, with

$$\langle {\Phi }_m^N|H_{\mathrm{T}}|{\Phi }_n^N\rangle ={ϵ}_n^N{\delta }_{n,m}.$$

(3)

For scattering, the total wave function of the scattering system is substituted into the Schrödinger equation. This equation is then solved through expansion of the total scattering wave function with the obtained target pseudostates. For scattering, the total wave function of the scattering system is expanded in the cases of the pseudostates, substituted into the Schrödinger equation, which is then transformed through the Green’s function approach into the coupled Lippmann–Schwinger equations for the T matrix:

$$\begin{array}{cc}\hfill \langle {\mathit{k}}_n& {\Phi }_n^N\left|T\right|{\Phi }_i^N{\mathit{k}}_i\rangle =\langle {\mathit{k}}_n{\Phi }_n^N\left|V\right|{\Phi }_i^N{\mathit{k}}_i\rangle +\hfill \\ \hfill & \sum _{m=1}^N\int d\mathit{k}\frac{\langle {\mathit{k}}_n{\Phi }_n^N\left|V\right|{\Phi }_m^N\mathit{k}\rangle \langle \mathit{k}{\Phi }_m^N\left|T\right|{\Phi }_i^N{\mathit{k}}_i\rangle }{E+i0-{ϵ}_n^N-k^2/2}.\hfill \end{array}$$

(4)

Here, $|{\mathit{k}}_n\rangle$ is a plane wave with energy $k_n^2/2$, E is the total energy, and V is the interaction potential between the positron and the target.

The coupled Lippmann–Schwinger Equations (4) are then solved via a partial-wave expansion technique. For efficiency of partial-wave convergence, analytical Born completion is utilized. We obtain the T-matrix elements for each of the pseudostates, from which relevant cross sections can be extracted. The calculation of mass stopping power and mean excitation energy within the CCC formalism is described in Refs. [34,35,42].

2.2. CCC-Scaled Complex Model Potential

To extract direct ionization and positronium formation cross sections from the single-center CCC calculation, we rely upon the implementation of a complex model potential given by,

$$V_{\mathrm{opt}}(r,E_i)=V_{\mathrm{st}}\left(r\right)+V_{\mathrm{pol}}+i{V}_{\mathrm{abs}}(r,E_i).$$

(5)

Here, $V_{\mathrm{st}}$ is the static potential, $V_{\mathrm{pol}}$ is the polarization potential, and $V_{\mathrm{abs}}$ is the absorption potential. The exact form of these is provided in Ref. [36], but we note that $V_{\mathrm{st}}$ is obtained directly from our atomic structure model, and $V_{\mathrm{pol}}$ contains a parameter that is modified for each incident energy to produce an elastic cross section that is the same as our single-center calculation.

To model positronium formation, we utilize the delta variational technique, with our delta equivalent to that of Chiari et al. [43] but with the adjustable parameter $E_m$ being set to 10 eV. Following the calculation of the required cross sections, the positronium formation and direct inelastic component of this calculation are scaled to agree with the single-center CCC calculation as described in Ref. [36]. This process is referred to as the CCC-scaled complex model potential (CCC-pot). Due to the direct ionization threshold of boron being significantly lower than hydrogen, it is necessary to scale the direct inelastic component to be equal to the CCC results at 100 eV instead of for energies above 500 eV. This slight modification is required to prevent inconsistencies between the CCC-pot and CCC results for the inelastic cross section. This modification has no impact on the accuracy of this approach, as positronium formation is negligible by 100 eV.

For energies greater than 10 eV above the direct ionization threshold, which is the point where single-center and two-center calculations are expected to be equivalent, the direct ionization cross section is obtained by subtracting the positronium-formation cross section from the electron-loss cross section. At lower energies, the complex scattering potential ionization contribution (CSP-ic) method [44] is utilized to extract the direct ionization, and the bound-state excitation cross section, from the direct inelastic component of the complex model potential.

2.3. Calculation Details

The current model contains orbitals with $N_{\ell }=20-\ell$ and ${\alpha }_{\ell }=1.0$. We include in the structure configurations of the form $1s^{2}2s^{2}n\ell$ and $1s^{2}2p^{2}n\ell$ for $\ell \le {\ell }_{\max }$, where ${\ell }_{\max }$ is the maximum angular momentum of the model. Also included are $1s^{2}2s2pn\ell$ for $\ell \le 4$, $1s^{2}2p^3$, and $1s^{2}2sn{\ell }^2$ for all orbitals between $2p$ and $3d$. Above the direct ionization threshold, a 1559-state ${\ell }_{\max }=9$ is used up to 15 eV. Above 15 eV, a 1353-state ${\ell }_{\max }=8$ model is used. Both of these models contain all states with energies up to 100 eV relative to the ${\mathrm{B}}^{+}$ ionization threshold. For energies below the positronium formation, a 1673-state model with ${\alpha }_{\ell }=3.0$, ${\ell }_{\max }=20$, and energies up to 350 eV above the direct ionization threshold are used. This larger model is required to obtain convergent results at low energies due to the slow convergence with ℓ for energies below the positronium formation threshold. The electron-loss cross section is extrapolated above 50 eV using a 3832 state Born model containing the above configurations, in addition to $1s^{2}2s2pn\ell$, $1s^{2}2s3sn\ell$ and $1s^{2}2s3pn\ell$ for $\ell \le 8$, $1s^{2}2s3dn\ell$, and $1s2s^{2}2pn\ell$ configurations.

2.4. Convergence Study

A convergence study of the electron loss (positronium formation plus direct ionization) cross section for different ${\ell }_{\max }$ for positron boron is shown in Figure 1. This cross section converges quickly with calculations fully converged for ${\ell }_{\max }\ge 8$ for energies above 10 eV, and ${\ell }_{\max }\ge 4$ above 50 eV. In Figure 2, a convergence study is presented for the scattering length for the B models used below the positronium formation. Convergence of this value is found to occur by ${\ell }_{\max }=20$. The slow convergence at low energies in relation to ${\ell }_{\max }$ leads to the requirement of very large scattering calculations. This is also found to be the case in CCC calculations for positron scattering from magnesium [45], which also has a low positronium formation threshold (0.8 eV).

2.5. CCC-SCAR

The IAM-SCAR approach [37] utilizes screening coefficients ($s_i$) to account for the geometry of the molecule to obtain molecular cross sections ${\sigma }_{\mathrm{m}}$ from the constituent atomic cross sections ${\sigma }_i$

$${\sigma }_{\mathrm{m}}=\sum _i{s}_i{\sigma }_i.$$

(6)

The screening coefficients have values $0\le s_i\le 1$ and, therefore, reduce the summed cross section for the atomic components of the molecule. The equations required to calculate these screening coefficients are provided by Blanco and García [37]. Once we obtain $s_i$, we apply it to each component of the total cross section to obtain electron loss, direct ionization, positronium formation, elastic, and electronic excitation cross sections for the relevant molecule. Here, we refer to IAM-SCAR calculations completed using CCC cross sections as CCC-SCAR.

The thresholds for electronic excitation, direct ionization, and positronium formation are different in the molecules compared to their constituent atoms. To correct for this, after completing IAM-SCAR calculations, the results for these transitions are shifted to the molecular thresholds shown in Table 1. From previous calculations conducted with the CCC-SCAR approach [46], we expect the uncertainty of the current molecular results to be 20% for energies below 50 eV and within 10% for energies above 100 eV.

The presence of a permanent dipole in polar molecules results in significant contributions to the total cross section from inelastic rotational transitions, particularly at low energies. The permanent dipole for each of the considered molecules are presented in Table 1. To account for this, we can sum the rotational excitation cross section of a molecule to the CCC-SCAR result to obtain CCC-SCAR+rot values. We calculated the J = 0 →$J=1$ rotational excitation cross section within the Born approximation for BH and BF using

$$\sigma (J\to J+1)=\frac{8\pi }{3k^2}{\mu }^2\frac{J+1}{2J+1}ln\frac{k+k^{\prime }}{k-k^{\prime }}$$

(7)

From Ref. [47], here, $\mu$ is the dipole moment of the target molecule, and $k/k^{\prime }$ are the initial/final momentum of the projectile. For these molecules, we utilized potential energy curves [48,49] within the MCCC formalism [50] to calculate the rotational excitation energies. Although ${\mathrm{BF}}_2$ also contains a permanent dipole moment, the energy of the rotational excitation threshold for this molecule is not available within the literature and, therefore, we have not calculated the rotational results for this molecule. Vibrational excitations are not accounted for in the current calculations.

Table 1. Direct ionization and electronic excitation thresholds for the considered molecules. Also shown are the dipole moments of these molecules.

Molecule	Direct Ion. (eV)	Exc. (eV)	Dipole (D)
BH	9.77 [51]	2.87 [52]	1.33 [49]
BF	11.12 [51]	6.34 [53]	0.86 [27]
${\mathrm{BF}}_2$	8.0 [54]	4.66 [55]	0.58 [27]
${\mathrm{BF}}_3$	15.7 [51]	6.99 [56]	0.0

Table 1. Direct ionization and electronic excitation thresholds for the considered molecules. Also shown are the dipole moments of these molecules.

Molecule	Direct Ion. (eV)	Exc. (eV)	Dipole (D)
BH	9.77 [51]	2.87 [52]	1.33 [49]
BF	11.12 [51]	6.34 [53]	0.86 [27]
${\mathrm{BF}}_2$	8.0 [54]	4.66 [55]	0.58 [27]
${\mathrm{BF}}_3$	15.7 [51]	6.99 [56]	0.0

3. Results for Atomic Boron

3.1. Structure

Excitation energies from the current CCC calculation, BSR [10] and NIST [57] are presented in

Table 2. Excitation energies (eV) for B doublet target states.

	State	Term	CCC	BSR [10]	NIST [57]
1	$2s^{2}2p$	${ }^2{P}^o$	0.000	0.000	0.000
2	$2s^{2}3s$	${ }^{2}S$	4.898	4.916	4.964
3	$2s2p^2$	${ }^{2}D$	5.931	5.939	5.933
4	$2s^{2}3p$	${ }^2{P}^o$	6.006	5.974	6.027
5	$2s^{2}3d$	${ }^{2}D$	6.761	6.725	6.790
6	$2s^{2}4s$	${ }^{2}S$	6.773	6.752	6.820
7	$2s^{2}4p$	${ }^2{P}^o$	7.117	7.097	7.165
8	$2s^{2}4d$	${ }^{2}D$	7.399	7.374	7.438
9	$2s^{2}4f$	${ }^2{F}^o$	7.397	7.370	7.443
10	$2s^{2}5s$	${ }^{2}S$	7.427	7.387	7.457
Ion.	Limit		8.383	8.222	8.298

. There is excellent agreement between these results, with all excitations being within 0.1 eV of the NIST values. In

Table 3. Oscillator strengths for ground state of B.

Lower Level	Upper Level	CCC	BSR [10]	NIST [57]
$2s^{2}2p$${ }^2{P}^o$	$2s^{2}3s$${ }^{2}S$	0.0770	0.0803	0.0785
	$2s^{2}3d$${ }^{2}D$	0.167	0.0172	0.170
	$2s^{2}4s$${ }^{2}S$	0.0132	0.0162	0.0154
	$2s^{2}4d$${ }^{2}D$	0.0759	0.0762	0.0723
	$2s^{2}5s$${ }^{2}S$	0.0069	0.0117	0.0082

, we present the current oscillator strengths for transitions from the ground state of B alongside the BSR [10] and NIST [57] results. Close agreement is typically observed between the calculations for the presented oscillator strengths, with minor discrepancies present for the weaker transitions.

The dipole polarizability (${\alpha }_D$) of both models utilized for B is 20.3 $a_0^3$, within 1% of the accepted value of 20.5 $a_0^3$. This quantity must be accurate to calculate cross sections at low energies correctly. Due to the relatively low ionization energy of boron, the majority of the dipole polarizability originates from the continuum rather than the bound states. Therefore, to accurately model boron, the impact of the continuum must be correctly accounted for. The close agreement of ${\alpha }_D$, oscillator strengths, and energy levels with past theory and experiment provides strong evidence for the accuracy of the current B structure model.

Higher multipole polarizabilities are expected to have a negligible impact on the current cross sections; however, the ${\alpha }_Q$ and ${\alpha }_O$ results of our small-energy model are included for completeness. Our calculation of B has ${\alpha }_Q=135.89$ ${\mathrm{a}}_0^5$ and ${\alpha }_O=2243.2$ ${\mathrm{a}}_0^7$. The current ${\alpha }_Q$ result is within 8% of the CEPA-NO result of 145.7 ${\mathrm{a}}_0^5$ [58].

3.2. Total Cross Section

In Figure 3, the CCC-pot results for the total cross section of positron scattering on B are presented alongside the electron scattering BSR calculation of Wang et al. [10]. Typically, for atomic targets below 500 eV, the electron and positron total cross sections differ significantly due to the different scattering dynamics present for each projectile. In this case, however, we see unexpected agreement between the electron BSR and positron CCC-pot results for energies above 15 eV. From considering the components of ionization (Figure 6), elastic scattering (Figure 7) and excitation (Figure 10), the reason for this agreement is because the difference in elastic scattering between the two projectiles is almost equivalent to the difference in inelastic scattering. The significantly larger elastic cross section for the electron case is a result of electron exchange, a process which does not occur in positron scattering.

3.3. Positronium Formation and Ionization Cross Sections

In Figure 4, positron boron CCC-pot results are presented without comparison, as there is no existing theory or experiment for this transition. Direct ionization results for boron are shown in Figure 5. Due to the absence of previous positron results, we present the current CCC results alongside the BSR calculation of Wang et al. [10] and BEB calculation of Kim and Stone [11]. At high incident energies, the CCC results become equal to that of the Born approximation, which is equal for positrons and electrons. We find this to be valid for energies above 300 eV, with close agreement between the electron BEB calculation and the current CCC result above this energy. The results for the electron-loss cross section are shown in Figure 6.

3.4. Elastic Cross Section

In Figure 7, we present the elastic cross section for boron. The positron results from the CCC calculations and Dapor and Miotello [9] are shown alongside the electron scattering BSR calculations of Wang et al. [10] and NIST [59]. As previously mentioned, the electron results for elastic scattering are significantly greater than the positron calculation across most of the energy range due to the electron exchange process. Above 500 eV, near-perfect agreement is found between the CCC calculation and the results of Dapor and Miotello [9]. For energies above 1000 eV, the electron elastic results of NIST [59] are close to the positron results, with them being nearly the same by 5000 eV.

3.5. Momentum-Transfer Cross Section

The CCC momentum-transfer cross section for boron is presented in Figure 8 alongside the only other calculation of Dapor and Miotello [9]. As with the elastic cross section, excellent agreement is observed between these calculations. Above 1000 eV, the NIST [59] results are close to the positron results.

3.6. Low Energy Scattering

Due to the absence of other theoretical results, we present only CCC low-energy results for both momentum-transfer and elastic scattering in Figure 9. This figure shows that a Ramsauer–Townsend minimum is not present in either the momentum-transfer or elastic cross section. As scattering becomes isotropic for low energies, the momentum-transfer and elastic cross section become equal, in this case, for energies below ${10}^{-2}$ eV.

We find the scattering length of boron to be 53.85$a_0$. As this scattering length is significantly greater than the geometric size of the boron atom, a positron virtual state exists. This virtual state forms due to boron’s large dipole polarizability, creating an attraction between the positron and boron atom. The existence of this virtual state results in an increase in both the elastic and annihilation cross sections at low incident energies [60]. We calculate the energy of this virtual state to be 4.69 $\times {10}^{-3}$ eV through

$$ϵ=\frac{1}{2A^2},$$

(8)

where A is the scattering length.

3.7. Excitation Cross Sections

The single-center CCC formalism is unable to obtain convergence in the small energy region (6.8 eV) between positronium formation and the ionization threshold. This is due to the inability to approximate positronium formation with the positive-energy atomic pseudostates, which are closed in this energy region. In our experience, running small eigenstate-only calculations are likely to provide more accurate results in this energy region.

The total and first nine individual bound-state excitation cross sections for boron are shown in Figure 10. For the total excitation cross section, there are no other results to compare against within the literature. The sharp spike at the lower energies in the CCC result for the total excitation is unphysical and a result of a lack of convergence in the single-center approach. The CCC-pot total excitation result does not exhibit this unphysical behavior; however, this approach cannot discretize the individual excitations. Therefore, for the lower energies, the current results for individual excitations are instead from a 10-state CCC model. For the $2s^{2}4f$${ }^2$${\mathrm{F}}^o$ and $2s^{2}5s$${ }^2$S excitations, this smaller model is significantly larger than the 1353-state CCC model. This discrepancy is likely due to the strong coupling of these states to the continuum in the larger model; therefore, the 10-state model is uniformly scaled to agree with the larger model at 18 eV for these excitations. This process better approximates the low-energy behavior of these excitations by minimizing the unphysical behavior that results from positronium formation in the single-center technique.

For low energies, the results for each projectile are not expected to be equal due to the impact of the different Coulomb forces. However, there is close agreement between the current positron results and the electron BSR [10] results by 100 eV for almost all transitions. The one exception is the $2s^{2}5s$${ }^2$S excitation, which differs significantly at high energies between the CCC and BSR due to the different oscillator strengths found by these models for this transition.

3.8. Stopping Power and Mean Excitation Energy

The current results for the positron stopping power of boron are shown in Figure 11 alongside the ESTAR database [61] results for the electron case. We also present the positronium formation component of the stopping power which is estimated from the CCC-pot result through the method described in Ref. [34]. At 1000 eV, these electron results are slightly below the current calculation, but the difference between the electron and positron calculations becomes negligible by 5000 eV.

The CCC results for the mean excitation energy of boron are shown in Figure 12. The mean excitation energy generally slowly rises across the entire energy range before plateauing by 5000 eV. There are no previous positron or electron results within the literature.

4. Results for Molecular Targets

4.1. BH

The total cross section and its components for $e^{+}$-BH are presented in Figure 13. These results are presented without comparison due to the lack of positron and high-energy electron results in the literature. For energies below 6 eV, the dominant inelastic process is rotational excitation. For energies between 6 eV and 20 eV, it is positronium formation. Above this energy, direct ionization is the dominant process.

4.2. BF

The direct ionization cross section for $e^{+}$-BF is shown in Figure 14. Current CCC-SCAR results are shown alongside the electron calculations of Gupta et al. [27] and Kim and Irikura [28]. Both of these electron calculations are in close agreement across the presented energy range. The current positron results are significantly higher than the electron results until 3000 eV. Above this energy, good agreement is found between the presented calculations.

In Figure 15, the CCC-SCAR and CCC-SCAR+rot results for the total cross section for $e^{+}$-BF are shown, along with its components. These are presented without comparison, as other electron results for these cross sections either have not been calculated or are limited to low energies.

4.3. ${BF}_2$

The direct ionization CCC-SCAR results for $e^{+}$-${\mathrm{BF}}_2$ are shown in Figure 16. As with BF, these results are shown alongside the electron calculations of Gupta et al. [27] and Kim and Irikura [28]. Unlike BF, the two electron calculations disagree for intermediate energies. The current positron results are above both of the electron calculations until 200 eV. Above 200 eV, the current results are slightly lower than the calculation of Kim and Irikura [28] to 5000 eV. Compared to the calculations of Gupta et al. [27], the CCC-SCAR is slightly lower for energies above 1000 eV. The CCC-SCAR results for the total cross section and its components for $e^{+}$-${\mathrm{BF}}_2$ are shown in Figure 17. The current results do not include the rotational transition; however, as the dipole moment of ${\mathrm{BF}}_2$ is smaller than the other considered molecules this transition is expected to only be significant below the positronium formation threshold.

4.4. ${BF}_3$

The total cross section of $e^{+}$-${\mathrm{BF}}_3$ scattering is shown in Figure 18. The current CCC-SCAR results are shown alongside those for electrons calculated by Vinodkumar et al. [22]. At high energies, where we would expect results between projectiles to be the same, the electron results descend more rapidly than the current calculation. The results for elastic scattering from the same calculations are shown in Figure 19. As with the total cross section, the elastic results of Vinodkumar et al. [22] descend more rapidly above 1500 eV than the current CCC-SCAR results.

We present the direct ionization cross section for $e^{+}$-${\mathrm{BF}}_3$ in Figure 20. As there are no existing positron results, we present our CCC-SCAR calculation alongside the electron calculations of Vinodkumar et al. [22] and Kim and Irikura [28] and the measurements of Kurepa et al. [29]. For energies above 700 eV, there is close agreement between the current calculation and both electron results.

The total inelastic cross section is presented in Figure 21, alongside its positronium formation, direct ionization, and total electronic excitation components. The positronium formation is the most significant process below 30 eV, with a maximum over twice that of the direct ionization. Above 30 eV, direct ionization is the dominant inelastic process to 5000 eV. The total electronic excitation peaks at 15 eV, then steadily declines with increasing energy.

5. Conclusions

A comprehensive set of cross sections is calculated for positron scattering from boron within the single-center CCC approach. Good agreement is observed for atomic boron with the only existing positron calculation for elastic and momentum transfer cross sections available at high energies only. Excellent agreement is found with the past theory and experiments for the polarizabilities, oscillator strengths, and excitation energies, with differences only present for weak transitions. Due to the absence of positron results, other cross sections are compared to existing electron theoretical results. In particular, excellent agreement is found for the direct ionization and electron-loss cross section for energies above 300 eV. There is also good agreement between the previous electron and current positron results for most of the first nine bound-state excitations by 100 eV. Above the positronium formation threshold, there is an unexpected agreement between the current positron results and previous electron calculations for the total cross section. This is a result of the differences between the elastic cross section of the different projectiles being approximately equal to their differences in the inelastic cross section.

Using the IAM-SCAR approach and previous CCC calculations, we have also calculated the total, electron loss, positronium formation, electronic excitation, and elastic cross sections for BH, BF, ${\mathrm{BF}}_2$, and ${\mathrm{BF}}_3$. Good agreement is found at high incident energies for the direct ionization cross section between the current positron calculations and electron results for these molecules. Due to the absence of previous results within the literature, several results are presented without comparison. Further theoretical and experimental work is encouraged for these targets at intermediate and high energies.