Positron scattering and transport in liquid helium

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 (Paul1957, ; Charlton2001, ; Charlton2009, ) and to field-dependent measurements at 3.5 amagat of Davies et al. (Davies1989, ). The general consensus of the zero-field effective atomic number for room temperature is $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}\approx 3.9$ and our value of $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}=3.84$ 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, $f(\bm{v})$ in equation (2), is a non-equilibrium 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 $f(\bm{v})$ 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 (Green2017, ; Boyle2014, ) that enough positrons survive to accurately represent the steady-state distribution.

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 $\rho_{L}=\rho_{0}+\rho_{1}$ corresponds to the focus atom’s charge density and $\rho_{S}$ denotes the surrounding average. Note that the $r_{m}$ lower limit on the outer integral of $\rho_{S}$ 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 $r_{m}\rightarrow\infty$ and $\rho_{S}\rightarrow 0$ as required.

With the total averaged charge distribution defined, we can easily extend the definition of $Z_{\mathrm{eff}}$ to include the total contribution from the focus and surrounding atoms:

We have found that in our current focus of helium, the contribution of $Z_{\mathrm{eff}}^{S}$ to the total $Z_{\mathrm{eff}}$ is negligible, however this may not be true for larger atoms.

The value of $r_{m}$ represents the region of space “owned” by the focus atom and distinguishes it from the rest of the bulk. In comparison to our previous works involving electrons in dense fluids, a different definition for $r_{m}$ is required for positrons in dense fluids. In our previous works, we followed the lead of Lekner (Lekner1967a, ) to choose $r_{m}$ as a turning point of the potential. However, in the current case of positron scattering, where the sign of the static potential is reversed, this definition results in a much larger value of $r_{m}$ which appears to be physically invalid. Hence we make an alternative choice of setting $r_{m}$ to the Wigner-Seitz radius, $r_{\mathrm{WS}}=(4\pi N/3)^{-1/3}$. In the dense gas phase of helium at $N=35.7$ amagat, this is $r_{\mathrm{WS}}=6.29\,\text{\r{A}}$ and in the liquid phase at $N=0.0188\,\text{\r{A}}^{-3}$, this is $r_{\mathrm{WS}}=2.33\,\text{\r{A}}$. These radii, compared with the relevant potentials of the liquid problem, are shown in figure 3. We have also explored an alternative choice for the Wigner-Seitz radius (Evans2010, ), called the “local Wigner-Seitz radius” $r_{\mathrm{LWS}}=(4\pi Ng_{\mathrm{max}}/3)^{-1/3}$, where $g_{\mathrm{max}}$ is the maximum of the pair correlator $g(r)$. This quantity attempts to account for the increased density that the positron would feel in the majority of collisions. In our case, this results in a value of $r_{\mathrm{LWS}}=6.24\,\text{\r{A}}$ in the dense gas phase, and $r_{\mathrm{LWS}}=2.08\,\text{\r{A}}$ in the liquid phase.

We have explored the choice of $r_{\mathrm{WS}}$ 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.

Even as the positron velocity approaches zero, it will feel a background energy in the presence of a liquid or dense gas (Iakubov1982, ). 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 (Evans2010, ; Borghesani2006, ) 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 $U_{2}(r\rightarrow 0)$, which corresponds to the potential calculated from the average of the surrounding atoms at the origin, and the second surrogate is $V_{\mathrm{WS}}$, a Wigner-Seitz calculation in the style of (Evans2010, ), which we will describe in more detail in an upcoming paper. In short, the $V_{\mathrm{WS}}$ 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 (White2018, ). Note that the value of $V_{\mathrm{WS}}$ itself depends on the value of $r_{m}$ and we will refer to $V_{\mathrm{WS}}$ and $V_{\mathrm{LWS}}$ as the potential shift from using the regular ($r_{\mathrm{WS}}$) and local ($r_{\mathrm{LWS}}$) Wigner-Seitz radii respectively.

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 $T=4.2\ \text{K}$, we use the pair-correlator and structure factor derived from experiments by (Sears1979, ). For the dense gas case at $T=295.65$ K, we have calculated the pair-correlator from Monte Carlo simulations with $N=10000$ atoms using an untruncated Lennard-Jones potential with parameters (Oh2013, ) $\epsilon_{LJ}/k_{B}=5.465$ K and $\sigma_{LJ}=2.628\,\text{\r{A}}$. These pair correlators are shown in 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.

There are several measurements of the zero-field annihilation rate, see (Fox1977, ; Nieminen1980, ; Charlton2001, ) for a compilation, which allow us to assume a value of approximately $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}\approx 3.9$ for the dense gas phase and $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}\approx 3.6$ for the liquid phase. Our calculations, using several different choices for $\Delta V$, span a range of different increases/decreases in the zero-field $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$. In both phases, $\Delta V=0$ shows an unusual increase in $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$ 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 (Davies1989, ) for the dense gas phase and (Pepe1995, ; Pepe1994, ) for the liquid phase. In both cases, there is a decrease in $\langle Z_{\mathrm{eff}}\rangle(E)$ 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 $\langle Z_{\mathrm{eff}}\rangle(E)$ 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 $\langle Z_{\mathrm{eff}}\rangle(E)$ is much larger than experimentally observed. The similarity between the $\Delta V=V_{\mathrm{WS}}$ and $\Delta V=V_{\mathrm{LWS}}$ 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 $\langle Z_{\mathrm{eff}}\rangle$ values.

It is also possible for us to choose different values for our simulation parameters of $r_{m}$ and $\Delta V$, 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 $\langle Z_{\mathrm{eff}}\rangle(E)$, 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 $Z_{\mathrm{eff}}$. The differences between the $\Delta V=U_{2}(0)$ 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 (Charlton2020, ).

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.

From figure 5, we can see that the various choices of $\Delta V$ allow us to tune the value of $\langle Z_{\mathrm{eff}}\rangle(E)$ at low fields. However, these choices all result in the same behaviour of $\langle Z_{\mathrm{eff}}\rangle(E)$ 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 $\langle Z_{\mathrm{eff}}\rangle$ 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 $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$. 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 $\langle Z_{\mathrm{eff}}^{\mathrm{imp}}\rangle$ by a cross section that is proportional to that of ethane, i.e. $Z_{\mathrm{eff}}^{\mathrm{imp}}(\epsilon)=CZ_{\mathrm{eff}}^{\mathrm{C}_{2}\mathrm{H}_{6}}(\epsilon)$, and reinterpret $\tilde{x}=xC$ 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 $\langle Z_{\mathrm{eff}}\rangle(E)$, so it is important to limit the number of these as much as possible.

Our simulations, after fitting for $\tilde{x}$ 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 $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$. However, in the case of $\Delta V=0$, no amount of impurity will lower the $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$ 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 $\epsilon_{\mathrm{inel}}$. We again reinterpret this quantity as the magnitude $\tilde{A}=xA$ 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: $\tilde{x}$, $\tilde{A}$ and $\epsilon_{\mathrm{inel}}$. The value of $\tilde{x}$ 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 $\Delta V=V_{\mathrm{WS}}$ includes an additional additional peak at around $E/N=2\times 10^{-3}$ Td not seen in the experimental data, while the $\Delta V=U_{2}(0)$ 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 $\langle Z_{\mathrm{eff}}\rangle(E)$ values relative to the zero-field $\langle Z_{\mathrm{eff}}\rangle_{T_{0}}$. This is shown in figure 9, where the $\Delta V=V_{\mathrm{WS}}$ and $\Delta V=V_{\mathrm{LWS}}$ appear to give the closest fit, although all choices are not unreasonable.

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 $\Delta V=U_{2}(0)$ 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 $\Delta V=V_{\mathrm{WS}}$ case as suitable to add an impurity.

The effect of the impurity in the $\Delta V=V_{\mathrm{WS}}$ case is shown in figure 10, with and without 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 (Pepe1995, ) 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 $\langle Z_{\mathrm{eff}}\rangle$ 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 (Manninen1978, ). 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 $Z_{\mathrm{eff}}$.

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 (Chase1961, ) of helium is $1.05\approx 1$.

Again, we have made a comparison with the relative difference to the zero-field $\langle Z_{\mathrm{eff}}\rangle$ value, shown in figure 11. In contrast to the similar comparison in figure 9, there is a much bigger difference in the choices of $\Delta V$ for calculation, but the $\Delta V=V_{\mathrm{WS}}$ and $\Delta V=V_{\mathrm{LWS}}$ 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 (Davies1989, ) 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.