Physics Letters A 377 (2013) 463–466

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Physics Letters A

www.elsevier.com/locate/pla

Empirical estimation of the equilibrium charge distribution of heavyions in solid and gaseous media

N.V. Novikov ∗, Ya.A. Teplova

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 November 2012Received in revised form 24 December 2012Accepted 24 December 2012Available online 28 December 2012Communicated by V.M. Agranovich

Keywords:Ion–atom interactionCharge fractionsCharge exchange cross sectionsDensity effect

Based on the analysis of experimental data on equilibrium charge distributions of ions in gases and solidmedia, the analytical parameterization of the charge distribution width as a function of the average ioncharge and nuclear charges of the ion and the target atom and that of the asymmetry as a function ofthe average ion charge are obtained. The proposed relations are applicable in a broad energy interval anddescribe the existing experimental data satisfactorily.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The charge distribution of particles passed through the matteris one of the main characteristics of the ion beam. Taking thisdistribution into account is important for the description of theproperties of the scattered beam and as a measure of the influenceof the ion on the medium. As the ion charge increases, the num-ber of excited atoms and electron–hole pairs in the medium andthe probability of the potential electron emission also increase [1].

The problem consists in the estimation of a relative numberof ions with the charge q and the energy E passed through themedium if the charges of ion nuclei Z and the target atom Zt areknown. It is assumed that the target thickness exceeds the equi-librium thickness; i.e., in the case of the propagation through themedium, the equilibrium charge distribution is established, whichcan be characterized by dimensionless parameters:

qav =∑

q

qFq, d2 =∑

q

(q − qav)2 Fq,

s =∑

q

(q − qav)3 Fq/d3, (1)

where Fq is the relative number (the charge fraction) of ions withthe charge q, qav is the average equilibrium charge, and the param-eters d and s characterize the width and asymmetry of the chargedistribution, respectively.

* Corresponding author.E-mail address: nvnovikov65@mail.ru (N.V. Novikov).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.12.019

One of the methods for describing the charge distribution is toestimate cross sections of electron loss and capture by the ion andto use the system of equations of charge exchange [2] in orderto calculate Fq . However, there is not enough experimental dataabout charge changing cross sections in solid. The cross sectionvalues are known mainly for gas. Methods for calculating crosssections of single electron loss and capture by the ion are welldeveloped for the simplest cases (small Z and Zt ), and to takeinto account interactions between many-electron ions and mediumatoms even in the case of gas, a correction, which can be expressedin the form of the ratio between theoretical and experimental crosssections, is required [3]. To calculate cross sections for solid, cor-rections for the density effect must also be taken into account.There are several programs [4,5] used to estimate Fq and chargeexchange cross sections for gases and solid, but their applicabilityis limited to fast collisions (E > 1 MeV/amu).

Another direction in the description of the charge distributionof ions passed through the matter involves obtaining empirical de-pendence (1) of parameters based on experimental data on Fq .The parameters of charge distribution in empirical method are de-termined from averaging of the available experimental data. Thisestimation neglects the shell effects and, therefore, the agreementwith some experimental data can be satisfactory. On the otherhand, the advantages of empirical method are a universality of so-lution in wide range of parameters and a possibility to estimatethe charge distribution, when an experimental data are absent.Several approximations for qav(Z , Zt , E) are known [6–9]. Expres-sions for qav(Z , Zt , E) in gas and in solid are assumed to be mostexact [9]; they give the averaged dependence without consider-ing oscillations related to the shell structure of the ion and the

464 N.V. Novikov, Ya.A. Teplova / Physics Letters A 377 (2013) 463–466

medium atom. The number of empirical relations used to estimatethe width of the charge distribution is significantly smaller. One ofthem [7]

d2 = {1 − (qav/Z)5/3}qav/4 (2)

was applied in gas and in solid. The width of the charge distri-bution in approximation (2) is independent of Zt ; however, as Eincreases, the quantity d increases for slow collisions d2 ∼ qav anddecreases in accordance with the power law for fast collisions. Theauthors of [10] also indicated that the parameter d for fast ionspassed through a carbon foil (Zt = 6) can be represented in termsof the function of the number of electrons d = d(Z − qav). The de-pendence d(Zt) in this approximation was not studied. There areno empirical relations for estimating s.

The aim of this Letter is to obtain the empirical dependencesd(Z , Zt , E) and s(Z , Zt , E) in gas and in solid based on existingexperimental data.

2. Results and discussion

In empirical estimation the parameters of charge distributionare the result of large number averaging of ion-target combina-tions. The experimental data with various error are taken intoaccount with identical weight. To describe the width of the chargedistribution of ions passed through the matter, we propose the pa-rameterization of d and s with respect to the ratio qav/Z ratherthan with respect to the ion velocity and energy. Here we usethe expressions for qav in solid and in gases from [9]. The do-mains of definition of the functions d(qav/Z) and s(qav/Z) arelimited (0 � qav/Z � 1), which is convenient for the approxima-tion. The dome-shaped dependence d(qav/Z) is described by thefunction

d(qav/Z) = C{

1 − exp[−(qav/Z)α

]}

× {1 − exp

[−(1 − qav/Z)β]}

, (3)

where the parameter C characterizes the “plateau” height for in-termediate energy and the parameters α and β characterize adecrease in d for slow (qav/Z → 0) and fast (qav/Z → 1) collisions,respectively:

d ∼ (qav/Z)α for qav/Z → 0, (4)

d ∼ (1 − qav/Z)β for qav/Z → 1. (5)

The dependence on E in (3)–(5) is taken into account inqav(Z , Zt , E). The widths of the charge distributions in gases andsolids are different, because the values of qav(Z , Zt , E) [9] for gasesand solids are different.

In the general case, the result of approximating the experimen-tal dependence by the function with the three parameters (α,β, C)

in the case of correlation between them cannot yield a unique so-lution. Such correlations between α and C and between β and Cexist. To determine the values of the parameters in (3) unambigu-ously, it is assumed that α and β only depend on the aggregatestate and are independent of Z and Zt . The values of α and β

in (3) were determined using all existing data (Z � 92) separatelyfor gases (Zt = 2,7,18,36) and solids (Zt = 6,13,47,79). The val-ues of the parameter C were calculated by approximating the ex-perimental dependence d(qav/Z) [11] by relation (3) for fixed Zand Zt . As a result, it was established that the value of the parame-ter C increases with increasing Z and Zt for both solid and gaseousmedia. Neglecting oscillations related to the shell structure of thetarget atom and restricting ourselves to the linear dependence ofC on Z and Zt , we obtained

Fig. 1. Width (d) of the charge distribution of Ar ions in carbon. Experimental data:(F) [12], (a) [13], ( ) [14], ( ) [15], (∗) [16], ( ) [17], ( ) [18], ( ) [19]. The-oretical estimates: 1 – corresponds to the data obtained using Eqs. (3) and (6);2 – [7]; 3 – [4]; 4 – [10].

Fig. 2. Width (d) of the charge distribution of I ions in carbon. Experimental data:( ) [20], ( ) [21], (∗) [22]. The notation of theoretical estimates is the same as inFig. 1.

α = 0.23, β = 0.32,

C = 2.66939 − 0.0098 × Zt + 0.05802 × Z

+ 0.00048 × Z × Zt (6)

for solids and

α = 0.40, β = 0.75,

C = 3.01904 − 0.01821 × Zt + 0.07817 × Z

+ 0.00132 × Z × Zt (7)

for gases. The applicability of these relations outside the limits ofthe interval of Zt used (i.e., Zt > 79 for solids and Zt > 36 forgases) was not considered. Relations (6) and (7) are more exactthan (2), because they were obtained using the data interpolationseparately for gases and solids and by taking into account the de-pendence of the width of the charge distribution on Zt .

The dependence of the width parameter d of the charge distri-bution (Figs. 1–3) shows that d(qav/Z) is a slowly varying function

N.V. Novikov, Ya.A. Teplova / Physics Letters A 377 (2013) 463–466 465

Fig. 3. Width (d) of the charge distribution of N ions in krypton. Experimental data:( ) [23], ( ) [24]. The notation of theoretical estimates is the same as in Fig. 1.

for average values of qav/Z . The decrease in d as qav/Z → 0 (4)is explained by the neutralization of ions in the limit of smallvelocities. For fast collisions, the width of the charge distributiondecreases as qav/Z → 1 (5) because of the total ion “stripping”. Inboth limiting cases, the charge distribution is degenerate (d → 0),because the ion is only in one of the states: either q = 0 or q = Z .The approximation of d(qav/Z) by relation (3) describes the ex-perimental data satisfactorily in the entire interval of energy E .If the dependences α(Z , Zt) and β(Z , Zt) are taken into account,then relation (3) makes it possible to reach the best agreementwith the experimental data. However, owing to the correlationsbetween α and C and between β and C , the dependences of theparameters in (3) on Z and Zt become nonmonotonous, whichcomplicated their interpolation. Estimate (2) reproduces the exper-imental dependence d(E) qualitatively, but decreases the quantityd by a factor of 1.5–2 for slow collisions. The calculation using theETACHA program in [4] and the estimation in [10] agree with theexperimental data only for fast collisions E > 1 MeV/amu in solid(Figs. 1, 2) and E > 4 MeV/amu in gases (Fig. 3).

The experimental dependence s(qav/Z) is approximated by thefunction

s(qav/Z) = A exp[−λA(qav/Z)

] − B exp[−λB(1 − qav/Z)

], (8)

where the parameters A and λA characterize an increase in theasymmetry |s| for slow collisions (qav/Z → 0), and the parametersB and λB for fast collisions (qav/Z → 1). The dependences on Eand Zt in (8) are taken into account in qav(Z , Zt , E).

The values for A, λA , B , and λB in (8) were determined usingall existing data (Z � 92) separately for gases and solids. It was as-sumed in the data processing that s > 0 for qav/Z < 0.5 and s < 0for qav/Z � 0.5. As a result, we obtained

A = 3, λA = 25, B = 6, λB = 50, (9)

for solids (Fig. 4) and

A = 2, λA = 20, B = 7, λB = 45 (10)

for gases (Fig. 5).The error of each of the parameters A and λA is about 50%,

and that of the parameters B , λB is about 10%. Relation (8) de-scribes the experimental data satisfactorily in the entire intervalof equilibrium charges (Figs. 4 and 5). For 0.3 < qav/Z < 0.7 the

Fig. 4. Dependence of the asymmetry parameter (s) of the charge distribution insolids. Points (") correspond to the experimental data for ions (Z = 5–92) in C,Al, Ni, Ag, and Au. The curve corresponds to the result of approximating (8) byparameters (9).

Fig. 5. Dependence of the asymmetry parameter (s) of the charge distribution ingases. Points (2) correspond to the experimental data for ions (Z = 5–92) in He, N,Ne, Ar, Kr, and Xe. The curve denotes the result of approximating (8) by parame-ters (10).

asymmetries of the charge distributions of ions in solid and gas al-most coincide, and the distribution is close to the symmetric one,s → 0. For qav/Z < 0.2 and qav/Z > 0.8 the asymmetry with theinclusion of the sign of the parameter s is larger in solid than ingases, because the values of the parameter A in (9) and the param-eter B in (10) are larger. This is related to the fact that, in solids,the electron loss cross section is larger and the electron capturecross section is smaller than in gases.

3. Conclusions

Based on the analysis of the existing experimental values ofcharged fractions for Z � 92 in gases and solids, we have ob-tained relations for the width and asymmetry of the equilibriumcharge distribution of ions passed through a medium. When d(E)

function varies slowly and s → 0, the problem of the descriptionof charge distribution can be solved with a minor number of pa-rameters, and the best agreement with the available experimentaldata can be achieved. But the conditions of applicability of thisapproximation depend on Z and Zt . The choice of trial functions(Eqs. (3) and (8)) allows to estimate the parameters of charge dis-tribution in all range 0 � qav/Z � 1. Using the ratio qav/Z as an

466 N.V. Novikov, Ya.A. Teplova / Physics Letters A 377 (2013) 463–466

argument makes it possible to describe charge distributions forslow (qav/Z → 0) and fast (qav/Z → 1) collisions.

The proposed relation for the width of the charge distribu-tion (3) approximates the experimental data significantly betterthan (2), owing to the gas–solid correction and the inclusion of thedependence on Zt . If oscillations related to the shell structure ofthe target atom are neglected, then d(Zt) turns out to be a slowlyvarying function.

We have proposed a relation describing the asymmetry of thecharge distribution. In the ranges qav/Z < 0.2 and qav/Z > 0.8, theasymmetry in solids (the sign of s is taken into account) is greaterthan in gases, which is explained by the dependence of the elec-tron loss and capture cross sections on the medium density.

The obtained relations for d and s can be used in a broadinterval of E , Z , and Zt , where they describe the existing exper-imental data satisfactorily and can be used to empirically esti-mate the equilibrium charge distribution of ions passed througha medium.

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