The Bloch correction, key to heavy-ion stopping

In 1913, after his arrival at Rutherford’s laboratory in Manchester, Bohr¹ published his first paper on atomic physics entitled “On the theory of the decrease of velocity of moving electrified particles on passing through matter,” that dealt with the penetration of alpha and beta particles through gases and solids. Bohr’s paper was not the first attempt of a theoretical analysis, but it became the one that directed the course of this field of research for more than a century.

Today, knowledge about the slowing-down and coming to rest of all kinds of energetic charged particles in matter is essential in the study of numerous radiation phenomena and all applications of particle accelerators. Examples range from cosmic-ray physics to surface coating and nanotechnology, from ion-beam therapy to plasma physics and fusion technology, from space travel to materials modification and analysis, from health science to the nuclear industry, and presumably many more.

Bohr’s theory was based on classical mechanics but was followed by Bethe’s quantum theory² in 1930, and in 1933, Bloch³ provided a link between the two formulations which, in principle, defined respective ranges of validity in terms of the type of projectile and the velocity range. The Bethe regime describes, broadly spoken, light particles at high velocities. This was the dominating application for several decades. The Bohr regime is defined by the criterion⁴ 

where $Z1$ and $v$ denote the atomic number and speed of the projectile, respectively. Practical applications were not in sight for many years, with the exception of early studies of the slowing-down of fission fragments.⁵ By the time heavy-ion accelerators arrived in the 1950s, Bethe’s theory had been widely accepted as the universal reference standard. Bohr’s theory had made its way into textbooks, whereas Bloch’s name represented the Bloch relation, a Thomas–Fermi estimate of the $I$-value, the material parameter entering the Bethe formula.⁶ 

A revival of the Bloch correction⁷ came in the 1970s, when experimental techniques had developed to the point of allowing measurement of small deviations in the energy loss of light particles from predictions of the Bethe theory.^8,9 Despite the fact that with the advent of heavy-ion accelerators, major limitations of the Bethe theory became obvious,^10–12 the existence of the Bloch correction had little effect on the generation of stopping tables.^13–16 When the function finally received interest for heavy ions, it was primarily for application in the relativistic regime.^17,18

In the present note, we describe the practical use of the Bloch formula in the understanding of the stopping of heavy ions over a wide range of energies, mainly in the nonrelativistic regime, as well as its limitations and the competition with the ion charge. Although some illustrations are based on our PASS code, we believe that our findings are relevant for a broader range of theoretical schemes.

The Bloch formula originates in the stopping of point charges which, therefore, is the first aspect to be explored here, whereafter we include the screening by electrons bound to the projectile. Meaningful comparisons with experiment require consideration of several other effects, which are included where necessary and summarized in  Appendix B.

For an introductory survey on stopping of ions, we refer to Ref. 20. Penetration of charged particles in matter is a mature field with more than 100 years’ tradition. Basic concepts like scattering and stopping cross section, energy-loss straggling, as well as several measures of the penetration depth are well established and generally applied in theory and experiment. Activities are to some degree concentrated in clusters inspired by applications in materials science, radiology, and fusion technology, but all of them draw knowledge from nuclear, atomic, and plasma physics.

Characteristic in the field is a tight connection between experiment and theory. On the one hand, theory is indispensable in a field that deals with the interaction between about 92 types of ions and as many types of atomic targets, to which numerous compounds in several aggregation states can be added. On the other hand, most theoretical predictions can be tested experimentally, where suitable equipment is accessible.

Figure 1 lists ion-target combinations for which measurements of stopping cross sections have been compiled in the IAEA database.¹⁹. Measurements with protons and helium ions dominate heavily and cover a significant fraction of all target elements. This has both historical and practical reasons. With regard to the topic of the present article, wide areas are seen in the remaining parts of the ion-target plane covered with only a few data or none at all. At the same time, tabulations of stopping data have, until 20 years ago, been based on interpolation between experimental data without much support from theory. Providing a solid basis for tabulations of stopping cross sections and related data such as straggling is the main task for theoreticians in the field.

On theoretical side, Ref. 21 offers a status report by 2014, with characterizations of popular programs like TCS-EFSR,²² CasP,²³ SWLP,²⁴ PASS²⁵ and several schemes based on Fermi-gas-like targets. To this adds the LS correction¹⁸ as an important contribution to the relativistic regime.

More recently, traditional Fermi-gas theory has found a revised theoretical basis,²⁶ the consequences of which deserve attention. Previous unsuccessful attempts to determine stopping cross sections from cross sections for inelastic collisions in gases have found an interesting revival.²⁷ In the long run, calculations by TDDFT (time-dependent density functional theory) and related ab initio methods^28–32 may become the most common tool, not the least due to substantially improved ability to account for properties of the stopping material. At present, applications in heavy-ion stopping seem scarce: Most computations address hydrogen and helium ions, the preferred geometry is the center of a channel in a single crystal, and computation times are typically too long to allow for statistics.

In accordance with common practice, we write the mean energy loss per path length in the form $−dE/dx=NS$⁠, where $N$ denotes the number of target atoms or molecules per volume. The electronic stopping cross section $S$ can be written as

in both Bohr’s and Bethe’s stopping theories, where $Z1$ and $Z2$ denote the atomic numbers of the projectile and target, respectively. The essential physics enters the stopping number $L$ which, in Bohr theory, reads

in the nonrelativistic regime, where $nj$ is the number of electrons in the jth target shell and $ωj$ its resonance frequency. Individual target electrons are modeled as classical harmonic oscillators. $C$ is a dimensionless constant $C=2/eγ$⁠, where $γ=0.57722$ is Euler’s constant.

The corresponding expression in Bethe theory reads

where $fj$ stands for a dipole oscillator strength and $Ij$ the mean excitation energy for electrons in the $j$th target shell. In the following, $Ij$ will be denoted by $Ij=ℏωj$ for comparison with the Bohr formula. In applications, it is often justified to identify the oscillator strengths $fj$ with the occupation numbers $nj$⁠.

With the Bloch correction

where $ψ$ denotes the digamma function³³ and $v0=e2/ℏ$ the Bohr speed, we finally arrive at the Bloch stopping formula

In studying the interplay between the Bloch formula and the Bohr or Bethe formula, we do not need to split up Eqs. (3) and (4) into shells. Instead, we operate with one shell and a resonance frequency $ω$⁠.

The Bethe formula is based on the leading term in a perturbation expansion (Born series) in terms of the interaction between a projectile ion and target electrons. Higher-order terms must be proportional to powers of $Z1$⁠. Actually, even within the leading order, Eq. (4) represents the leading term in an asymptotic expansion³⁴ in powers of the parameter $1/B$⁠, where

An obvious dimensionless parameter proportional to $Z1$ as a basis of the perturbation expansion is Bohr’s $κ$⁠, Eq. (1). Another quantity of interest is the ratio $κ/B=Z1e2ω/mv3$⁠, which is the reciprocal of the parameter

entering the Bohr stopping formula, Eq. (3).

Expansions in powers of $1/B$⁠, $1/ξ$⁠, and $κ$ denote the shell, Barkas–Andersen and Bloch correction, respectively. The leading terms in these expansions are useful as long as they are small, i.e., at not too low projectile speed, where they allow estimates of the relative importance of the underlying effects.

Before considering the relative importance of the above three corrections, we first note that higher-order terms in these expansions—which are rarely calculated—contain coupling terms. Therefore, also for this reason, these parameters only give a hint on relative importance, whereas quantitative estimates require alternative approaches avoiding series expansions.

Next, we note that the parameter $1/B$ determining the shell correction is independent of $Z1$⁠. It reflects the orbital motion of the target electrons,^34,35 and its magnitude, compared to the Bethe formula, will be similar for light as well as heavy ions at the same projectile speed. Therefore, the shell correction is the first effect to be considered for light-ion stopping.

For heavier ions, i.e., increasing atomic number $Z1$⁠, the Bloch correction, which reflects the difference between quantal and classical Coulomb scattering, and the Barkas–Andersen correction, which reflects the influence of binding and screening on free-Coulomb scattering, will increase in importance. In estimating the relative importance of the two corrections at a given atomic number and speed, we need to note that the expansion parameter in the Bloch correction⁷ is actually $κ2$⁠. Thus, the Barkas–Andersen correction will be dominating for $1/ξ>κ2$ or

Roughly speaking, this will be below the stopping maximum (Bragg’s peak) for light ions such as protons and antiprotons, and at increasingly small velocities for heavier ions.

In summary, we may conclude that for heavy ions, the leading deviation from the Bethe formula is the Bloch correction.

The Bloch formula has received considerable interest among theoreticians.^{3,4,18,23,26,36–38} Here, we sketch a simple derivation and note some mathematical properties that are useful in applications.

In Bethe as well as Bohr stopping theory, interactions are classified into close and distant collisions. Distant collisions are treated by quantal or classical perturbation theory to first order. Close collisions are treated as free-Coulomb scattering in the Bohr theory and by the first-order perturbation theory in Bethe’s theory. The “weak” point in Bohr’s theory is the assumption of classical scattering, which is questionable for $κ<1$⁠. The weak point in Bethe’s theory is the negligence of nonlinear terms in free-Coulomb scattering. The Bloch formula is the result of repairing these handicaps. The classification into close and distant interactions is not strictly necessary in this procedure,^26,37 but we shall follow it here to keep the connection with Bohr and Bethe theories. The closest comes a derivation by Lindhard and Sørensen,¹⁸ which also offers a relativistic generalization that is found to be of central significance for swift heavy ions.

The stopping cross section for straight Coulomb scattering is known to diverge because of the long range of the interaction. But scattering at large impact parameters is weak and well described by the Born approximation. Therefore, in the difference between exact quantal Coulomb scattering and Born approximation, the divergence at large distances disappears, and what remains is the deviation from the exact stopping cross section for Coulomb scattering from the one for the Born approximation. This difference, which is the Bloch correction, picks up contributions from close collisions, which are only weakly affected by the binding forces acting on the target electrons.

Therefore, adding the Bloch correction to the Bethe stopping cross section delivers an expression with correct (finite) contributions for distant collisions. At the same time, this yields a more appropriate description at lower velocities by accounting for higher-order $Z1$ contributions.

In both Bethe’s and Bohr’s stopping theory, the divergence is removed by taking into account the binding of target electrons. Coarsely speaking, this means that the interaction is truncated at the adiabatic radius $aad=v/ω$⁠. This produces the Bethe logarithm in Eq. (4).

In Ref. 18, the divergent stopping number for Coulomb scattering is written as

where

For $Z1=0$⁠, this reduces to

so that the Bloch correction reads

which is equivalent to Eq. (5) and is seen to represent a power series in $y2$ with the leading term $∝Z12$⁠. This, together with the factor $Z12$ in the front part of Eq. (2) identifies (5) as a $Z14$ correction to the stopping cross section.

An asymptotic expansion³³ in terms of negative powers of $y$⁠,

allows us to write the Bloch formula as

Thus, in the limit of $(v/Z1v0)2/12≪1$⁠, the Bloch formula reduces to the classical Bohr formula, whereas quantum mechanics enters only into a small correction.

The division of the $(Z1,v)$ plane into a Bohr and a Bethe regime is in agreement with Bohr’s kappa criterion, Eq. (1) for the range of validity of a description of the scattering process by a wave packet. One important point of Bloch’s theory is that, without the use of a classical argument, Bohr’s stopping formula emerges as a valid description of the stopping of heavy ions at not too high velocities, independent of Planck’s constant.

Figure 2 shows measured stopping cross sections of Al for Kr ions compared with the predictions of the Bohr, the Bethe, and the Bloch model. Complete agreement is seen between the Bohr and the Bloch curve except for a minute difference above the classical limit which, for a krypton ion, lies at $∼300MeV/u$⁠. Below the classical limit, the Bethe formula leads to a dramatically overestimated stopping cross section. Also, the Bohr and the Bloch results lie noticeably above the measured values. More serious, however, is the fact that all curves turn negative at 1 and 0.1ṀeV/u, respectively. This is not an inefficiency of the physical model but solely the result of a high-velocity approximation, leading to the logarithmic expressions in Eqs. (3) and (4).

Clearly, errors of the magnitude seen in Fig. 2 in the energy range below 10 MeV/u are unacceptable and need to be repaired. Since the problem seems to be related to the Bethe logarithm, we may try to remove it by finding a better approximation to the Bethe stopping cross section. An exact evaluation of the first Born approximation was presented in Ref. 39.

Figure 3 shows the result of switching the Bethe logarithm with the exact stopping cross section for a harmonic oscillator. Evidently, the Bethe curve does not any longer turn negative and shows a smooth behavior toward an effective threshold near $E=0.01$ MeV/u, although it still lies far above the experimental values. The Bloch curve does not any longer coincide with the Bohr logarithm. While that was one goal of the exercise, it was not a goal to let that point move to even higher energy, as it actually happened.

In other words, by generating a more satisfactory representation of the Bethe stopping cross section at low velocity, we achieved an even less satisfactory behavior at low-velocities of the Bohr stopping cross section. This despite the fact that the Bohr model should be superior to Bethe at low velocities. Hence, rather than improving the Bethe formula, our next choice is to find a better representation of the Bohr model at low speed.

Equation (6) can be rearranged to

where the first term represents the Bohr logarithm and the rest what we call an inverse-Bloch correction,⁴⁰ 

where $y$ is defined in Eq. (11). Equation (14) shows that the first term in a series expansion in powers of $1/y$ is $−1/12y2$⁠, which goes as $∝v2/Z12$⁠.

Now, instead of operating with the Bohr logarithm, we need to find an expression of the physics contained in the Bohr model without an unphysical behavior such as a drop-off to negative values. Such an expression has been found in Ref. 41, where the integral over the energy loss vs impact parameter in the Bohr model was evaluated numerically. The only difference to Bohr’s determination of $LBohr$ is avoiding asymptotic expansion of Bessel functions in the integral over the impact parameter,

Figure 4 illustrates the situation in the Kr–Al system. Now also the Bohr curve goes smoothly toward zero instead of turning negative. The curve Bohr+inverse-Bloch shows only a very small difference to the Bohr curve. This is evidently due to a very small inverse-Bloch correction, which indicates that this system is essentially classical in the depicted energy range. At the high-energy end of the graph, the Bohr curve rises slowly above the Bethe curve, and here the Bloch curve follows Bethe, as it is supposed to do. The slight difference at the low-energy end is real and increases with decreasing energy. In fact, it reflects the logarithm in the inverse-Bloch term which, however, is overrun by a shell correction of the type shown in Fig. 7 below.

Evidently, the agreement between the theory as it stands and experiments is far from satisfactory. However, the main lesson is the reduction of the huge discrepancy with the Bethe theory to a level that is compatible in accuracy with those effects that have been ignored so far.

With the above definitions,

Equation (6) can be written in the form

As found by Bloch,³ this relation is satisfied when the Bethe and Bohr logarithms are inserted for the stopping numbers. However, these logarithms are approximate solutions showing an unphysical behavior at low projectile speed. The question arises, whether a generalization of the Bloch formula can be found that would satisfy Eq. (19) with realistic expressions for $LBohr(ξ)$ and $LBethe(B)$⁠. This is, of course, easily achieved if $LBloch$ is allowed to depend on $ξ$ and/or $B$ together with $y$⁠, but as pointed out in Sec. I C, the Bloch correction is a property of Coulomb scattering and, hence, cannot depend on material parameters such as $ω$ and/or $Z2$⁠.

Equation (19) is obeyed because the sum of two logarithms is the logarithm of the product.  Appendix A proves the reverse relation: If (19) is satisfied, the $L$-functions are logarithms, and the most general solution for Eq. (19) can be written as

with arbitrary dimensionless parameters $α,β$⁠, and $γ$⁠. Hence, Eq. (6) can only be valid if all three $L$ functions are determined by the same logarithmic dependence. For the Bloch term, this means that the relation is only fulfilled for $y≫1$⁠, see Fig. 5, i.e., in the Bohr regime, as it must be.

More interesting is the fact that the relation breaks down as soon as $LBohr$ and/or $LBethe$ cannot be expressed by logarithms, and it will not be possible to find a Bloch-like expression that is independent of the material parameters $Z2$ and/or $ω$⁠.

In other words, the Bloch formula is a relation combining classical and quantal stopping cross sections in the limiting case only when both stopping numbers can be expressed as a logarithm. If we adopt a non-logarithmic form of the Bethe stopping cross section, as we have done in Sec. II B, we necessarily arrive at an unacceptable form of the Bloch stopping cross section. Alternatively, if we adopt a non-logarithmic form of the Bohr stopping cross section, we arrive at an unacceptable form of the Born stopping cross section. But that result does not do any harm, since the low-velocity behavior is determined by the Bohr scheme.

We conclude that for heavy ions, where $(Z1,v)$ falls into the Bohr regime, the convenient way to describe stopping in both the Bohr and the Bethe regimes is the Bloch scheme made up by an improved Bohr function and the inverse-Bloch correction. The opposite case, treating a system where $(Z1,v)$ falls into the Bethe regime, represents a well established routine.

Figures 6 and 7 show measured stopping cross sections for $Z1$ in Si from the literature, compiled in Ref. 19. Figure 6 is in conventional units, whereas the abscissa unit in Fig. 7 is the Bohr parameter $ξ=mv3/Z1e2ω$⁠. In the latter case, curves for different $Z1$ are nearly proportional, confirming that we are in the Bohr regime.

The two plots also show theoretical curves. In Fig. 6, we inserted output from PASS,⁴² which is explained briefly in  Appendix B. In Fig. 7, broken curves represent Bloch stopping for a bare ion, as discussed in Sec. II C. Here, the target is characterized by a single shell with $I=ℏω$⁠, and instead of the Bohr logarithm we plot the equivalent of the curve from Ref. 41 as broken curves.

Solid curves in Fig. 7 incorporate a shell correction, which accounts for the internal motion of the target electrons. This correction is well established in Bethe stopping theory and included in standard tables such as Ref. 43. It was introduced into the Bohr theory only a few years ago,⁴⁴ but here it has been calculated via the common kinetic transformation,³⁵ 

where $S0(v)$ represents the uncorrected stopping cross section (broken lines in Fig. 7) and $f(v)$ the velocity distribution of the electrons in an undisturbed target.

Figure 7 confirms that avoiding logarithmic expansion of the stopping number and taking into account the shell correction reproduces experimental results within better than a factor of two over the entire depicted energy range. On the other hand, the agreement between the solid lines is clearly inferior compared with the PASS output in Fig. 6. This leaves space for further contributions and modifications to the calculated stopping cross sections, but at the same time, it suggests that the influence of the ion charge on the stopping cross section is smaller than what one might have expected.

Table I shows the Thomas–Fermi speed $vTF=Z12/3v0$ and the Bohr limit $v1=2Z1v0$ in units of energy per nucleon for several ions. The Thomas–Fermi speed defines the transition from near-neutral to near-bare ions via relations like

whereas $v1$ determines the borderline between the Bohr and the Bethe stopping regime. Evidently, the Thomas–Fermi energy is lower than the classical limit in all cases, indicating that with increasing atomic number $Z1$⁠, an increasing portion of the Bohr regime is characterized by highly charged or bare ions.

This confirms what may be extracted already from Fig. 2: The large discrepancy between the Bethe stopping cross section and experiments is primarily a signal of the transition from Bethe to Bohr stopping and much less a consequence of decreasing ion charge.⁴⁵ An equivalent conclusion has been reached in Ref. 46 on the basis of nonlinear quantum stopping theory.³⁸ 

To further elucidate the matter, we consider the stopping cross section for a frozen charge. Information from measurements is scarce: Interesting results from Ogawa et al.^47,48 refer mostly to high beam energies, where only the highest charge states play a role. However, measurements by Blazevic et al.⁴⁹ on Ar in C over a limited energy range show good agreement with calculations by three theoretical schemes.^25,50,51 We take this as a justification for relying exclusively on theory in the present context.

Stopping cross sections for partially stripped ions can be determined theoretically by replacing the Coulomb interaction with the target electrons by a pertinent screened-Coulomb potential.⁵² Explicit treatments of both Bohr’s and Bethe’s stopping theories for screened-Coulomb interaction were presented in Ref. 53. This scheme was subsequently expanded into binary stopping theory,²⁵ which allows for the inclusion of a number of processes, of which we mention the Barkas–Andersen correction and the shell correction.

Figure 8 shows stopping cross sections of silicon for Xe ions in frozen charge states from $q1=0$ to 54 according to PASS. It is instructive to compare this graph with Fig. 6, which refers to the same target material. Evidently, where stopping cross sections in Fig. 6 depend sensitively on the atomic number $Z1$⁠, the dependence on the ion charge $q1$ in Fig. 8 is much weaker.

Figure 9 provides insight into the variations in charge-dependent stopping by separation into contributions from individual target shells. It is seen that the contribution of the target K shell to the stopping cross section is the same for a bare and a neutral ion. The contribution of the L shell is slightly reduced for the neutral ion, while that of the M shell is almost absent with a reduction of more than an order of magnitude for 3s electrons and almost two orders of magnitude for 3p.

This demonstrates the fact that an essential part of the energy loss is due to inner shells of the target atom, the interaction with which is only insignificantly affected by the charge state.

In Fig. 8, the position of the Bragg peak lies at a significantly higher energy for charge 0+ than for 54+. This is likewise a consequence of the reduced contribution of distant collisions: The position of the Bragg peak depends on the binding energy and moves to higher energies from outer to inner shells. With a decreasing contribution of outer shells to stopping, the Bragg peak moves upward in energy.

Discrepancies between the predictions of Bethe stopping theory and measurements of heavy-ion stopping of the type shown in Fig. 2 have been known for more than half a century. The fact that heavy-ion stopping belongs into the Bohr regime was likewise known⁴ and emphasized in reviews and summary talks, but we have not, until the turn of the last century, encountered a graph of the type of Fig. 2 that would have illustrated the power of Bohr theory compared with the problems of Bethe theory.

Bohr,⁵ studying ranges of fission fragments in a cloud chamber right after the discovery of nuclear fission, introduced the concept of an effective charge by replacing the atomic number in the stopping formula (3) by the charge number $q1$⁠. Bohr operated with the classical stopping number $LBohr$⁠, as is appropriate for heavy ions in the lower MeV range. Northcliffe,⁵⁴ as a part of a major experimental program, modified this concept by using the Bethe stopping number instead of $LBohr$ to define an effective-charge factor $γ2$⁠, both in measurements with lighter ions such as neon at 10 MeV/u, where this was justifiable, and at lower energies well down in the Bohr regime.

Figure 10 shows the effective-charge ratio $γ$⁠, defined by

for $Z1$ in Si, extracted from PASS.

This scheme was then expanded into a table of heavy-ion stopping cross sections¹³ on the basis of data for protons compared with stopping cross sections measured for selected heavy ions and subsequent interpolation. The minimum energy in these tables is 0.0125 MeV/u, corresponding to a velocity $v/2v0$⁠, which lies in the Bohr regime for all ions from He upward. This scheme was subsequently taken over by other authors,^14–16,55 and in a modified version, Ref. 56, protons were replaced by He as the basis for determining $γ2$⁠.

It was recognized from the beginning that the effective charge so defined was not identical with the ion charge. While this stimulated increased interest in an early study⁵⁷ of ionic charge states in solids and even its definition,⁵⁸ there are obvious pitfalls associated with the concept:

• Taken literally, replacing $Z1$ by the ion charge $q1$ would predict a vanishing stopping cross section for a neutral ion, in contrast with intuition, theory, and experiment.

• The model predicts the position of the Bragg peak to be independent of $Z1$⁠, in contrast to experiment, as shown in Fig. 6.

• The Thomas–Fermi relation Eq. (24), which describes the ion charge reasonably well for all ions and targets, bears no relation to the extracted effective-charge parameter⁵⁴ $γ2=1−1.85e−2v/vK$⁠, where $vK$ denotes the velocity of target K electrons.

The effective charge has become accepted as a tool in tabulating and interpolating measured stopping cross sections. With this in mind, any ion species with an adequate number of stopping data could serve as the basis. There is a connection to the actual ion charge, which can be studied theoretically,^46,59 but despite its name, the effective charge is not a charge.

Our main objection against the concept is lack of consistency: The ion charge is a determining factor in the velocity range around and below the Thomas–Fermi speed $vTF$⁠. The Bohr kappa parameter then reads

which is $>1$ for all $Z1$ and thus implies that, for $Z1≫1$⁠, we are in the Bohr regime.

More than half a century ago, measurements with mesons⁶⁰ and, subsequently, protons and alphas⁸ revealed the existence of a $Z13$ correction to the Bethe stopping formula (4). A Taylor expansion of the stopping number $L$ in powers of $Z1$ led to a term

on the basis of Bohr stopping theory.⁶¹ This correction, now called Barkas or Barkas–Andersen correction, was found to lie in the few-% regime in the first experiments, but a glance at Eq. (27) indicates severe limitations at low velocities.

Later on, comparison between the stopping of protons and antiprotons⁶² has stimulated much theoretical work. For a summary, we refer to Ref. 20. More important in the present context, the fact that a correction proportional to $Z1$ might be a significant issue in heavy-ion stopping has received much less attention.

In Sec. I, the $(Z1,v)$ plane was divided up into a Bohr and a Bethe regimes according to the magnitude of the Bohr parameter $κ=2Z1v0/v$⁠. In the Bohr regime, for $κ>1$⁠, the leading term in (27) reads

implying that this correction decreases with increasing $Z1$ and decreasing $ω$⁠.

As observed by Lindhard,⁷ the stopping number in the Bohr model must necessarily be a function of the parameter $ξ=mv3/Z1e2ω$ for the simple reason that this is the only way to compose a dimensionless variable from the pertinent parameters, $v,m,Z1,$ and $e$ under the assumption of unscreened Coulomb interaction between the ion and a target electron. Moreover, writing the equation of motion in terms of dimensionless variables shows that this function is universal, as long as shell and screening correction, projectile excitation, and charge exchange are neglected. Therefore, for a bare ion, at energies within the Bohr regime, a rigorous classical theory will result in a universal expression for the stopping number in terms of the variable $ξ=mv3/Z1e2ω$⁠. This implies that the Barkas ratio should be a universal function of $ξ$⁠, as long as the projectile is a point charge.^7,63,64

A classical estimate of the Barkas–Andersen correction may be extracted from binary stopping theory discussed in  Appendix A, which allows us to invert the sign of the ion-electron interaction potential so that in addition to the stopping cross section $S+$ of a positive ion, one can find the cross section $S−$ of the mirror ion, i.e., an atom made up by an antinucleon and the corresponding number of positrons. Figures 11 and 12 show average stopping cross sections $(S++S−)/2$ and relative differences $(S+−S−)/(S++S−)$ for the Kr–Al system.

Figure 11 shows the dependence of the two quantities on the charge state. Up to about 0.1 MeV, the effect is only weakly dependent on the charge but quite high, $∼30−40$%. The effect decreases with increasing energy, most rapidly for the highest charge. The slow decrease in the case of the neutral ion is a manifestation of the significance of screening: After all, there is no Barkas–Andersen effect for pure Coulomb scattering. However, the neutral component in the charge distribution at high energy is negligible, as is evident from the fact that the curve for charge equilibrium follows closely the one for $S36+$ over a wide energy range.

Figure 12 shows similar data for charge equilibrium separated into contributions from individual target shells. Again, up to 0.1 MeV, the relative effect does not depend significantly on either energy or shell number, but at higher energies, it is the relative effect in the M shell that drops most rapidly to zero, whereas in the K shell there is still an $∼1%$ effect at 1000 MeV/u. This is a manifestation of the dependence on the $I$-value $ℏω$ in Eq. (28). It does not only hold in the relative effect but also in the absolute difference (not shown).

All direct measurements and most theoretical studies of the Barkas–Andersen effect refer to the difference in the stopping cross section between protons and antiprotons, where the relative effect in the H–Al system is $∼1%$ at 1 MeV/u and increasing up to $∼40%$ in the keV/u range.⁶⁵ Figures 11 and 12 suggest similar values for the Kr–Al system.

Two findings are not necessarily expected

• Although the stopping cross section increases with increasing ion charge, the Barkas–Andersen effect decreases on a relative scale.

• The contribution of inner shells to the Barkas–Andersen correction may be substantial, in particular, for the K shell because of high values of $I=ℏω$⁠.

• The Bloch correction is known to be a significant addition to the Bethe stopping formula for swift light ions. The resulting Bloch formula may be rearranged as the sum of the Bohr formula and an inverse-Bloch correction following Ref. 40. This expression is a useful tool in the analysis and prediction of stopping cross sections for heavy ions.

• This divides the $(Z1,v)$ plane into a Bohr regime and a Bethe regime with the border line determined by Bohr’s kappa criterion.

• We have found that the specific form of the Bloch formula is closely connected to the well-known approximations of the Bohr and Bethe stopping numbers by logarithms. This has consequences on the low-velocity behavior of either scheme.

• Bethe’s theory is known to overestimate the stopping cross section of all but the lightest ions in the region around the Bragg peak. This feature has traditionally been repaired by the introduction of an effective charge. As pointed out previously,⁴⁵ rather than being related to the ion charge, the effective-charge factor $γ2$ primarily expresses the transition from the Bethe to the Bohr stopping regime.

• Consequently, the influence of the ion charge on the stopping cross section is found to be smaller, typically within a factor of two, and comparable with other effects like Barkas–Andersen correction, electron capture and loss, projectile excitation, and shell correction, which have not been discussed here.

• Within these limits, knowledge of the equilibrium projectile charge as well as related aspects appears important.

• There appears to be a consensus that equilibrium charges not only depend on the projectile speed and atomic number $Z1$ but also on the target species $Z2$ and, especially, the aggregation state.⁶⁶ Drastic differences are known when charge states in plasmas are compared with neutral gases, but significant differences are also documented between gases and solids.

• While part of our results have been illustrated by output from PASS, we believe that our conclusions are relevant for a wider variety of stopping theories.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Which differentiable functions $f,g,h:]0,∞[↦R$ fulfill the scaling relation

for all $x,y∈]0,∞[$ ?

First, we substitute $xy$ for $x$⁠,

Now, setting $y=1$ yields the equation $h(x)=f(x)+g(1)$ that allows us to eliminate $h$ from the problem,

Setting $x=1$ results in $g(y)=−f(y)+f(1)+g(1)$⁠, identifying $g$ as $−f$ plus a constant. This allows us to also eliminate $g$ from the scaling relation,

Differentiating this equation with respect to $y$⁠, and setting $y=1$⁠, yields

with the obvious solution,

These are the only differentiable functions $f,g,$ and $h$ fulfilling the original scaling relation, with free parameters $f(1)$⁠, $g(1)$⁠, and $f′(1)$⁠.

In Sec. I A we have mentioned several theoretical schemes, which have been developed during the past 20 years to compute stopping cross sections and related data for heavy ions. Table II summarizes features⁶⁷ of PASS and CasP. The main difference between them is the starting point, which is Bohr’s theory for PASS and Bethe’s theory for CasP. In both systems, there is a built in bridge across the border between the Bohr and the Bethe regime, which for PASS is the inverse-Bloch correction and for CasP the Bloch correction. Figure 13 shows stopping cross sections for Fr in La and La in Fr, which are truly predicted, since experimental data are not known to us for either combination. We have added output from the popular SRIM code,⁵⁵ which is based primarily on interpolation of experimental data and does not involve the Bloch formula. We mention two special features:

• According to the reciprocity principle,⁶⁹ the two stopping cross sections should coincide at low velocities. This is indeed found for the PASS data, and approximately also for the CasP data, while there is a major discrepancy in the SRIM data.

• According to standard theory,⁷⁰ and in agreement with numerous measurements, the stopping cross section at low velocities should be approximately proportional to the velocity, i.e., to $E$⁠. This is found for SRIM and for PASS, while CasP predicts a steeper slope. We find it conceivable that this is related to the effect shown in Fig. 3 which was directly related to the use of the Bloch correction, i.e., the passage from the Bethe to the Bohr regime.