1. INTRODUCTION
The energy loss of charged particles in a free electron gas is of considerable interest to actual slowing-down problems. This is a topic of relevance to understand the beam-target interaction in the contexts of particle driven fusion (Deutsch, Reference Deutsch1984; Hoffmann et al., Reference Hoffmann, Weyrich, Wahl, Gardes, Bimbot and Fleurier1990; Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001; Deutsch & Popoff, Reference Deutsch and Popoff2006; Nardi et al., Reference Nardi, Fisher, Roth, Blazevic and Hoffmann2006; Nardi et al., Reference Nardi, Maron and Hoffmann2007). Energy losses of ions moving in an electron gas can be studied through the stopping power of the medium (Eisenbarth et al., Reference Eisenbarth, Rosmej, Shevelko, Blazevic and Hoffmann2007). Dielectric formalism has become one of the most used methods to describe this stopping power. The use of this formalism was introduced by Fermi (Reference Fermi1940). Subsequent developments made it possible to extend the dielectric formalism to provide a more comprehensive description of the stopping of ions in matter (Lindhard, Reference Lindhard1954; Lindhard & Winther, Reference Lindhard and Winther1964). For dilute plasmas, the dielectric formulation of the energy-loss rate was first studied by Pines and Bohm (Reference Pines and Bohm1952), Akhiezer and Sitenko (Reference Akhiezer and Sitenko1952), and other scientists. Large number of calculations of electronic stopping forces of ions and electrons in plasmas has been carried out since then using the random phase approximation (RPA) (see Zwicknagel et al., (Reference Zwicknagel, Toepffer and Reinhard1999) for a complete list). The RPA is usually valid for high-velocity projectiles and in the weak coupling limit of an electron gas. But for partially coupled plasmas, which are subject of much interest for current studies of inertial confinement fusion (ICF), RPA it is not sufficient and the electronic interactions have to be taken into account. The coupling parameter, Γ, is a measure of target electron interactions. It is defined, in a degenerate electron gas, as the ratio between potential and kinetic energies of the electrons, Γ ≡ e 2/r smv F2, where r s is the Wigner-Seitz radius and v F is the Fermi velocity. In this article, the coupling will be treated through two different ways: the Mermin function or the local field corrections (LFCs).
Mermin (Reference Mermin1970), and later Das (Reference Das1975), derived an expression for the dielectric function taking into account the target electron interactions and also preserving the local particle density. Recently, extended dielectric function has been considered which conserves momentum and energy (Selchow et al., Reference Selchow, Ropke and Morawetz2000; Morawetz & Fuhrmann, Reference Morawetz and Fuhrmann2000; Atwal & Ashcroft, Reference Atwal and Ashcroft2002), but it is somewhat involved and it has only small differences with Mermin dielectric function. Mermin dielectric function has been successfully applied to solids (Barriga-Carrasco & Garcia-Molina, Reference Barriga-Carrasco and Garcia-Molina2003, Reference Barriga-Carrasco and Garcia-Molina2004) and to plasmas (Barriga-Carrasco, Reference Barriga-Carrasco2006a, Reference Barriga-Carrasco2006b, Reference Barriga-Carrasco2007; Barriga-Carrasco & Maynard, Reference Barriga-Carrasco and Maynard2006; Barriga-Carrasco & Deutsch, Reference Barriga-Carrasco and Deutsch2006).
On the other hand, some authors (Hubbard, Reference Hubbard1957; Singwi et al., Reference Singwi, Tosi, Land and Sjölander1968; Vashishta & Singwi, Reference Vashishta and Singwi1972; Vaishya & Gupta, Reference Vaishya and Gupta1973; Pathak & Vashishta, Reference Pathak and Vashishta1973; Ichimaru & Utsumi, Reference Ichimaru and Utsumi1980) have introduced the local field corrections to improve previous results based on the RPA theory. Mostly static approximations have been proposed (SLFC), as it was considered that greater part of the local field corrections will succeed for the static limit. Hubbard obtained an explicit expression for the static local field correction, which takes into account the exchange effects but neglects correlations (Hubbard, Reference Hubbard1957). Next step was made by Singwi et al. (Reference Singwi, Tosi, Land and Sjölander1968) which related the local field corrections with the static structure function. This last result had some deficiencies, as it violates the compressibility sum rule. Latter deficiency was removed by Vashishta and Singwi (Reference Vashishta and Singwi1972), and improved by Pathak and Vashishta (Reference Pathak and Vashishta1973) demanding that the response function fulfills the third-order frequency sum rule. Ichimaru and Utsumi (Reference Ichimaru and Utsumi1980) present a simple fitting formula for the static local field correction of coupled electron gas. But in Vaishya and Gupta (Reference Vaishya and Gupta1973) it was shown that one cannot construct a SLFC which fulfills both limits, the compressibility and the third-order sum rule. Therefore the concentration was mostly focused on the construction of dynamical local field corrections (DLFC). Then Yan et al. (Reference Yan, Tanaka, Mitake and Ichimaru1985) proposed a parametrization of the DLFC that takes into account the asymptotic behaviors in their frequency dependence. We will use in this work the Ichimaru and Utsumi, and the Yan et al. parametrizations for SLFC and DLFC functions, respectively.
Mermin dielectric function is derived in Section 2 while local field corrections are in Section 3. This two methods are compared in Section 4 for the electronic stopping of a proton traversing a degenerate plasma.
2. MERMIN DIELECTRIC FUNCTION
In this section, we are going to develop the Mermin dielectric function ɛM(k, ω) in terms of the wave number k and of the frequency ω provided by a consistent quantum mechanical analysis. First the dielectric response of the electronic medium is calculated in the random phase approximation (RPA). We use atomic units (a.u.), e = ħ= m e = 1, to simplify formulas. The RPA analysis yields to the expression (Lindhard, Reference Lindhard1954)
where 
. For degenerate plasmas, the distribution function is 
for k > k F and 
for k < k F, where k F is the Fermi wave number. In this part of the analysis, we assume the absence of collisions so the damping constant approaches zero, ν → 0+.
Dielectric function can be separated into its real and imaginary parts
ɛr(k, ω) can be directly obtained from Eq. (1), (Arista & Brandt, Reference Arista and Brandt1984)
where g(x) corresponds to
and u = ω/kv F and z = k/2k F are the common dimensionless variables (Lindhard, Reference Lindhard1954). 
is Fermi velocity in a.u.
The function ɛi(k, ω) also follows from Eq. (1), (Arista & Brandt, Reference Arista and Brandt1984)
We will see in Section 4 that for ion stopping considerations, it is worth defining the energy loss function (ELF)
![\hbox{ELF} \equiv \hbox{Im} \left[{-1\over \varepsilon \lpar k\comma \, \omega\rpar }\right].](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S0263034608000402_eqn6.gif?pub-status=live){kind=small}
As mentioned in the introduction, the RPA it is not sufficient for partially coupled plasmas and the target electron interactions have to be taken into account. Mermin dielectric function (1970) is derived taking care of these interactions and also preserving the local particle density
where ɛ(k, ω) is the RPA dielectric function from Eq. (2). Electron collisions are considered through their collision frequency, ν. It is easy to see that when ν → 0, the Mermin function reproduces the RPA one.
The collision frequency ν in solids can be determined experimentally, but in plasmas, nowadays, it must be calculated theoretically (Barriga-Carrasco, Reference Barriga-Carrasco2008). It is known that in a fully ionized plasma, the collision frequency is determined by electron-electron (e-e) and electron-ion (e-i) Coulomb collisions (if we do not consider impurities). We can assert that the total effective frequency can be obtained as the sum of these collisions ν = νee + νei (this is an extension of the Matthiessen rule to partially degenerate plasmas, Cassisi et al., Reference Cassisi, Potekhin, Pietrinferni, Catelan and Salaris2007). Then ν can be easily divided into e-e collisions and e-i collisions to study their effects separately.
The e-e collision frequency of nonrelativistic degenerate electrons was first analyzed by Lampe (Reference Lampe1968a, Reference Lampe1968b) using the formalism of the dynamic screening of the e-e interaction. After that, Flowers and Itoh (Reference Flowers and Itoh1976) obtained the expression for the relativistic degenerate electrons. Recently, Shternin and Yakovlev (Reference Shternin and Yakovlev2006) obtained an analytical formula for nonrelativistic and relativistic electrons at high degeneracy
where 
, βr = x/(1 + x 2)1/2, x = p F/m ec is the relativistic parameter of degenerate electrons, ωp is the plasma frequency, and α is the fine-structure constant. On the other hand, I(βr, y) function is
where A = 12.2 + 25.2βr2, B = Aexp[(0.123636 + 0.016234βr2)/C], C = 8/105 + 0.05714βr4, and D = 0.1558y 1−0.75βr.
The effective e-i collision frequency for degenerate plasmas was also derived by Flowers and Itoh (Reference Flowers and Itoh1976) and lately by Shternin and Yakovlev (Reference Shternin and Yakovlev2006)
where n i is the ion density, Z i is the ion atomic number, and Λei is the Coulomb logarithm. Then the total electron frequency results from
Figure 1 shows RPA and Mermin energy loss function dependence with ω/ωp when k/k F = 1, for a n e = 1023 cm−3 degenerate plasma. The collision frequency used in Mermin case is ν = 3.6 fs−1; it is obtained from the last procedure. Solid line represents RPA ELF while dashed line represents Mermin ELF. When collisions are considered through Mermin dielectric function, the ELF increases around ω = 0. Also its maximum and the edge at ω/ωp = 2 smooth in a great deal. In next sections, Mermin method will be compared with the local field corrections.
Fig. 1. (Color online) RPA, Mermin, SLFC and DLFC energy loss function dependence with ω/ωp when k/k F = 1, for a n e = 1023 cm−3 degenerate plasma.
3. LOCAL FIELD CORRECTIONS
If LFCs are considered the dielectric function reads
where ɛ(k, ω) is the RPA dielectric function and G(k, ω) is the local field corrections of the electron gas. Mostly static approximations (SLFC), G(k) = G(k, 0), have been proposed in the past, as it is considered that greater part of the local field corrections will succeed for the static limit, ω = 0. It has started with the pioneering work of Hubbard (Reference Hubbard1957) who first introduced the notation of local field corrections and took into account the exchange contributions
This expression has established a remarkable improvement of the RPA but it was insufficient due to its self-inconsistency which leads the pair correlation function still to unphysical negative values. This has been repaired by Singwi et al. (STLS) (1968) by using the correlation contribution
where the static structure factor is
and V(k) = 4π/k 2 is the Coulomb potential. This provides a self-consistent problem in solving the dielectric function and the static structure factor. Eq. (13) has been improved further by Pathak and Vashista (1973) demanding that the response function should fulfill the third-order frequency sum rule, which resulted in
leading to the improved small-distance limit. At the same time, there have been different improvements to the derivation of LFC from the virial formula (Vashishta & Singwi, Reference Vashishta and Singwi1972; Vaishya & Gupta, Reference Vaishya and Gupta1973) which have resulted in expressions known from density variations
for the degenerate electron liquids at metallic densities a = 2/3.
The self-consistent (Singwi et al., Reference Singwi, Tosi, Land and Sjölander1968; Pathak & Vashishta, Reference Pathak and Vashishta1973) and the variational (Vashishta & Singwi, Reference Vashishta and Singwi1972; Vaishya & Gupta, Reference Vaishya and Gupta1973) formulations need of nonlinear integral equations and computer simulations to obtain the SLFC. For coupled degenerate electron liquids it will be useful to derive a parametrized expression which accurately fits the results of the self-consistent formulation as well as the variational calculations. On the suggestion of their microscopic calculations, Ichimaru and Utsumi (IU) (1980) adopted the formula
![\eqalign{G_{IU}\lpar k\rpar &= {A k^{4} \over k_{F}^{4}} + {B k^{2} \over k_{F}^{2}} + C + \left[{A k^{4} \over k_{F}^{4}} + \left(B + {8 \over 3}\right){k^{2} \over k_{F}^{2}} - C\right]\cr &\quad \times {4k_{F}^{2} - k^{2} \over 4k k_{F}} \ln \left\vert {2k_{F} + k \over 2k_{F} - k}\right\vert.\cr}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S0263034608000402_eqn17.gif?pub-status=live)
The parameters are A = 0.029, B = 9/16γ0−3/64[1−g 0] −16/15A and C=−3/4γ0 + 9/16 [1−g 0]−16/5A, where
![g_{0} = {1 \over 8} \left[{z \over I\lpar z\rpar }\right]\comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S0263034608000402_eqn18.gif?pub-status=live){kind=small}
and I(z) is the modified Bessel function of the first order of z = 4(αr s/π)1/2, with α = (4/9π)1/3 and r s = (3/4πn e) me 2/ħ2. Also γ0 is defined as
where b 0 = 0.0621814, b 1 = 9.81379, b 2 = 2.82224, and b 3 = 0.736411.
But in Vaishya and Gupta (Reference Vaishya and Gupta1973) it was shown that one cannot construct a SLFC which fulfills the compressibility and the third-order sum rules. Therefore the concentration was mostly focused on the construction of dynamical local field corrections (DLFC), G(k, ω). The formulation of the DLFC for a coupled degenerate plasma is a difficult task, then Yan et al. (Reference Yan, Tanaka, Mitake and Ichimaru1985) proposed a parametrization that takes into account the asymptotic behaviors of the DLFC in their frequency dependence
The proposed formula for G(k, ω), satisfying these two constraints, is
Figure 2 shows the LFC as a function of k/k F, for a n e = 1023 cm−3 degenerate coupled plasma (Γ = 0.686). Solid lines represents IU parametrization, G IU(k), and PV function, G PV(k). Other curves in Figure 2 represent Yan et al. (Reference Yan, Tanaka, Mitake and Ichimaru1985) parametrization, G Y(k, ω), for different frequency values; dashed line, ω = 0, dotted line, ω = ωp, and dashed-dotted line, ω = 10ωp. As it is seen G Y(k, ω) tends to G IU(k) for low frequencies while it tends to G PV(k) for high frequencies.
Fig. 2. (Color online) G Y(k, ω) as a function of k/k F, for a n e = 1023 cm−3 degenerate coupled plasma (Γ = 0.686). It tends to G IU(k) for low frequencies while it tends to G PV(k) for high frequencies.
The corresponding SLFC y DLFC energy loss functions are drawn in Figure 1 with dotted and dashed-dotted lines, respectively. The SLFC is based on the UI parametrization and the DLFC is based on the Yan et al. one. When collisions are considered in both cases, the ELF increases for low frequencies up to the frequency at maximum, even with higher values than in the Mermin case. Then it decreases suddenly at ω/ωp = 2 as in the RPA case. Differences between SLFC and DLFC corrections are minimal. This is because our calculated DLFC function has a low frequency dependence, as it can be seen in Figure 2; for enough high k values it changes only by 15% along all frequency range, 0 ≤ ω < ∞.
4. ELECTRONIC STOPPING
In the dielectric formalism, the formula to calculate the ion electronic stopping in any target is very well known. The electronic stopping for a swift pointlike ion with charge Z travelling with constant velocity v through a target plasma defined by its energy loss function is
![S_{e}\lpar v\rpar = {2Z^{2} \over \pi v^{2}} \vint_{0}^{\infty} {\hbox{d}k \over k} \vint_{0}^{kv} \hbox{d}\omega \omega \,\hbox{Im} \left[{-1 \over \varepsilon_{\rm x} \lpar k\comma \, \omega\rpar }\right]\comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S0263034608000402_eqn21.gif?pub-status=live){kind=small}
where Im[−1/ɛx(k, ω)] is the energy loss function of any dielectric function stated before. Then it is easy to compare the electronic stopping that results from the use of the Mermin or the local field correction dielectric function.
Figure 3 represents the proton electronic stopping as a function of the proton energy in a n e = 1023 cm−3 degenerate plasma normalized to S 0 = (Zk F)2. The coupling parameter value is obtained from plasma electron density Γ = 0.686 ≲ 1 which indicates that we are in the limit of coupled plasmas. The electronic stoppings are contrasted with Bethe formula at high energies. Solid line corresponds to the calculation with the RPA dielectric function, i.e., not considering collisions. Dashed line is the calculation with the Mermin dielectric function, Eq. (7), where the collision frequency is ν = 3.6 fs−1. Dotted line is the calculation with the LFC dielectric function, Eq. (11), with the static IU parametrization, Eq. (17). Finally, dashed-dotted line is the electronic stopping obtained with the LFC dielectric function, Eq. (11), but with the Yan et al. (Reference Yan, Tanaka, Mitake and Ichimaru1985) DLFC function, Eq. (20). Also relative deviations, (S x − S RPA)/S RPA, are shown to see clearly the differences between methods. Both kinds of LFC, static and dynamic, produce an enhancement in the stopping at low energies. But for the static case, this enhancement arrives at the maximum while for the dynamic case, this enhancement disappears before the maximum. After the maximum, both LFC cases recover the RPA values tending to the Bethe limit at high energies. Mermin method produces a high increase at very low energies, higher than the one produced by the LFC ones, but this increase is less significant than the LFC one at lower energies than the energy at maximum. Around and just after the maximum, Mermin values drop below RPA values. Finally, they also tend to Bethe limit at high energies. We see important differences between Mermin and LFC methods; they are very important at very low energies and by 30% around the maximum. Similar results for LFC approach have been recognized for nondegenerate cases (Yan et al., Reference Yan, Tanaka, Mitake and Ichimaru1985).
Fig. 3. (Color online) Proton electronic stopping as a function of its energy, in a n e = 1023 cm−3 degenerate plasma, normalized to S 0 = (Zk F)2. Solid line corresponds to RPA calculation, dashed line to Mermin calculation, dotted line to SLFC calculation and dashed-dotted line to the DLFC one. Relative deviations, respect to the RPA calculation, are also shown.
5. CONCLUSIONS
In this work, the effects of target electron collisions on the electronic stopping of protons in degenerate plasmas have been examined by means of two methods: the Mermin dielectric function or the local field corrections. The electronic stopping is due to the free electrons as the plasma target is considered fully ionized. Its electronic density is around solid values n e ≃ 1023 cm−3, which are very interesting for ICF studies. To calculate the electronic stopping, we have used the random phase approximation for degenerate plasmas, i.e., the Lindhard dielectric function. Then we have considered electron collisions through two methods: the Mermin dielectric function and the local field corrections. The LFC methods produce an enhancement in stopping before the maximum. But for the static case, this enhancement arrives at the maximum while for the dynamic case, this enhancement disappears before the maximum. On the other hand, Mermin method produces a high increase at very low energies, higher than the one produced by the LFC, but this increase is less significant than the LFC at lower energies than the energy at maximum. Around and just after the maximum, Mermin values are damped below RPA values. Finally, all of them tend to Bethe limit at high energies. Differences between Mermin and LFC methods are very important at very low energies and by 30% around the maximum.
But it is not easy to decide which method is better. LFC methods usually fulfil the sum rules as LFC functions are defined in order to fulfil them. But it was demonstrated that one cannot construct a SLFC which fulfills the compressibility and the third-order sum rules. That is why we propose to use a DLFC which is parameterized between one SLFC, which takes into account the compressibility sum rule, and another SLFC, which takes into account the third-order sum rule. On the other hand, Selchow and Morawetz (Reference Selchow and Morawetz1999) showed that Mermin dielectric function carries out the strongest sum rules; the longitudinal frequency, the conductivity, the compressibility and the screening sum rules, and recovers Drude formula for long-wavelength limit. Then it is not easy to manifest which method is better from this point of view. Difference between the Mermin approach and the LFC approach has been tested for nondegenerate, classical plasmas by comparing with computer simulations (Pschiwul & Zwicknagel, Reference Pschiwul and Zwicknagel2003). They found that LFC method works only for low coupling but fails for strong one, while Mermin method works also for strong coupling if an appropriate collision frequency is applied. One can thought that it will be the same for degenerate plasmas, but the comparison of our results with computer simulations is out of our possibilities.
ACKNOWLEDGMENTS
This work was financed by the Spanish Ministerio de Educación y Ciencia (under project FIS2006-05389) and the Consejería de Educación y Ciencia de la Junta de Comunidades de Castilla-La Mancha (under project PAI-05-071).
