Low and intermediate energy stopping power of protons and antiprotons in canonical targets

TL;DR

This paper introduces a non-perturbative model for calculating the electronic stopping power of protons and antiprotons in various solid targets, achieving good agreement with experimental data across a wide energy range.

Contribution

It combines a central screened potential approach with Lindhard dielectric formalism and shellwise local plasma approximation to accurately model stopping power from low to high energies.

Findings

Good agreement with experimental data at low energies.

Effective modeling across low, intermediate, and high energy regimes.

Neglects collective plasmon excitations in the low to intermediate energy range.

Abstract

In this work we propose a non-perturbative approximation to the electronic stopping power based on the central screened potential of a projectile moving in a free electron gas, by Nagy and Apagyi. We used this model to evaluate the energy loss of protons and antiprotons in ten solid targets: Cr, C, Ni, Be, Ti, Si, Al, Ge, Pb, Li and Rb. They were chosen as canonicals because they have reliable values of the Seitz radio, rS=1.48-5.31 a.u., which cover most of the possible metallic solids. Present low velocity results agree well with the experimental data for both proton and antiproton impact. Our formalism describes the binary collision of the projectile and one electron of the free electron gas. It does not include the collective or plasmon excitations, which are important in the intermediate to high velocity regime. The distinguishing feature of this contribution is that by using the present model for low to intermediate energies (below the appearance of plasmon excitations), the Lindhard dielectric formalism (including plasmons) for intermediate to high energies, and the shellwise local plasma approximation to account for the inner shell contribution, then a good full-theoretical description of the available experimental data is obtained in an extensive energy range, covering the low, intermediate and high energy region.

Tables

Table 1. Table 1: The 10 solid targets studied here, their Wigner-Seitz radii, r S subscript 𝑟 𝑆 r_{S} , and Fermi velocity, v F subscript 𝑣 𝐹 v_{F} Isaacson , the calculated minimum velocity for plasmon excitation, v P subscript 𝑣 𝑃 v_{P} , given by ( 12 ), and the present results for the friction in the limit v → 0 → 𝑣 0 v\to 0 for proton, Q + 1 ​ ( 0 ) subscript 𝑄 1 0 Q_{+1}(0) , and antiproton Q − 1 ​ ( 0 ) subscript 𝑄 1 0 Q_{-1}(0) impact, as defined in ( 16 ). Atomic units are used throughout this table.

	$r_S$	$r_S^{exp}$	$v_F$	$v_P/v_F$	$Q_{+1}\left(0\right)$	$Q_{-1}\left(0\right)$
$Cr$	1.48	1.55	1.30	1.29	0.307	0.176
$C$	1.60	1.66	1.20	1.30	0.295	0.163
$Be$	1.87	1.78	1.03	1.32	0.269	0.141
$Ti$	1.92	1.93	1.00	1.33	0.264	0.137
$Si$	2.01	1.97	0.955	1.33	0.256	0.131
$Al$	2.07	2.12	0.927	1.34	0.250	0.127
$Ge$	2.09	2.02	0.918	1.34	0.248	0.126
$Pb$	2.30	2.26	0.834	1.36	0.229	0.113
$Li$	3.27	3.21	0.587	1.43	0.150	0.075
$Rb$	5.31	5.45	0.361	1.54	0.041	0.048

Equations

$$V_Z(r)=-\frac{Z}{r}\left(V_1{e}^{-{\mu }_{1}r}+V_2{e}^{-{\mu }_{2}r}\right),$$

$$\left\{\begin{array}[]{cc}V_{1}&=\frac{(\alpha+\beta)^{2}}{4\alpha\beta},\ \ \ \ \ \ \mu_{1}=\alpha-\beta,\\ &\\ V_{2}&=-\frac{(\alpha-\beta)^{2}}{4\alpha\beta},\ \ \ \ \mu_{2}=\alpha+\beta\end{array}\right.,$$

$$n_i(r)=Z\frac{({\alpha }^2-{\beta }^2{)}^2}{16\pi \alpha \beta r}(e^{-{\mu }_{1}r}-e^{-{\mu }_{2}r}).$$

$$\int d\overrightarrow{r}{n}_i(r)=Z,$$

$$n_i(r)=\frac{Z}{8\pi }({\alpha }^2-{\beta }^2{)}^2\left(\frac{1}{\alpha }-r\right)+O(r^2).$$

$$\alpha$$

$$\beta$$

$$b=\frac{v_F^2/3}{\frac{1}{2}+\frac{v_F^2-v^2}{4v{v}_F}\log \left|\frac{v+v_F}{v-v_F}\right|},$$

$$-2Z=\underset{r\to 0}{\lim }\frac{\frac{d}{dr}{n}_i(r)}{n_i(r)+n_0}.$$

$$b\rightarrow\left\{\begin{array}[]{cc}v_{F}^{2}/3,&\ \ \ as\ \ v\rightarrow 0\\ \\ v^{2}\ \ \ ,&\ \ \ as\ \ v\rightarrow\infty\end{array}\right.$$

$$V(r)\underset{v\to \infty }{\to }-\frac{Z}{r}{e}^{-\frac{{\omega }_P}{v}r},$$

$$n_i(r=0)+n_0\underset{v\to \infty }{\to }{n}_0\left(1+2\frac{Z}{v}\right).$$

$$v_P/v_F\simeq 1+(3\pi v_F{)}^{-1/2}.$$

$$\frac{dS}{dx}(v)=2\int \frac{d{\overrightarrow{k}}_i}{{\left(2\pi \right)}^3}\theta (k_i-k_F)\frac{ds}{dx}({\overrightarrow{k^{\prime }}}_i)$$

$$\frac{ds}{dx}({\overrightarrow{k^{\prime }}}_i)=2\pi \frac{k_i^{\prime }}{v}\overrightarrow{k_i^{\prime }}\cdot \overrightarrow{v}{\sigma }_{tr}(\overrightarrow{k_i^{\prime }})$$

$${\sigma }_{tr}(k)=\frac{4\pi }{k^2}\sum _{l=0}^{\infty }(l+1){\sin }^2\left[{\delta }_l(k)-{\delta }_{l+1}(k)\right]$$

$$\frac{dS}{dx}=Z^{2}v{Q}_Z(v),$$

$$Q^{LRT}(v\to 0)=\frac{2}{3\pi }\left(\ln \left(1+\frac{6.03}{r_S}\right)-\frac{1}{1+\frac{r_S}{6.03}}\right).$$

$$v/v_F\ge Z_P{r}_S.$$

$$Q_0(v)=\underset{Z\to 0}{\mathrm{Lim}}{Q}_Z(v)$$

Full text

stopping power, antiproton, proton

I Introduction

The energy loss of ions in solids has historically been a subject of interest due to its importance in different fields of technological and biological interest, such as ion beam analysis, radiation damage, and range of ions in matter. The relevance of this subject can be noted in the extended compilation of experimental data in Paul , very active up to the present time.

The theoretical developments cover from the Bethe theory for high energies in the 30s Bethe up to the time-dependent DFT for very low impact velocities DFT ; Ech07 ; Kohanoff12 ; Shukri16 , going through the dielectric formalism by Lindhard Lindhard ; LW ; LS and later models Abril98 ; MM1 ; Montanari02 , the free electron gas models ferrell77 ; Eche90 , and the binary theories Sigmund ; Grande ; Arista02 ; Bailey15a . Very effective too are the semiempirical descriptions and codes, the most extended of which is SRIM SRIM . Many reviews on this subject have been published, see for example the classic ones by Fano Fano63 and Inokuti Inokuti71 , or more recently by Arista Arista04 and Sigmund Sigmund17 .

In the last decade, the stopping power has had a revival due to the requirement of more accurate experimental data, and to the possibilities and precision of the up to date techniques SDB_HCI16 . Perhaps the most challenging ones are the low-energy antiproton experiments at CERN and the future prospects of the Facility for Antiproton and Ion Research Widmann ; FAIR at Darmstadt.

Based on the publications of experimental works compiled in Paul , the number of measured ion-target systems has increased from 74 in the period 2005-2008, to 96 in 2009-2012 and 158 in 2013-2016. The studied targets are approximately two-third compounds (mainly oxides and polymers) and one-third atomic targets, with special interest in the very low velocity range (i.e. $v<1$). This revival is related not only to the direct interest in the stopping powers but also to the inclusion of these values in simulations with different purposes Geant4 ; Barradas . It must be mentioned that most of the values included in these simulations come from SRIM SRIM , or the ICRU reports icru , and important discrepancies have been reported Barradas ; SDB_HCI16 ; Wittmaack .

The impulse of the new experimental measurements of stopping by low energy projectiles (i.e. by antiprotons Moller3 ; Moller1 ; Moller2 and by protons Vs94 ; HB06 ; Fm02 ; Roth13 ; Pr11b ) beards the theoretical developments. The expected linear dependence with the velocity, the influence of d-electron excitation and the density of electrons involved in the projectile loss of energy, have attracted many of the stopping power experimental efforts in the last years Fig07 ; Ca09 ; Mar09 ; Go13 ; Go14 ; Cel15 ; Bauer17 . The theoretical work on low energy stopping is extensive, such as by the groups of Echenique Eche81 ; Zaremba95 , Nagy Nagy98 ; Nagy96 ; Nagy94 , Arista Arista02 ; Arista98 ; FV_Arista ; FV_Arista2 , Cabrera-Trujillo Cab00 , and more recently Kadyrov Bailey15a ; Bailey15b and Grande Grande2016 .

The accuracy of the new experimental techniques and the necessity of full theoretical data, lead us to wonder which is the highest theoretical precision to describe these low-energy new experimental measurements. For this purpose, we present here a non-perturbative binary collisional model to describe the electronic stopping power $dS/dx$ of heavy charged projectiles in a free electron gas (FEG). The description at low impact velocities $v$ is amplified by calculating the friction parameter $Q=(dS/dx)/v$. In order to have a clear view of the problem to solve, we analyze the case of protons and antiprotons (no charge state considerations), and targets of well-established Wigner-Seitz radius, $r_{s}$.

We define the canonical metallic solids as those of reliable $r_{S}$, thus any doubts arising from their description can be dispelled. The criterion we followed is that the theoretical $r_{S}$ obtained considering the atomic density and the number of valence electrons do not defer more than $5\%$ with respect to the value deduced from the experimental plasmon energy Isaacson . In this way, we state these canonical targets as settings for future theoretical and experimental comparisons.

We present in this work a non-perturbative binary collisional model to describe the electronic stopping power of heavy charged projectiles in a free electron gas (FEG). The present model is based on the central screened potential of a projectile moving in a free electron gas, by Nagy and Apagyi Nagy98 , corrected in order to verify the cusp condition. Thus, we have successfully faced the theoretical problem of negative induced density by negative charge intruders Singwi68 .

The main characteristics of our proposal are: i) the use of a central potential $V_{Z}(r)$ that is Coulombic at the origin and decays exponentially at large distances; ii) an induced density that verifies the closure relation, which is finite at the origin and never becomes negative (as it happens if we use the Yukawa potential); iii) the cusp condition is imposed through an additional parameter $\lambda$. This strategy is valid at low energies, or at least where plasmons play a minor role. It only accounts for the outer electrons, so that inner-shell contributions have to be included in an independent form.

The goal of this work is to describe the stopping power of ions in solids in an extended energy region. For this purpose, we resort to two different descriptions for the valence electron contribution: the present binary non-perturbative model for low and intermediate energies, and the perturbative dielectric formalism in the intermediate to high energy region. The dielectric formalism is perturbative but contains not only binary but also the collective excitations. The inner-shell contribution is included by using the proved shellwise local plasma approximation (SLPA) MM1 ; Cantero09 ; Pb .

We chose ten canonical targets, Cr, C, Be, Ti, Si, Al, Ge, Pb, Li and Rb, of Wigner-Seitz radius $r_{S}=1.48-5.31$, covering most of the metallic solids. These targets belong to the groups of alcaline metals (Li, Rb, Be), the post transition metals (Al, C, Si, Ge, Pb), and the first groups of the transition metals (Ti and Cr). Among the transition metals, those elements with few d-electrons also have well-established $r_{s}$ values (groups 3 to 6 of the periodic table of elements). Instead, we skip here groups 7 to 12, where $d$-electrons play a quasi FEG role depending on the impact velocity. There are very interesting targets that have been an object of extensive experimental research in the last decade, focused on targets such as Pd, Pt, Cu, Ag, Cu, Au, or Zn Fig07 ; Ca09 ; Mar09 ; Go13 ; Go14 ; Cel15 ; Bauer17 .

In this contribution we only consider targets for which there are low energy experimental data in the literature. For proton impact we use the compilation in Paul , for antiproton impact the measurements by Möller and collaborators Moller3 ; Moller1 ; Moller2 .

We describe the present formalism in section II. In section III we show the scope of the model by comparing it with the low energy measurements for protons and antiprotons. We extend the theoretical-experimental comparison from 0.25 to 500 keV by combining the binary and the dielectric formalisms for the FEG, and the SLPA for the inner shells. The experimental needs and future prospects are discussed in section IV. Atomic units are used in all this paper, except when it is explicitly stated.

II Theory

II.1 Potential and density

Consider a heavy bare Coulomb projectile of charge $Z$ and velocity $v$ travelling within a FEG. Let us model the projectile-electron interaction by means of the central effective potential $V_{Z}(r)$ introduced by Nagy and Apagyi Nagy98 :

[TABLE]

with

[TABLE]

This screened potential tends exponentially to zero at large distances and has the correct limit $V_{Z}(r)\rightarrow-Z/r$ as $r\rightarrow 0$ for any value of $\alpha$ and $\beta$.

The induced density $n_{i}$ can be determined by using the Poisson equation to get

[TABLE]

It can be easily checked that $n_{i}$ verifies the desired closure relation

[TABLE]

as far as $Re(\alpha+\beta)>0$, and that it is finite at $r=0$,

[TABLE]

Following Nagy and Echenique Nagy93 , the parameters $\alpha$ and $\beta$ are defined as

[TABLE]

where $b$ can be related to the real part of the Lindhard dielectric function,

[TABLE]

with $v_{F}$ being the Fermi velocity, $v_{F}=1.917\ r_{S}^{-1}$, and $\omega_{P}=1.732\ r_{S}^{-3/2}$ is the plasmon frequency. As in Nagy93 , we leave $\lambda$ as a free parameter determined by imposing the cusp condition to the density,

[TABLE]

Note that Eq. (7) introduces the dependency of the potential with the ion velocity. The asymptotic limits of (7) are

[TABLE]

The cusp condition greatly improves the behavior of $n_{i}$ at the origin, erasing nonphysical negative electronic densities. For example, in the case of antiprotons at rest in a FEG, $\lambda=1$ gives negative density, $n_{i}(0)+n_{0}<\ 0$, for certain values of $r_{S}$. Instead, by imposing the cusp condition (8) to get $\lambda$, very reasonable values are obtained, i.e. $-n_{0}<n_{i}(0)<n_{0}$. It is worth mentioning that, no matter the impact velocity, the electronic density always verifies the cusp condition at the origin.

In Fig. 1 we plotted the potentials and total electronic densities (solid lines) generated by protons ($Z=+1$) and antiprotons ($Z=-1$) at rest. We consider $r_{S}$ ranging from $1$ to $6$, which covers by far all the known values. As expected, the potential tends to be Coulombic as $r_{S}$ increases (the density of electrons decreases, and therefore the screening too). The induced densities displayed in Fig. 1 satisfy the cusp condition imposed by $\lambda$. As can be noted, even the densities originated by antiprotons never become negative. The shape of the density induced by a negative charge can also be interpreted as a pair distribution function. For the sake of comparison, we include in Fig. 1 the pair distribution function reported by Singwi et al. Singwi68 . It is the best Random Phase Approximation (RPA), including short-range correlation and exchange effect. Even so, this RPA pair distribution function presents negative densities at the origin for $r_{S}>4$.

At high velocities ($v>>v_{F}$), $\lambda\rightarrow 1$, and the following expected limits are verified:

[TABLE]

and

[TABLE]

Beyond the theoretical validity of these limits, the physics involved in the present model only describes binary collisional processes. The collective electronic excitations, also known as plasmon excitations, are not included. The plasmon contribution is important at high energies. Within the dielectric formalism, the minimum impact velocity $v_{P}$ to excite plasmons can be approximated as MM00

[TABLE]

We will return to this point in section III where we compare our results (using the present non-perturbative model and using the Linhard dielectric formalism) to the existing experimental data.

II.2 Stopping power and friction

The calculation of stopping power, or energy loss per unit path length, $dS/dx$, implies the integration of the electron momentum $\overrightarrow{k}_{i}$ over all the Fermi sphere ferrell77 ,

[TABLE]

with

[TABLE]

and $\overrightarrow{k_{i}^{\prime}}=\overrightarrow{k_{i}}-\overrightarrow{v}$ the relative velocity. The transport cross section $\sigma_{tr}(k)$ is

[TABLE]

with $\delta_{l}(k)$ being the phase shifts generated by the potential $V_{Z}(r)$. The central potential given by (1) is expressed in terms of exponentials, so the first Born approximation to $\sigma_{tr}(k)$ can be calculated straightforwardly.

An alternative expression to the transport cross section has been recently proposed by Grande Grande2016 , derived from the retarding force due to an asymmetrically-induced charge density acting on the projectile. This non-central density is calculated from the spherically symmetric potential using the partial-wave expansion in a frame fixed to the ion. The results by Grande Grande2016 are interesting, mainly in the intermediate energy region, which is the conflictive zone where the validity of the high energy models (including plasmons) and the low energy ones (binary) compete.

The stopping power can also be expressed in terms of the friction parameter $Q_{Z}(v)$ as

[TABLE]

According to Fermi and Teller, at low impact velocities the stopping power is expected to show a linear dependency with the velocity, and so $Q_{Z}(v)$ tends to a constant. In the perturbative regime the stopping power is proportional to $Z^{2}$, so $Q_{Z}(v)$ is independent of $Z$.

The linear response theory (LRT) by Ferrel and Ritchie ferrell77 , predicts

[TABLE]

This is a first perturbative approximation, therefore insensitive to the charge $Z$ of the intruder.

Taking into account the projectile charge and velocity and the screening by the FEG, a reasonable criterion for the perturbative regime is

[TABLE]

In fact, as $v_{F}=1.917/r_{S}$, then this criteria is equivalent to $v\geq 1.917\ Z_{P}$. We will return to this in the next section in view of the theoretical-experimental comparison.

III Results and comparison with the experimental data

In this section we display the results of the present formalism for antiproton and proton impact in Cr, C, Be, Ti, Si, Al, Ge, Pb, Li, and Rb. We chose them because they are typical canonical metals, i.e. their valence electrons act as a free electron gas of constant and well known value of $r_{S}$. As we will comment later in this work, there are many more metallic targets that could be described using the present model, but there are no experimental measurements at low energies to compare with (see section IV).

In table I we list the ten targets, their $r_{S}$ and $v_{F}$ values Isaacson , the calculated $v_{P}$ MM00 , and our non-perturbative results for $Q_{+1}(v)$ and $Q_{-1}(v)$ in the limit $v\to 0$. These values may be used as predictions for future low energy measurements by proton and antiproton impact.

III.1 Proton and antiproton energy loss at low impact velocities

In what follows we compare our friction $Q_{Z}(v)$ at low impact energies with the experimental data available in the literature. We focus on this energy region in order to have only the contribution of the valence electrons. In Fig. 2 we report our results for proton $Q_{+1}$, and for antiproton impact $Q_{-1}$, in the limit $v\to 0$ as a function of $r_{S}$. The theoretical values displayed in table 1 for specific targets are also included in Fig. 2. These results confirm the experimental evidence that protons cede more energy to the FEG than antiprotons.

We also include in Fig. 2 (dotted line) the prediction of the linear response theory (LRT) by Ferrel and Ritchie ferrell77 , Eq. (17), which is independent of the projectile charge. The comparison of our non-perturbative values and the linear ones is very interesting. Our results correctly tend to the $Q_{LRT}$ as $r_{s}\rightarrow 0$, where we can consider that the screening of the projectile is so high that it can be described as a perturbation. On the contrary, as $r_{S}$ increases, the projectile becomes a huge perturbation to the FEG so the linear models cannot describe it.

To explore the perturbative limit we also calculated the friction for $Z\to 0$,

[TABLE]

and plotted it in Fig. 2 (black solid line). For $r_{S}>1$, $Q_{0}$ divides the region between $Q_{+1}$ and $Q_{-1}$, the known Barkas effect Sigmundv1 .

The description of the energy loss by antiproton impact is a challenge for any model. But it has the advantage that there is no possibility of charge exchange Widmann . The measurements by Moeller et al. Moller3 ; Moller1 ; Moller2 for antiprotons in several targets let us to test our theory with the Coulomb sign of the intruder. In Fig. 3 we display the present values for the friction as a function of the impact velocity for antiprotons in C, Si, and Al. Note that the agreement is very good in a linear-scale plot. We focused on the low velocity region in order to have only the valence electron contribution. Inner shells, however, may be contributing for Al above $v=1.5$. The theoretical description of low-energy antiproton measurements is an open path for future experimental research Widmann . Energy loss investigation is part of the physics program of the next-generation antiproton source FLAIR (Facility for Low-energy Antiproton and Ion Research) FLAIR , planned for the next five years.

At low impact energies the stopping power depends only on the value of $r_{S}$, so it should be the same for different metals of similar $r_{S}$. In Fig. 4 we plot together the experimental data for protons in three targets of $r_{S}\simeq 2$, Al, Ge and Si. As can be noted in this figure, all the low energy measurements are quite close within the experimental spread. We also display in this figure our theoretical results for Al, Si and Ge, which are actually very close and nicely describe the low energy measurements in the three targets. This confirms that $r_{S}$ is the only relevant parameter at low energies.

The experimental frictions displayed in Fig. 4 are $Q_{+1}^{exp}=0.25\pm 0.07$, and so are our theoretical results, with $Q_{+1}\to 0.25$ as $v\to 0$. Similar values of the friction at low energies are expected for other metals of $r_{S}\simeq 2$, such as Zn, Ga or Te. The recent low energy measurements for protons in Zn Go14 confirm this. But the most interesting point is the prediction of the value of $Q_{+1}$ for future measurements in Te, with no data of stopping at all, or in Ga, with no low energy measurements Paul .

III.2 Low to intermediate energy region

Above certain impact velocity $v_{P}$, the energy loss implies not only binary but also collective excitations (plasmons) ferrell77 . Though the present non-linear binary theory has the correct high energy limit, in the intermediate energy region it lacks the collective contribution. The extension to impact velocities $v>v_{P}$ can be performed by using the well-known Lindhard dielectric formalism Lindhard ; LW . This formalism includes both binary and collective excitations, and tends to the Bethe limit at high energies. But it is a linear response approximation, therefore valid within the perturbative limits.

A detailed comparison of the friction as a function of the impact velocity by using the present model (non-perturbative) and the Lindhard dielectric formalism (perturbative) is presented in Fig. 5. The results for protons in Cr, C, Be, Ti, Si, Al, Ge, Pb, Li and Rb, are displayed in this figure from top to bottom. Note that the lowest the $r_{S}$ the largest the stopping power. Three regions I, II and III, are indicated in Fig. 5, separated at $v_{P}$ and $2\ v_{P}$. In region I ($v<v_{P}$), the present collisional formalism (solid lines) is very appropriate because it includes all the perturbative orders and only binary collisions are involved. Instead, the dielectric formalism (dotted lines) is very poor, it underestimates the stopping power in this region. On the contrary, in region III ($v>2\ v_{P}$), which is clearly perturbative, the dielectric formalism is correct since it includes both plasmons and single-electron excitations. The non-perturbative results are below the dielectric ones in this region. Region II is the intermediate one, in which non-linear effects and plasmon contribution compete in importance. The validity of each formalism in this region is subject to the comparison with the experimental data, as will be shown in section III.4.

We also include in Fig. 5 the isolated plasmon contribution (dashed lines). Note that the impact velocity $v_{P}$ above which the plasmon excitation starts contributing agrees quite well with the values in table I. It can be observed that for $v>v_{P}$ plasmon excitation is not negligible at all. As predicted by Lindhard and Winter LW , at very high energies the equipartition rule holds and the binary stopping equals to the plasmon one.

III.3 Extension to higher energies, the inner-shell contribution

At sufficiently high impact energies the impinging projectile will be able to remove sub-valence electrons. To extend the theoretical description to intermediate and high velocities we include the inner-shell contribution by resorting to the SLPA MM1 ; Cantero09 ; Pb . During the last years we have developed this model based on the dielectric formalism and the local plasma approximation by Lindhard and Scharff LS . The contribution of each sub-shell of target electrons is described including screening, collective response and correlation in the final state. The inputs are the densities and binding energies of each sub-shell. For non-relativistic atoms, i.e. atomic numbers up to 54, they can be obtained from the Hartree-Fock wave functions by Bunge et al. Bunge . For targets of higher atomic numbers, the relativistic Dirac equation must be solved. The most interesting characteristic of the SLPA is that it is a density-based model, therefore capable of being used for molecular targets as far as a good description of the different shells electronic density is available TiO2 ; ZnO . The great limitation is that it is a perturbative model.

In Fig. 6 we display the SLPA results for the stopping power due to the inner-shells of Cr, C, Be, Ti, Al, Si, Ge, Li and Pb. For Pb (Z=82, relativistic target) we used the results obtained by employing the GRASP code in Pb . For the rest, we used the atomic wave functions by Bunge Bunge . As we are dealing with solids, the binding energies are slightly different from those of the gas phase. We use the experimental binding energies relative to the top of the Fermi level for metals compiled by Williams Will95 , instead of the theoretical values for single atoms, which correspond to gases.

It can be noted in Fig. 6 that the inner-shell contribution falls down several orders of magnitude when going from high to low energies. We are fully aware of the inability of the perturbative SLPA to describe the low energy region, but inner-shell contribution is relatively negligible in this energy region. On the contrary, as velocity increases, the relative importance of the inner-shells grows and, at the same time, the validity of the SLPA starts to hold.

III.4 Comparison with the experiments in an extended energy range

We performed an extensive comparison of the present theoretical results and the experimental data in the IAEA database Paul . We analyzed the stopping of protons in Cr, C, Be, Ti, Si, Al, Ge, Pb and Li. We did not include Rb in this comparison because there are no measurements in the low energy region, which is our main interest. By combining our non-perturbative and perturbative calculations in different energy regions, we managed to cover an extended range of $(0.25-500)$ keV. The extension to higher impact energies by using the dielectric formalism and the SLPA has already been demonstrated MM1 ; Cantero09 ; Pb .

In Figs. 7-9 we compare our theoretical results with the available experimental data Paul for the nine targets mentioned above. We display the friction for the lowest velocities in order to heighten the low stopping values. Instead, for the highest velocities we plotted the stopping power.

We show in these three figures the total values using the non-perturbative approximation for the FEG (red solid lines), and the perturbative model for the FEG (blue dashed lines). The total stopping power was obtained by adding the SLPA results for the inner-shell contribution. The experimental data in Figs. 7-9 follow the same notation using different letters as symbols as in Paul .

We separate three regions as in Fig. 5. Both boundaries, at $v_{P}$ and $2\ v_{P}$, are displayed with vertical dashed-lines. These energy regions involve different physical regimes. In the low energy region I, valence electrons are the main contribution and a non-perturbative description is mandatory. The high energy region III corresponds to the perturbative regime. As mentioned before, the intermediate region II is very interesting because plasmon excitation starts to occur and the validity of the perturbative description will depend on each case. It is worth to note that the stopping maximum is in this region, for impact velocity $v\gtrsim v_{P}$.

Figure 7 displays the present results for Cr, C and Be ($r_{S}=1.48,1.6$ and $1.87$, respectively). In the upper plot, for protons in Cr, our non-perturbative results clearly describe the experimental measurements in the whole energy range. The data by Eppacher and Semrad Ep92 of 1992 (represented by letter ”F” in this plot) is the most recent one and covers an extended energy region from $20-700$ keV. Only the low energy values, i.e. impact velocity below 1.2, seem to be too large. There is no experimental data for $v<1$. New low energy measurements for this system are welcome. On the other hand, the FEG of Cr has the highest electronic density, or equivalently, the smallest $r_{S}$ considered here. This implies a large screening of the projectile, and almost a perturbative regime in the whole velocity range. This explains the agreement of the perturbative calculations down to impact velocities $v\geq\ 1.7$.

Also displayed in Fig. 7 are the present results for stopping of protons in amorphous Carbon. This is one of the targets with more experimental measurements due to its different applications. However, the complexity of carbon (amorphous or cristal phases) also introduces dispersion among different sets of measurements. We show in this figure the available data since 1980. It can be noted that our non-perturbative friction reproduces the experiments in regions II and III, but overestimates a little in region I. As predicted, the perturbative results are reasonable for $v\geq 1.917\ Z_{P}$. Note that for carbon we also reproduce very well the antiproton impact measurements, even for very low velocities (see Fig. 3).

Finally, for Beryllium, at the bottom part of Fig. 7, we begin to note the difference between the non-perturbative and the perturbative descriptions, the latter including plasmons. The separation between the curves is clear for $v>v_{P}$. The agreement of the present non-perturbative results with the low-energy data is very good. However for $v\geq 1.5$ the non-perturbative results are too low, while the perturbative ones describe well the experiments. This difference is explained by the lack of plasmon contribution in the binary model. The experimental values represented by letters ”E” and ”I” correspond to the measurements by Warshaw warshaw and by Kahn kahn in the 50s, which are below the general tendency of much recent data Paul . It can be said that the combination of the present non-perturbative model for $v\leq v_{P}$ and the dielectric formalism for $v\geq v_{P}$ gives a good description of the energy loss of protons in Be in the whole energy range.

Figure 8 displays the present results for Ti, Si and Al. Again the vertical dashed lines separate the three energy regions mentioned above. In the upper figure, the energy loss in Ti is nicely described in the whole energy range, showing a very good agreement with the experiments. For low velocities, $v<1.8$ a.u., the present non-perturbative formalism describes the data correctly. Only the low energy measurements by Arkhipov et al. in 1969 Ar69 (letter ”F”) are higher than the rest. This detail would not be noticed if we plotted the stopping power instead of the friction coefficient at low energies. Some doubts on the normalization of Arkhipov’s data have been stated by Paul in Paul . Titanium is a target of technological importance that deserves new stopping measurements, not only in the low energy region, but also around the stopping maximum, where only one set of data is available (by Ormrod in 1971 Or71 , letter ”G” in this figure). In the intermediate region II, the binary model clearly underestimates the experimental values for $v\geq 2$. This can be adjudicated to the lack of plasmons, included in the dielectric formalism.

The energy loss of protons in Si have more than 600 experimental values for the different energies. Among all these data, we show in Fig. 8 those measured since 1990. It is a criteria to have a clearer view of the experimental tendency and to avoid the great dispersion among the oldest measurements. The agreement of our results for protons in Si shown in Fig. 8 is very good from the very low to the high energies. This is more noticeable if we focus on the latest experimental measurements: the low energy data by Hobbler et al. in 2006 HB06 (letter ”a” in region I), by Fama et al. in 2002 Fm02 (letter ”W” in regions I and II), and the high energy data by Abdesselam et al. in 2008 Ab08 (letter ”b” in regions II and III). Instead, the measurements by Konac et al. in 1998 Kc98 (letters ”Y” and ”Z”) are too low for $v<0.7$. Again this difference in the friction would be very small if we plot the stopping power in the low velocity regime too. The friction plot acts as a magnifier of the low velocity behavior, which is very demanding for any theoretical description. We describe Si as a free electron gas with no energy gap, and we do not extend the calculations below $v=0.1$. We do not discuss here the threshold of the Si as semiconductor Roth13 ; Archubi16 , which is below $v=0.03$. For $v>v_{P}$ the perturbative calculation improves the binary one. The combination of formalisms, the non-perturbative one in region I and the dielectric one in regions II and III, lead us to correctly describe the experimental data of stopping power of protons in Si in the whole energy range. This includes the stopping maximum, which is around 52 keV ($v=1.5$). For Si, we obtain very good agreement with the experiments, not only for protons (Fig. 8) but also for antiprotons (Fig. 3).

In the bottom plot of Fig. 8 we display the theoretical-experimental comparison for Aluminum, which is one of the most studied targets. As for the previous case, we restrain the comparison to modern experiments (1990 up to now). The agreement at low energies is quite nice, specially with the newest low energy data by Primetzhofer et al. Pr11b in 2011 (with letters ”E” and ”D” in Fig. 8). Clearly the perturbative model underestimates the friction in this region. The results in regions II and III show that for $v>v_{P}$ the non-perturbative formalism (binary) underestimates the measurements, while the dielectric results (binary and plasmos) nicely describe the data. Our proposal is that the combination of both models allows to describe the stopping of H in Al in the whole energy range, from the very low up to the high energies.

Finally, in Fig. 9 we plot the energy loss of protons in Ge, Pb and Li. We can note in these cases that all the physics involved is enhanced, i.e. the importance of a non-linear description as compared to the contribution of plasmons. For the case of Ge, in the upper part of Fig. 9, the validity ranges of the non-perturbative binary formalism and the dielectric formalism (binary+plamons) are very clear. The description of the experimental data by the non-perturbative calculation in region I is quite good. It nicely links the data by Eppacher and Semrad Ep92 , (with letter ”K”), and the most recent measurements (letters ”N” and ”M”) by Bauer and collaborators in Linz Roth13 . On the other hand, the data by Arkhipov and Gott in 1969 Ar69 (letter ”J”) is clearly above the rest. Present results also agree with the TDFDT values for protons in Ge Ullah15 , which apply only to very low energies (i.e. $v\leq 0.6$). In regions II and III, our binary formalism underestimates the measurements showing the importance of plasmon excitations in these energy regions. Instead, the perturbative results describe nicely the measurements for $v>v_{P}$, and clearly separate from the binary results. This behavior is the expected one, it has been already found for Si and Al in Fig. 8, but for Ge the difference is more pronounced.

The medium graphic in Fig. 9 corresponds to protons in Pb. This case is special because we are dealing with a relativistic target, 82 bound electrons, including the K-L-M-N-O shells and the 6s2-6p2 electrons as free electron gas. We follow Pb to calculate the inner-shell contribution to the energy loss by using the SLPA together with the relativistic densities of electrons of each subshell (spin-orbit split) obtained with the GrasP code. In the present calculations, the experimental binding energies were used Will95 . This improves our previous results in Pb in the high energy region.

The $r_{S}=2.3$ of Pb is higher than in the previous targets, hence it is more non-perturbative. This can be noted in the comparison of both models (solid and dashed curves) in the intermediate region II. We can say that the non-linear contributions are more important than the plasmons for Pb. The binary non-perturbative calculations for Pb clearly improve the perturbative ones for $v\leq\ 1.5\ v_{P}$, with very good agreement with Eppacher data Ep92 . Unfortunately, there are no measurements of stopping of protons in lead below 25 keV. Our results indicate an almost linear tendency of the stopping of protons in Pb for $v\leq 0.7$, with $Q_{+1}\sim 0.23$. Low energy measurements would be a good test for this prediction.

Finally, we display in the bottom part of Fig. 9 our results and the experimental data for protons in Li. This is the target with the largest value of $r_{S}=3.27$ we here considered, and so the smallest electron density and the lowest Fermi velocity, $v_{F}=0.59$. This makes Li a very interesting test of our model because it is highly non-perturbative around the stopping maximum, i.e. between $20-40$ keV. Only two sets of data are available for this system Paul , so more measurements are welcome, mainly for $v<1.5$. The present non-perturbative model describes properly the experimental values in the intermediate region II, but it overestimates the data for $v<1$. It is worth to mention that different theoretical calculations for H in solid Li by Kaneko K93 , and by Cabrera Trujillo et al Cab08 are also above the measurements by Eppacher et al. Ep95 around the stopping maximum. In the high energy region III, the perturbative model is valid, and the good description of the experiments for impact velocities $v>1.7$ shows the importance of plasmon excitations.

IV Experimental scarcity and future prospects

The great absent in the comparisons of section III.4 is Rubidium, with a FEG characterized by $r_{S}=5.31$ and a very low Fermi velocity, $v_{F}=0.361$. Protons introduce a huge perturbation to such a FEG, which can test any non-perturbative theoretical model to the limit. Unfortunately, the available measurements are for $v\geq 1.1$, which is 3 times $v_{F}$. This turns Rb a very interesting target to be studied, experimentally and theoretically, and an opportunity of future research.

We expect the predictions for low-velocity friction as a function of the $r_{S}$ presented here (Fig. 2 and table 1) to be benchmarks for future measurements. There are more canonical metals Isaacson . In general, these targets belong to the s and p-blocks of the periodic table of elements (alkaline metals and earth metals, with valence s-electrons; post transition metals and metalloids, with valence p-electrons). However, for many of them there are no experimental stopping powers at low impact energies. For example, for proton impact in the s-block elements there is no data for impact energies $E<20$ keV for Mg ($r_{S}$=2.66), Ca ($r_{S}$=3.27), and Sr ($r_{S}$=3.59), and there is no data at all for Na ($r_{S}$=3.99). Also some relativistic targets, such as Cs ($r_{S}$=5.75) and Ba ($r_{S}$=3.74) have no stopping data at all. Among the elements of the p-block of metals, there is no low energy data for protons in Ga ($r_{S}$=2.19) for $E<70$ keV, in Sn ($r_{S}$=2.4) for $E<20$ keV, and there is no data at any impact energy for protons in Se ($r_{S}$=1.84) and Te ($r_{S}$=2.09). Note that the latter is a very interesting case to test the universal predictions of Fig. 4 for the $r_{S}\simeq 2$ elements.

The transition metals of groups 7 to 12 of the periodic table have been the focus of attention for the low energy experimental research during the last fifteen years. Unexpected experimental changes were found in friction when d-electrons start to be active in the collisions. It can be thought as an inhomogeneus $r_{S}$, depending on the impact velocity. However, even for the transition metals, those elements of groups 3 to 6 (the d sub-shells mostly empty) have canonical $r_{s}$ values and could be tested by our non-perturbative model if low energy stopping data were available. Some examples are V ($r_{S}$=1.66), with no data for $E<30$ keV, Nb ($r_{S}$=3.07) with no data for $E<20$ keV and great dispersion of the experimental data around the maximum of the sopping power, and Mo ($r_{S}$=1.61) with no data for $E<70$ keV. Even the relativistic W ($r_{S}$=1.62) has no data for $E<80$ keV. An interesting case is Ta, with very recent measurements for $E<10$ keV by Bauer et al Bauer17 , and an unexplained high density of valence electrons. This target requires the relativistic treatment to determine the shell to shell electronic densities and binding energies for the SLPA calculation of the stopping by inner shells.

All the targets mentioned above are interesting aims for future experimental and theoretical research. Knowing their stopping values is important, not only as atomic solids, but also because they are known partners in compounds of technological interest SDB_HCI16 , and most of the stopping calculations in compounds are obtained from their components, with bond corrections in some cases. So reliable predictions of their values would be very useful.

V Conclusions

In this work we propose a non-linear model to deal with low and intermediate impact stopping based on a central screened potential for a projectile moving in a free electron gas. This potential induces a density of electrons that verifies the cusp condition at the origin, independently of the impact velocity, and the charge sign of the intruder.

In order to test this model for proton and antiproton impact we chose canonical solid targets (reliable value of the Wigner-Seitz radius $r_{S}$), with experimental data available in the low energy region: Cr, C, Be, Ti, Si, Al, Ge, Pb, and Li. The comparison at low impact velocities was done in terms of the friction (stopping power per impact velocity), which is a very sensitive parameter, and let us to test the linear dependency with the velocity.

We proved that the present non-perturbative model gives a good description of the low energy data for antiprotons in C, Si and Al, and for protons in Cr, Be, Ti, Si, Al, Ge and Pb. For protons in C and Li some small overestimation is found as discussed in the text.

By combining the present model for low to intermediate energies, and the dielectric formalism (including plasmons) for intermediate to high energies, a good description of the stopping power was obtained in an extended energy range. The inner shell contribution was included by using the perturbative SLPA model. A detailed theoretical-experimental comparison was performed considering all the data available. We analyzed our results in three energy regions: for low impact energies up to that of plasmon excitations (the non-perturbative regime); for high energies (the perturbative regime), and in the intermediate energy region. We showed that in this intermediate region the non-perturbative description and the plasmon excitation compete in importance, depending on the $r_{S}$. We suggested that the perturbative description is valid for $v/v_{F}\geq r_{S}\ Z_{P}$. However, we found that for $r_{S}<2.1$ the perturbative results are valid even for lower impact velocities, $v/v_{F}\simeq 1.3\ Z_{P}$.

We recall the importance of Rb as a highly non-perturbative case (very low $v_{F}$) with no low energy measurements. We have also detected at least thirteen elements of well-known $r_{S}$ but with unmeasured stopping power at low energies. These targets deserve future experimental and theoretical research.

Acknowledgements.

This work was supported by the following institutions of Argentina: Consejo Nacional de Investigaciones Científicas y Técnicas, Agencia Nacional de Promoción Científica y Tecnológica, and Universidad de Buenos Aires. The authors acknowledge Pedro Grande for useful comments on this work.