Positron scattering and transport in liquid helium

In previous papers we have proposed a method for the ab initio calculation of fully differential cross-sections for electron scattering in liquids and applied it to liquid argon, xenon and krypton. In this paper, we extend the procedure to the consideration of positron scattering in liquid helium, which is complicated by the annihilation process as well as the fact that the electron definition for the region"owned"by a target atom used previously does not have a positron analogue. We explore several physically motivated definitions to obtain effective positron scattering in the dense fluid. We find that our calculations of a pure helium system cannot precisely match experimental measurements, however by including a small admixture (<0.1%) of an impurity, we can obtain reasonable agreement in the dense gas phase. In contrast, our calculations do not match well to the liquid phase measurements. This provides motivation to explore further multiple scattering effects in the theory.


I. INTRODUCTION
Positrons are used in a variety of diagnostic applications, including the medical diagnostic of PET (positron emission tomography), materials analysis through PALS (positron annihilation lifetime spectroscopy) and DBS (Doppler broadening spectroscopy) [1]. Positrons can even be used as an indirect probe of the structure of the Galaxy [2]. These experimental techniques generally rely on the interpretation of gamma rays emitted from the annihilation of the positrons with electrons.
To be able to interpret these diagnostics, it is essential to understand how the positron propagates through the material under investigation. As the concentration of positrons is typically very low, this falls under the umbrella of swarm modelling [3]. In gases this is usually explored through kinetic theory simulations, which allows for a simple scaling with density for transport properties such as annihilation rates and drift velocities. As the density significantly increases, these scaling behaviours have historically been used directly, even for systems as dense as liquids where the scaling laws break down.
To model charged particle transport in liquids and dense gases, we must account for effects such as multiple scattering and interaction screening, using correlations between particles in the fluid. This was first described by Lekner [4] and we have since extended the procedure to calculate more accurate, ab-initio, fully-differential effective elastic cross sections for electrons propagating in liquid argon, xenon and krypton [5][6][7] using only the pair correlator for each fluid.
In this article, we investigate positron transport. On the one hand, this should share much of the same properties of electron transport through a fluid, as both the electron and positron are a light charged particle. On the other hand, the interaction of the positron with a single atom of the fluid is very different: it has no exchange interaction, the sign of the Coulomb interaction is reversed and loss processes are always present, even as the collisional energy approaches zero. Note, however, that the polarisation interaction is similar for both the positron and electron, as the induced dipole-charge interaction is independent of the sign of the charge.
An important feature of our approach is that the effective cross sections are calculated in an ab initio manner from an interaction potential. This is useful because, a) less is known about the positron elastic and annihilation cross sections as measurements are more difficult than the corresponding electron system, and b) it is the interaction between the charged particle and the atom that is modified in the fluid, whereas the isolated-atom cross sections are not so simply related to the effective cross sections in the fluid.
The structure of this article is as follows. We first describe the methods that allow us to obtain cross sections in the gas and dense fluid phases from scattering calculations and then how we can use these to obtain the transport coefficients in the gas and dense fluid phases. Comparison of calculated transport coefficient calculations under equilbirum and non-equilbrium conditions (driven out of equilibrium through the application of an applied field) with available experimental measurements represents a stringent test on the accuracy of our position cross-sections for dense gas and liquid phases. We validate our scattering calculations using gas phase data, for which both total cross section measurements and transport data are available [8,9]. Then we apply the dense fluid formalisms, for which we can compare to experimental measurements in the dense gas [10] and liquid [11] regimes.
The dense gas comparisons suggest, with reference to previous analysis [12], that there is an incompatibility with several of the measurements. We have been able to show that density effects are significant in the dense gas phase only at low reduced electric fields, by performing full calculations and through simple qualitative arguments. This has allowed us to suggest that an admixture of an impurity may resolve the discrepancies between our calculations and experimental measurements. We will then perform similar analysis for the liquid phase, and discuss the incompatibilities between the calculations and measurements.

II. KINETIC THEORY AND TRANSPORT PROPERTIES
The kinetic equation used here to describe a positron swarm subject to an external electric field E in a background of gaseous or liquid helium is Boltzmann's equation (BE) for the phase-space distribution function. As shown in our previous works [5,6], comparison with positron swarm experiments can be made with only the steady-state, spatially-homogeneous solution: by performing a Legendre polynomial P l decomposition of the distribution function: and the collision integral J(f ): Details of our calculation method can be found in [5,6], which include a specialized collision operator for the coherent elastic scattering. For the current investigation, we must also include the annihilation process for the positron. This requires the definition of the annihilation collision operator The BE allows a connection between microscopic scattering information, and macroscopic transport properties. The macroscopic transport quantity of interest in this work is the average annihilation rate α an , which can be calculated from the energy distribution function,

III. SCATTERING OF POSITRONS BY INDIVIDUAL HELIUM ATOMS
The theoretical procedures used in this paper to describe the elastic scattering of positrons from helium atoms, at energies below the positronium formation threshold at 17.79 eV, are given in [13] and are essentially the same as those used in [5,6] for electron scattering from argon and xenon. Thus, only a brief discussion of the overall method will be given here.
In the purely elastic energy region, only the static and polarization potentials need to be included in the interaction for positron scattering. The scattering of the incident positrons, with wavenumber k, by helium atoms can then be described in the gaseous phase by the integral equation formulation of the partial wave Dirac-Fock scattering equations (see [13] for details). In matrix form, these equations can be written as where f κ (r) and g κ (r) are the large and small components of the scattering wavefunction, G(r, x) is the free particle Green's function and U (r) is the local potential. In particular, U (r) contains the static as well as the dipole and quadrupole polarization interactions, with the latter being calculated by the the polarized orbital method [14,15]. The calculation of the momentum transfer cross section σ mt from these potentials is discussed in [5,6].
For positron scattering we also require a cross section for annihilation or its equivalent designation in terms of Z eff , the effective number of atomic electrons [16]: where r 0 is the classical electron radius, c is the speed of light (c = 1/α in a.u. where α is the fine-structure constant), v is the velocity of the incident positron and Here Ψ is the total scattering wavefunction and the r i are the coordinates (including spin) of the atomic electrons while x is the position vector of the incident positron. The quantity Z eff can then be expressed as where Here ρ 0 (r) is the unperturbed density of the atomic electrons and ρ 1 (r) is the first-order correction. In terms of the atomic wavefunctions ρ 0 (r) is given by where P nκ (r) and Q nκ (r) are the large and small radial components of the atomic wavefunctions while q nκ = 2|κ| is the occupation number of the nκ subshell of a closed shell atom.
The first-order charge density was determined by the non-relativistic polarized orbital method [14,17], as relativistic effects are essentially negligible in light atomic systems. In the polarized-orbital method the first-order radial distortion F νν nl (r, x) of each atomic orbital P nl (r) is calculated adiabatically in the field of a point charge at a series of fixed points x (c.f. equation (12) of [15]). The corresponding non-relativistic scattering wavefunction f l (r) is normalized at infinity according to  [13] used in this paper to various experimental measurements [8,9].
Here, k is the wavenumber of the incident positron while δ l is the partial wave phase shift.
The correction to the charge density is then found by keeping only terms to first order and is given by where q nl = 2(2l + 1) is the occupation number of the nl subshell of a closed shell atom.
A comparison of the single-atom elastic cross sections to single-scatter experiments is shown in figure 1.

A. Transport coefficients
In order to test our calculation procedure in the dilute gas case, we can compare to various experimental measurements of the thermal zero-field annihilation rate [1,18,19] and to field-dependent measurements at 3.5 amagat of Davies et al. [10]. The general consensus of the zero-field effective atomic number for room temperature is Z eff T 0 ≈ 3.9 . The averaged Z eff (E) for gas phase compared to measurements for two different gas densities [10]. Although the calculation is within the error bars of the measurements at 3.5 amg density, the higher density measurements are significantly different. and our value of Z eff T 0 = 3.84 at 300 K is in good agreement. Our field-dependent results, shown in figure 2, are also in agreement with experiment, although the large uncertainties provide some leeway for variation.
We should note that the steady-state distribution, f (v) in equation (2), is a nonequilibrium distribution, even in the non-equilibrium case owing to the "hole-burning" effect provided by the energy-dependence of the annihilation collision frequency. Furthermore, it is also conceivable that the time-dependent behaviour of the positron swarm, as it approaches steady-state, could result in too few positrons that survive to reach the true steady-state f (v) distribution. If this were the case, then the experimental measurements would correspond to an average over transient distributions instead of steady-state. Fortunately, it has been shown [12,20] that enough positrons survive to accurately represent the steady-state distribution.

IV. SCATTERING OF POSITRONS BY DENSE HELIUM FLUIDS
Our approach to calculating the transport through liquids and dense gases, referred henceforth as dense fluids, is presented in [5,6]. In these papers we detailed the procedure, originally proposed by [21] for constructing effective scattering potentials for electrons in dense media. The procedure is almost identical for positron scattering and we do not repeat the formalism here but describe only the changes we have made for the current application to positrons. These include a) a contribution to the annihilation cross section from the average over surrounding atomic charge densities, b) a different choice of the outer radius of the scattering calculation and c) a potential shift, similar to that applied in our investigation of liquid krypton [7].
A. The averaged electron density ρ eff Analogous to the effective total potential, one can define an effective charge density with contributions from both the target atom and an ensemble average contribution from the atoms in the bulk, which acts to increase the positron annihilation rate in dense systems: Here ρ L = ρ 0 + ρ 1 corresponds to the focus atom's charge density and ρ S denotes the surrounding average. Note that the r m lower limit on the outer integral of ρ S indicates that only the charge density outside the region owned by the target atom contributes to the averaged density of its surrounding atoms. This is complementary to the upper limit of r m in equation (10). In other words, we consider any charge density within a range r m of an atom to be "owned" by that atom; this is necessary to prevent "double counting" of the electrons for each atom. In the dilute gas limit r m → ∞ and ρ S → 0 as required.
With the total averaged charge distribution defined, we can easily extend the definition of Z eff to include the total contribution from the focus and surrounding atoms: We have found that in our current focus of helium, the contribution of Z S eff to the total Z eff is negligible, however this may not be true for larger atoms. We have explored the choice of r WS before in our investigation of electrons in liquid argon, but found it to worsen the agreement between our calculations and experimental measurements. However, at that time we did not also apply an energy shift, which we discuss in the following section.

C. ∆V
Even as the positron velocity approaches zero, it will feel a background energy in the presence of a liquid or dense gas [23]. This quantity is known as V 0 and has been obtained through a combination of measurement and calculation for electron scattering in various liquids, see [22,24] and references therein. As it is not possible to do these same experiments with positrons, we instead investigate two different substitute values for V 0 . The first is U 2 (r → 0), which corresponds to the potential calculated from the average of the surrounding atoms at the origin, and the second surrogate is V WS , a Wigner-Seitz calculation in the style of [22], which we will describe in more detail in an upcoming paper. In short, the V WS value is found as the minimum energy solution for a wavefunction that satisfies a "spherical Bloch wave" boundary condition. We have applied a similar surrogate value for the potential shift when performing calculations of electrons in liquid krypton [7]. Note that the value of V WS itself depends on the value of r m and we will refer to V WS and V LWS as the potential shift from using the regular (r WS ) and local (r LWS ) Wigner-Seitz radii respectively.

D. Pair correlators in helium
The essential input to perform the dense fluid calculations is the fluid pair-correlator and its Fourier transform, the static structure factor. For liquid-phase helium at T = 4.2 K, we use the pair-correlator and structure factor derived from experiments by [25].  figure 4. As the pair correlator for the dense gas is relatively flat, it can be expected that some of the dense fluid effects will be negligible, however that contributions from the surrounding average will still be significant.

A. Experimental measurements
There are several measurements of the zero-field annihilation rate, see [1,27,28] for a compilation, which allow us to assume a value of approximately Z eff T 0 ≈ 3.9 for the dense gas phase and Z eff T 0 ≈ 3.6 for the liquid phase. Our calculations, using several different choices for ∆V , span a range of different increases/decreases in the zero-field Z eff T 0 . In both phases, ∆V = 0 shows an unusual increase in Z eff T 0 which cannot be reconciled with the experimental measurements.
We are only aware of a few measurements of the non-equilibrium field-dependent annihilation rate. These are [10] for the dense gas phase and [11,29] for the liquid phase. In both cases, there is a decrease in Z eff (E) as the field is increased. While our calculations, shown in figures 5 and 6, also show a decrease it happens a) over a larger variation of Z eff (E) for both dense gases and liquids and b) with a shoulder at either too small or too large a field. In addition, the calculated variation of Z eff (E) is much larger than experimentally observed. The similarity between the ∆V = V WS and ∆V = V LWS results in the dense gas was expected, as the key input distinguishing these approaches is the maximum in the pair correlator, which is negligible for the dense gas case. However, their behaviour in the liquid case is surprising: despite a 15% difference in r m , the two cases share almost identical elastic and annihilation cross sections, leading to almost identical Z eff values.
It is also possible for us to choose different values for our simulation parameters of r m and ∆V , which are not necessarily physically motivated. We have done this by scanning a wide range of values but no particular choice allows us to obtain both the required magnitude and field-dependence of Z eff (E), even approximately.
While the differences between our results and the experimental measurements in figures 5 and 6 appear to be quite large, this is due to a relatively small variation in Z eff . The differences between the ∆V = U 2 (0) calculations and the measurements are within 5% for the dense gas case and 10% for the liquid case. This could be accounted for by assuming a systematic uncertainty in the measurements, but we will instead consider what modifications can be made to our model to reconcile experiment and theory in the following sections. It is worth pointing out that the analysis involved in these measurements may be complicated by the ordering of the lifetimes for free positron annihilation and o-Ps annihilation: in the low density 3.5 amagat case, o-Ps annihilation is faster, and in the high-density 35.7 amagat case, free positron annihilation is faster [30].
We first discuss the dense gas case below in greater detail, and propose some modifications that we can make to explain the differences. We will then apply those considerations to the liquid phase.

B. Dense gas comparison
From figure 5, we can see that the various choices of ∆V allow us to tune the value of Z eff (E) at low fields. However, these choices all result in the same behaviour of Z eff (E) at high fields. We believe this should be expected from modifications due to the dense fluid, as large kinetic energies overwhelm these effects. It is rather the difference in the experimental measurements at higher reduced fields between the 3.5 amagat and 35.7 amagat results that we find surprising.
We have explored some modifications to our model of the gas in order to obtain agreement with experiment. In terms of transport quantities, we require one or both of the following modifications: either a) an additional source of annihilation which is significant at higher energies, or b) a source of friction to reduce the mean energy at higher fields. A lower mean energy has the desired side effect of increasing the Z eff felt by the ensemble, as the annihilation cross section is larger at lower energies.
Both of these effects can be produced by a small admixture of an impurity in the gas. The dominant effects of a molecular species as an impurity can be represented by two additional processes: another annihilation pathway and an inelastic cross section. In order to separate these effects, we first consider the zero-field case. Here, the positron distribution (neglecting the small perturbation from annihilation) will remain close to a thermal distribution. In this way, the additional inelastic cross section can be neglected and only the additional annihilation pathway will affect the measured Z eff T 0 . This leaves us with where x is the ratio of impurity density to helium density.
As it is likely that a mix of different hydrocarbons can play the role of impurities, we substitute their combined Z imp eff by a cross section that is proportional to that of ethane, i.e. Z imp eff ( ) = CZ C 2 H 6 eff ( ), and reinterpretx = xC as an effective ethane impurity density.
While this introduces an ambiguity into the impurity, it removes one fitting parameter from our calculations. We emphasize that even a few fitting parameters can allow us to fit any measured Z eff (E), so it is important to limit the number of these as much as possible.
Our simulations, after fitting forx at E = 0, are shown in figure 7. We can see that only a very small admixture is required to match the experimental value of Z eff T 0 . However, in the case of ∆V = 0, no amount of impurity will lower the Z eff T 0 value.
We now turn to including the second-most significant aspect of an impurity, which is the introduction of inelastic collisions with lower threshold energies than helium. As we again want to consider a range of hydrocarbon impurities, we use a surrogate cross section of constant magnitude A and threshold inel . We again reinterpret this quantity as the magnitudẽ A = xA which is an effective inelastic cross section, indicating a magnitude relative to the density of helium. By doing this, there are only three parameters to characterise the impurity:x,Ã and inel . The value ofx is fixed by the zero-field annihilation rate, so we now vary the latter two parameters to obtain the best fits shown in figure 8.
In all cases, the fits perform reasonably well and provide good agreement over most of the range of experimental measurements. The fit for ∆V = V WS includes an additional additional peak at around E/N = 2 × 10 −3 Td not seen in the experimental data, while the ∆V = U 2 (0) curve does not follow the data as closely.
As an alternative, we can choose to believe that the comparison of absolute values from our calculation and measurement may not be well posed, and instead we can compare the Z eff (E) values relative to the zero-field Z eff T 0 . This is shown in figure 9, where the

C. Liquid comparison
As with the dense gas case, we can apply the same steps to include an impurity to better match the experimental measurements. There is less likelihood for the presence of an impurity in liquid helium, as it would be expected to freeze out of the liquid. In any case, we can consider the effect it would have.
For the liquid the ∆V = U 2 (0) case produces a zero-field value which is higher than the experiment, even without the inclusion of an impurity. This means that we can only consider the ∆V = V WS case as suitable to add an impurity.
The effect of the impurity in the ∆V = V WS case is shown in figure 10, with and without Although the inclusion of the impurity can adjust the zero-field rate to bring it into agreement with the experimental measurement, the rest of the field range is not in agreement, even with the inclusion of an inelastic process. The uptick in the experimental measurements at the higher fields has been shown to be due to Ps formation as the positrons reached a steady-state distribution.
the additional of an inelastic process. It is clear to see that we cannot obtain agreement.
This is somewhat surprising, as we have some free parameters to manipulate. We believe this suggests that there is a contribution missing from our calculations, which is due to multiple scattering at high densities.
We should also point out that we should not aim to fit the uptick in the experimental results at high fields. This has been shown [11] to be an apparent increase only, and is actually due to the formation of positronium with ionised electrons. This spur-enhanced Ps formation is estimated to be at most 1% and only occurs at higher fields. As the apparent Z eff is about 1.2% larger at the higher fields, this fits almost perfectly with this explanation.
We can also consider the possibility of positrons forming self-trapped clusters of higher density in the helium liquid [31]. However, these clusters have been found to only be present for densities less than that of liquid helium. Hence, we can ignore this mechanism as a source of increased Z eff .
Finally, we note that we have not accounted for a difference between the applied and effective electric fields due to the permittivity of the liquid. This is because the effect is negligible, as the dielectric constant [32] of helium is 1.05 ≈ 1.
Again, we have made a comparison with the relative difference to the zero-field Z eff value, shown in figure 11. In contrast to the similar comparison in figure 9, there is a much bigger difference in the choices of ∆V for calculation, but the ∆V = V WS and ∆V = V LWS choices remain closest to the experimental measurements.
While the inability to fit the liquid results is problematic for our calculation method, we still believe that our approach to obtain agreement for the dense gas case is valid. This is because the density of 35.7 amg in the measurements of [10] is a rather dilute density, so multiple-scattering effects should also be relatively weak. However, we cannot completely rule out the possibility that our dense gas calculations are also lacking some additional physical behaviour.

VI. CONCLUSIONS
We have modelled the transport of positrons under an applied electric field through dense fluids of helium and compared our predictions of annihilation rates to experimental measurements in the dense gas and liquid phases. Our model includes modifications due to coherent scattering and screening of the interaction potential between the positron and the helium atom, which have been discussed in previous articles. This article has introduced additional considerations for the annihilation rate due to electrons from the surrounding atoms, and we have shown that double counting should be avoided in the averaging process.
While our model does not provide results in complete agreement with experimental measurements, we are able to include a very small (≈ 0.1%) contribution of an impurity that is representative of a hydrocarbon to vastly improve this agreement. The impurity is motivated by the noticeable difference between the dilute gas and dense gas measurements and is represented using a model which includes three fitting parameters.
Our model has been extended from our previous articles, to include one adjustable pa- Seitz model using a "local" Wigner-Seitz radius [22]. In both the dense gas and the liquid, the effect of ∆V = V WS and ∆V = V LWS were found to be nearly identical. In the case of a dense gas of helium, any of these choices can be made to agree with the experimental measurements, using different choices of an impurity admixture. However, for the case of liquid helium, only the choices of ∆V = V WS or ∆V = V LWS were found to be compatible, yet there remained significant discrepancies between our calculated values and the experimental measurements.
Our results, using the Boltzmann equation description outlined in this article, have also been independently verified using a Monte-Carlo calculation. Details of that implementation are available in [33,34].
One of the reasons that impurities can play a large role in our current investigations is due to the very small Z eff of helium. In the future, we wish to model positron transport through fluids of larger atomic species. These atoms, with many more electrons, may provide a means to better test our calculations by suppressing the potential effects of impurities.
In addition, we wish to explore further choices of ∆V and determine a method to uniquely specify its value. One manner in which to do this is to consider species in which there are a larger range of densities with experimental measurements, such as krypton [35]. We also intend to include further multiple-scattering corrections to both ∆V and Z eff [36,37], to see if these can identify the current disagreement between our calculations and experimental measurement.

VII. ACKNOWLEDGMENTS
We would like to acknowledge valuable discussions with Prof. M. Charlton regarding the possibility of impurities in the dense gas experimental measurements.