Reduction of fast ion drag in the presence of ‘hollow’ non-Maxwellian electron distributions

It is argued that the electronic stopping power in a plasma should be expected to exhibit significant differences in the presence of effects that shift the electron distribution function away from a Maxwellian. This is potentially important for nuclear reactions produced by laser-driven ion beams, where non-Maxwellian effects may have to be considered. We have calculated the electronic stopping power for a number of model distributions. Importantly, comparisons with the Maxwellian are done under the condition of energy density parity. ‘Hollow’ electron distribution functions (e.g. f∝vnfmax ) could be expected to show a reduced stopping power (when vi/vt<1 ). We show that this is indeed the case and that the difference can become a factor of 70. The super-Gaussian electron distribution function, on the other hand, will always show a higher stopping power than the Maxwellian for orders greater than 2.


Introduction
The study reported in this paper is motivated by the considerable interest there has been in nuclear reactions driven by ultraintense laser irradiation, which has been a highly active area since the development of multi-terawatt (TW) chirped pulse amplification (CPA) laser systems [1][2][3][4][5][6][7][8][9][10][11][12].These nuclear reactions are generally thought to occur due to the generation of energetic proton or ion beams (a well established phenomena in laser-solid interactions [13][14][15][16][17][18][19][20]) which then propagate into the irradiated target, a secondary target, or other surrounding material.It should be noted that gamma rays generated in these interactions can also drive nuclear reactions.In the course of many investigations over several years, there have been a number of studies where it has been difficult to reconcile the experimental results with theoretical predictions.Examples of this Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.include work by Guiffrida [4], and Toupin [2].One possible way to resolve these problems would be for the ion range to be enhanced above that which is expected in these theoretical studies, and it is the potential for enhancing the ion range that is the subject of this paper.
In the case of laser-generated ion beams, the nuclear reaction yield is strongly limited by the drag on the fast ions by the background electrons in the target [4].The spatial range over which an ion will have high energy is limited by the drag, leading to a stopping distance (λ s ).The probability of a nuclear reaction occurring before the ion loses its energy will thus be determined by, where n t is the target ion density.For ions in the low MeV range (relevant to relativistic laser-matter interactions), the stopping range in a cold target is on the micron scale.Since nuclear cross sections for ion driven reactions tend to not exceed one barn, we can thus see that the stopping range strongly limits the nuclear reaction probability (P ≪ 1).
If there are physical effects that can increase the ion range by reducing the drag due to the electrons, then the reactions yield could be increased.It is generally accepted that strongly heating the electrons to the point where v e,th ≫ v i,b (where v e,th is the electron thermal velocity, and v i,b is the velocity of the beam ions), leads to substantial reduction in the electronic drag as described by the well-known Chandrasekhar correction [21,22] to the electronic stopping power.Strong target heating is quite possible in ultra-intense laser-matter interactions, even in dense regions of the targets as heating can occur due to the propagation of supra-thermal electron [23][24][25], or ion beams [26,27].
That said, it is somewhat natural to ask if there are any effects beyond straightforward heating that can affect ion ranges.It is known that non-Maxwellian electron distributions can be produced in laser-matter interactions.A prominent example is the generation of super-Gaussian distributions via the Langdon effect [28][29][30][31].In this paper we have evaluated the potential for non-Maxwellian distributions to cause further enhancement of ion ranges.We show that 'hollow' electron distribution functions (where there are few electrons at very low energy) will generally achieve much reduced stopping power in comparison to a Maxwellian distribution under the stipulation that we compare cases of equal energy density.In contrast electron distributions that have more electrons at low energy will have enhanced stopping relative to a Maxwellian distribution of the same energy density, such as a super-Gaussian.We have not evaluated the potential for these 'hollow' distributions to be produced in any specific experiment, rather we argue that the potential for these distributions to reduce the stopping power means that the possibility that they might exist needs to be considered when analyzing any extant or future experiments.On the basis of the results presented herein, it would be worthwhile for researchers to carefully consider how such 'hollow' electron distributions might be generated in laser-driven experiments.

Notation
In this paper we will make reference to the gamma function, Γ(x), i.e.
as well as the incomplete gamma function, γ(s, x) Note that throughout the paper we will make reference to the ratio y = v i /v e,th , where v e,th = √ 3eT e /m e for T e in units of eV.This means, for example, that a proton at approximately 500 keV will yield y = 1 when the background electron is around 190 eV.When the background electron temperature reaches 2000 eV, we reach y ∼ 0.3.Of course, y will need to be determined for the particular ion and energy of interest to any reader, however this may serve as a guide that can help with the later sections of the paper.

Concept
The central argument of this paper is that a non-Maxwellian distribution function should be expected to produce a significantly different ion stopping power, and more specifically that electron distribution functions where the maximum in f is shifted away from v = 0 (but still remains isotropic) should exhibit reduced stopping power relative to a Maxwellian.
It is widely accepted that the electronic stopping power of a Maxwellian plasma reduces as the electron thermal speed approaches and overtakes the ion velocity with increasing temperature.The root of this can be seen by considering the Coulomb scattering of a single electron.If we transform to the frame where the ion is stationary (but non-relativistic), then the momentum change parallel to the initial direction of the electron can be found to be approximately, where b is the impact parameter, and v 0 is the initial approach velocity of the electron.It is important to note here that v 0 is the relative velocity of the electron and ion.If the electron thermal velocity is very small compared to v i then v 0 ≈ v i .On the other hand, once the electron thermal velocity exceeds the ion velocity, we can expect v 0 > v i for many electrons and thus we should expect the stopping to become progressively smaller than the cold limit as the electron thermal velocity increases.
From this it would also appear to follow that we should expect different stopping powers for different distribution functions, perhaps even when the energy densities are equal.Consider an electron distribution function that approaches a spherical shell, but with low velocity spread across the shell.It is not unreasonable to conjecture that this might exhibit a smaller stopping power than a Maxwellian distribution of comparable energy density.We might term such a distribution function, and those like it, a 'hollow' distribution function.
Although the possibility appears to be present in qualitative terms, justification requires a proper calculation of the stopping power, which is what we shall proceed to do in the rest of this paper.

Theoretical framework
In order to evaluate the stopping power (or drag) for a given electron distribution function we have employed the 'dielectric formulism', which is very widely used throughout the literature [21,22].This model starts from the position of trying evaluate the drag on the ion in terms of a net electric field, i.e. ) after which it is assumed that the plasma response can be treated in terms of a linear dielectric response.Thus after casting the field in terms of a Fourier transform, one obtains, where D is the dielectric function which needs to be determined from a separate model of the plasma response.Finally, one can obtain an approximation for this in the form: where L is a 'stopping number' (with slow logarithmic variation), µ is the cosine of the angle between the k-vector and ion velocity, and Y is related to the dielectric function via, Thus the key part of the calculation is actually the determination of the dielectric function and extracting the imaginary part of this.There is a well-established procedure for obtaining a dielectric function through the linearization of the Vlasov equation.The details of this are given in Krall and Trivelpiece [32], and other texts [33].If we only consider the electron contribution, then the dielectric function from this approach is, where f e,0 is the unperturbed electron distribution function.In the case of a Maxwellian electron distribution, one obtains where η = ω/v t k (v 2 t = 2k B T/m e ), and, from which one can determine that ℑ(Z) = √ πe −η 2 .When this is used in equation ( 7), one obtains, where This is the standard stopping power for a warm plasma via the Chandrasekhar correction, as one would expect to obtain.In what follows we will apply this method to a number of hypothetical electron distributions in order to validate our hypothesis concerning 'hollowed' electron distribution functions.

Results
Although a number of different possible distribution functions were considered, we have chosen two to present here.There are two reasons for these choices.Firstly these will fully illustrate the point we wish to make in this paper with a minimum of complication, and secondly as the calculation is much more straightforward than many other possible choices.These two choices are: and, In both equations ( 14) and (15), n e is the electron density, v 2 t = 2k B T/m e , and α and β are parameters that are included to ensure that we can achieve energy density parity with a Maxwellian electron distribution at the same temperature.On calculating (by taking the moment) the energy density, then for equation (14) we have that U = 2α 2 n e k B T, so energy density parity is achieved for the choice of α = √ 3/2.For the distribution given by equation (15) we have U = 5  2 β 2 n e k B T, so energy density parity is achieved with a Maxwellian for the choice of β = √ 3/5.Otherwise both distribution functions are fully normalized, and yield the density n e on taking the zeroth moment.
Having defined the distribution functions of interest, we can proceed to determine the stopping powers using the methodology described in section 4. Firstly one needs to calculate the dielectric functions.As the distributions are isotropic, one only needs to evaluate one arbitrary choice of k.If, for example, this is chosen to be parallel to one axis of a Cartesian coordinate system for velocity space then this evaluation can be done in Cartesian.One can then integrate over the perpendicular velocity components first.This will eventually lead to a single integral that need not be evaluated.For the distribution given by equation ( 14) we obtain, where, Likewise for the distribution function given in equation ( 15) we obtain, with, With the dielectric functions one can proceed to obtain the stopping powers via equation (7).The Y-functions are extracted from equations ( 17) and ( 19) (using the relation of equation ( 8)) which yields: and The final integrals can be carried out after noting that η = µv i /αv t and η = µv i /βv t respectively, which follows from the dielectric function in equation ( 7) being evaluated at ω = k.vi .If we define y 1 = v i /αv t and y 2 = v i /βv t , then the stopping powers that we obtain can be written as, for the distribution function given by equation ( 14), and, ] . ( It can seen that in terms of comparing stopping powers, all that needs to be done (due to the pre-factors being identical) is to compare the 'correction' functions, which we can define as: for the distribution given by equation ( 14), and, We can now compare the stopping powers purely in terms of comparing the correction factors.Firstly let us just compare the raw functions.This is done in figures 1 and 2.
From figures 1 and 2, we can see that, as y falls, the correction factor shrinks faster for both of the 'hollow' electron distribution functions when compared to the Maxwellian case.This is not a small difference either: for the distribution function described by equation ( 15) the correction factor is more than 100 times smaller than the Maxwellian correction factor at y = 0.1.To properly compare the correction factors we must account for α and β.This is done in figures 3 and 4, where we plot G(y)/H(y/α) and G(y)/I(H/β) respectively using α = √ 3/2 and β = √ 3/5 to show the comparison under conditions of energy density parity between these distribution functions and a Maxwellian.
From figure 3, it can be seen that when y ∼ 0.1 the stopping power (under otherwise identical conditions) is about eight times lower in the case of the electron distribution function given in equation ( 14) compared to a Maxwellian.Likewise,  13), ( 24) and ( 25).13), ( 24) and (25).
in figure 4. it can see that when y ∼ 0.1, the stopping power is about 70 times lower in the case of the electron distribution function given in equation ( 15) compared to Maxwellian under otherwise identical conditions.
We can thus conclude that different (than a Maxwellian) electron distribution functions can give substantially lower drag even when the energy densities are equal.We have argued that this should occur for 'hollow' electron distribution, and the distributions functions of equations ( 14) and ( 15) are indeed such 'hollow' distribution functions which thus support and act to validate this argument.We can also provide validating evidence by considering an opposing case, namely the 'super-Gaussian', i.e.
Note that here m denotes the power of the super-Gaussian function and is not a mass.A key feature of the super-Gaussian is that the high-energy tail vanishes for m > 2, and there is a corresponding increase in the number of electrons around 1-2v t .Energy density parity with a Maxwellian can be achieved by choosing, By proceeding to use the same methodology, one arrives at, where, and, On proceeding to calculate the correction factor (which we denote as G (m) SG , one obtains, Note that by using γ(1/2, x) = √ πErf √ x, and γ(a + 1, x) = aγ(a, x) − x a e −x it is possible to show that the standard Maxwellian result is obtained for the case of m = 2.The leading term in the stopping power is identical to that obtained previously (i.e.equations ( 22) and ( 23)) so this correction factor can be directly compared to those we have obtained earlier.
The 'raw' super-Gaussian correction factors are plotted in figure 5 for the cases of m = 2 (Maxwellian), 3, and 4. From figure 5 it is clearly seen that the super-Gaussian distributions functions with m > 2 all have higher stopping power (larger correction factor) than a Maxwellian.Thus it can be seen that when we consider a distribution function that is distorted from the Maxwellian so as to concentrate more electrons at modest energies we obtain a higher stopping power.To assist the reader in discerning the difference in the correction factors, in figure 6, the ratio G (4) SG (y) is plotted.This shows that the difference in the correction factor at small y ranges from 1.4 to 1.8 for a fourth order super-Gaussian relative to a Maxwellian, thus showing that the difference is far from negligible.
Thus we have seen in this section that, when the stopping powers for various distribution functions are calculated, those distributions functions which are 'hollowed out' with respect to the Maxwellian exhibit a significantly lower stopping power than the Maxwellian.When one calculates the stopping power for super-Gaussian distribution functions where there is an increased concentration of electrons at moderate energies, and the hot tail of the distribution vanishes, this is found to be greater than the Maxwellian stopping power.All of this validates the hypothesis set out in section 3.

Conclusions
In this paper we have shown that non-Maxwellian electron distribution functions can have significantly different ion stopping powers compared to a Maxwellian distribution in the regime where v i /v t < 1.We have calculated the stopping powers for several different cases.Two are cases of 'hollow' electron distribution functions (equations ( 14) and ( 15)) and we have also calculated this for the general case of an mth order super-Gaussian.We have indeed shown, through these full calculations, that the stopping power can be significantly different for non-Maxwellian distribution functions.In the case of the distribution described by equation ( 15), one can have differences in the range of 10-70 when v i /v t is no smaller than 0.1, and under the condition of energy density parity with a Maxwellian.
As there does appear to be a strong case that a non-Maxwellian electron distribution function has the potential to make a large difference to the electronic stopping power and thus the possible role of non-Maxwellian effects should demand greater consideration in the interpretation of future experiments and the development of laser-driven nuclear reactions into various technological applications.In particular the results suggest that researchers should give serious consideration as to how 'hollow' electron distribution functions could be produced in laser-driven experiments.
In closing we would like to take the opportunity to point out that there is a possibility that 'hollow' electron distribution functions may be a natural consequence of very high intensity ion beams.The argument for this possibility stems from considering the electron kinetic equation in the presence of a sufficiently supra-thermal monoenergetic ion beam.The advective collisional term in the electron kinetic equation in this case, approximates to ( ∂f ∂t where, ψ = Z 2 i e 4 Ln ib 4πϵ 2 0 m 2 e v 3 i If we now expand the distribution function in spherical harmonics, we thus find that there is a term that corresponds to a spherical advection term, i.e.
( ∂f ∂t where, and n ib is the ion density of the incident ion beam, and v i is the ion velocity of the monoenergetic incident ion beam.This term would tend to produce a 'hollowed' distribution function.This potential effect requires much more consideration before any more definitive statements could be made, however this argument is put here just to point out that the concept of 'hollow' distribution functions is motivated by physical considerations beyond the effect that this has on stopping power.