Nuclear Instruments and Methods in Physics Research B 315 (2013) 76–80

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Nuclear Instruments and Methods in Physics Research B

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Electronic stopping for swift carbon cluster ions connected with averagecharge reduction

0168-583X/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nimb.2013.03.025

⇑ Corresponding author.E-mail address: kaneko@dap.ous.ac.jp (T. Kaneko).

Toshiaki Kaneko ⇑, Kohsuke Ihara, Mahsa KohnoGraduate School of Science, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama 700-0005, Japan

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

Article history:Received 6 November 2012Received in revised form 1 March 2013Accepted 3 March 2013Available online 24 March 2013

Keywords:Cluster ionStopping powerAverage chargeDielectric function

The cluster effect in the average charge and the electronic energy-loss of swift carbon cluster ions in lin-ear-chained and ring structures with equal separation with kinetic energy ranging from 0.3 to 30 MeV/atom is theoretically investigated on the basis of a recent average charge theory and the dielectric func-tion formalism together with the wave-packet model. The dependence of the cluster average charge onthe constituent atoms clearly shows the reductive feature, regardless of the speed, which is moreenhanced with increasing the number of constituent atoms. It is proposed that for ring structure, theaverage charge of constituent ions is determined by an unique self-consistent formula. In the high speedlimit it has an asymptotic formula and the reduction effect tends to be vanishing. Regarding the energyloss, the electronic stopping powers of aluminum and silicon targets are calculated for Cn clusters in lin-ear-chain and ring structures with inclusion of the average charge reduction in a bulk. It is found thatthey show the super-linear cluster effect almost over the range of investigated speed except for large ringclusters at low speed.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Recently cluster or polyatomic projectiles have attracted inten-sive interest in the field of investigating the interaction of high en-ergy cluster ions with solid materials both from the basic andapplied viewpoints. For example, C60 fullerene ions and highlycharged bio-molecules have been accelerated at high energies[1]. From the viewpoints of application, cluster impact has severaladvantages, e.g., reduction of the kinetic energy per atom and sup-pression of the charge up effect in ion implantation. From basicviewpoints, on the other hand, it is because the research subjectsusing polyatomic ion beams are widely ranging, e.g., fragmentation[2–4], multiple ionization [5], emissions of ions and neutrals [6],reduction of average charge [7–9], secondary electron emission[10–13], and energy-deposition to a target [8,14–19], etc. The keyterm ‘cluster effect’ originates from a new character of the clusterion beam, described by the number of constituent atoms and thespatial structure, in addition to the conventional character of themono-atom ion beam, described by the ion speed, the ion element(or, atomic number), and the distribution of the bound electrons.

The cluster effect was found in the average charge [7–9], the en-ergy-loss phenomena [8,14,16,19–21], and the secondary electronyield [10–13]. As for the average charge, Brunelle et al. [7] found atfirst the reduction of the cluster average charge per ion. Recently,

Chiba et al. [9] reported the structure dependence of the clusteraverage charge using coulomb explosion imaging technique. Theyhave extended this method to the divergence-angle measurementfor various charge-state combinations [22]. Regarding the energyloss of a carbon cluster, there exists the threshold energy around1 MeV per atom for carbon target. At the incident energies largerthan the threshold, the energy loss per ion for a Cn cluster, DE(n),displays the positive or super-linear cluster effect [19], i.e.,D � DE(n) � DE(1) > 0 or R � DE(n)/DE(1) > 1. On the other hand,at lower incident energies, we found a few cases where the nega-tive or sub-linear cluster effect was reported [16]. Recently, makinguse of a novel experimental method, Tomita et al. [23] clearly ob-served the sub-linear cluster effect in the energy loss of the0.5 MeV/atom carbon cluster Cþn ðn ¼ 1� 4Þ penetrating a thin car-bon foil. Quite recently, we investigated the cluster average chargeand the energy loss of MeV/atom linear-chained Cn clusters pene-trating a thin carbon foil in a refined model, which contains thecoulomb explosion, the dissipated force, the polarization force,and the average charge reduction in a bulk [24]. It shows thesub-linear and super-linear cluster effects in the energy-loss,respectively, at lower and higher incident energy than the thresh-old, though the average charge per ion of the clusters indicates thesub-linear effect regardless of high or low ion-speed.

The aim of this paper is to investigate comprehensively the speeddependence of the cluster average charge, and the electronic stop-ping powers of aluminum and silicon for a Cn cluster with the clusteraverage charge reduction. Through this paper, m; e and �h denote,

T. Kaneko et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 76–80 77

respectively, the electron rest mass, the elementary charge, and thePlanck constant divided by 2p. In addition, the Bohr radius and theBohr speed are denoted by a0 ¼ �h2

=ðme2Þ ¼ 0:529� 10�10 m andv0 ð¼ e2=�hÞ ¼ 2:19� 106 m=s, respectively.

2. Theoretical model

2.1. Cluster average charge

It is well known that when energetic ions pass through a solidtarget, emergent ions have a charge-state distribution, due to sub-sequent electron capture and loss processes. The average value ofthe charge states, i.e., average charge, for single-ion incidence ismainly determined by the moving speed, and weakly dependentof material elements. This is supported by the compiled data [25]and the scaling formula holds valid. As for single-ion projectilewith atomic number Z, there are several formulas to describe theaverage charge Q as a function of the speed V , e.g., Q/Z = a � bexp(�cV/(Zdm0)). Here a; b; c; d are constant, chosen to fitthe experimental data in a scaled form [25], and usually we havea = b = 1, c = 0.8–1.0 and d = 2/3. Insensitive dependence of theaverage charge on target elements is supposed to come from astrong screening effect of a target-atom potential acting on elec-tron-loss process in a bulk. Recently, based on a fluid-mechanicalmodel, we presented a theoretical formula of the average chargeof a single ion with atomic number Z moving in a foil at speed Vis given by

QZ¼ 2ffiffiffiffi

pp

Z y

0dt expð�t2Þ; y ¼

ffiffiffi38

rVVb; ð1Þ

where Vb = 1.045Z2/3m0 is the average speed of the electrons boundon the ion in a statistical model [10]. This formula implicitly dis-plays that Q/Z is a function of V/Z2/3m0. This expression was ex-tended to cluster-ion projectiles with a modified Vb, whichincludes the binding effect of surrounding ions via the potential en-ergy. The resultant expression for the average charge Qi of the ithion in the cluster is given by

Qi

Z¼ 2ffiffiffiffi

pp

Z y

0dt expð�t2Þ;

y ¼ffiffiffi38

rVm0

1:092Z4=3 þ 2Xn

jð–iÞVjiðRjiÞ

!�1=2

: ð2Þ

Here VjiðRjiÞ denotes the interaction potential energy in atomicunits per electron of the ith ion at Ri

!with the jth ion at Rj

!. If

Rji ð¼ jRj!� Ri

!jÞ is large enough, VjiðRjiÞ reduces to the point-chargevalue Qj

Rji. Here we give several comments. First, if the inter-atomic

separations are large enough, the interaction potential energy termvanishes and consequently the average charge of each ion is deter-mined by its own speed. This means that cluster effect should dis-appear. The formula (2) reflects this fact clearly. Second, in thelimit of very high speed, the average charge of the cluster asymp-totically approaches that of a single ion with the correspondingspeed. This also means that the cluster effect apparently tends todisappear.

In general, the average charge of each ion depends on the orien-tation of the cluster and the cluster structure in space. In order toderive general results, however, we neglect the cluster orientationwhile the cluster is assumed to keep the spatial structure. In spiteof this action, we believe that the derived result does not lose ageneral feature. Let us consider first a cluster in a ring structure,where a 2D ring cluster is composed of n atoms with equal separa-tion of R and the interaction energy is given by in a point chargemodel as

Vtot ¼Xjð–iÞ

VjiðRjiÞ ¼QR

f ðnÞ; f ðnÞ ¼Xn�1

j¼1

sinðp=nÞsinðjp=nÞ : ð3Þ

It is noted that every ions have the same average charge Q, as isunderstood from symmetry. Then the average charge in this struc-ture is determined by a self-consistent equation

QZ¼ 2ffiffiffiffi

pp

Z y

0e�t2

dt; y ¼ffiffiffi38

rVm0

1:092Z4=3 þ 2QR

f ðnÞ� ��1=2

: ð4Þ

This is characteristic of a ring-structure cluster. Later we willshow numerical results in a few cases for a Cn ðn ¼ 2—10Þ.

As for linear-chained clusters with equal separation of R, theinteraction energy is given by

Vi ¼Xjð–iÞ

VjiðRjiÞ ¼1R

Xjð–iÞ

Q j

jj� ij ; ð5Þ

so that the upper limit of the integral (2) should be replaced by

y ¼ffiffi38

qVm0

1:092Z4=3 þ 2R

Pnjð–iÞ

Qj

jj�ij

� ��1=2. Then one find there are more

than two different values, depending on the atom position, of theaverage charge exist for linear-chained Cn (n > 2) clusters. For exam-ple, one realizes that the average charge of a central ion in a linearcluster is lower that of an edge ion, comparing the interaction ener-gies for them. We newly call this ‘the position effect’. This was sup-ported by a sophisticated experimental data [9]. In this section, wegive a theoretical expression in detail for the average charge of aconstituent ion in ring and linear-chained clusters. Later, we willdepict in several figures the dependence of the calculated clusteraverage charges on the ion speed, and the position effect.

2.2. Structure factor

In estimating the stopping power of material for a cluster, thespatial structure of a cluster plays a significant role. In cluster im-pact, we commented that the spatial structure of a cluster is one ofimportant beam parameter. Therefore we briefly describe thestructure factor derived from a structure in real space. For simplic-ity, we represent individual atoms in a cluster composed of natoms by delta-function points. Then, the spatial distribution ofconstituent atoms is given by qð r!Þ ¼

Pnj¼1dð r!� Rj

!Þ, so that thestructure factor, i.e., the Fourier transform of spatial distribution,is obtained as

qð k!Þ ¼

Zd r!qð r!Þ expð�i k

!r!Þ ¼

Xn

j¼1

expð�i k!

Rj!Þ: ð6Þ

The square absolute of the structure factor qð k!Þ is defined here

by S0ð k!Þ, so that we obtain

S0ð k!Þ ¼ jqð k

!Þj2 ¼ nþ

Xn

j

Xn

‘ð–jÞexpði k

!R!

j‘Þ; ð7Þ

with R!

j‘ ¼ Rj!� R‘

!. Here we consider the orientation-averaged

quantity S(k) of S0ð k!Þ, then we obtain

SðkÞ ¼ nþXn

j

Xn

‘ð–jÞ

sinðkRj‘ÞkRj‘

: ð8Þ

This function reflects the spatial structure of a cluster in theFourier space. Later, one will see the relation between S(k) andthe electric stopping power in dielectric formalism.

Fig. 1 shows S(k) as a function of k in atomic units for Cn (n = 3,6) clusters in a linear-chain and ring structures with equal separa-tion of 2.4a0. The solid lines and dashed lines refer to the ringstructure and the linear-chain structure, respectively. One can dis-tinguish the curves for C3 or C6 by the starting value S(k = 0) = n2.

Fig. 1. Profile of S(k) as a function of k in atomic units for a Cn (n = 3, 6): the solidlines and the dashed lines denote respectively the ring structure and the linear-chain structure with R = 2.4a0.

78 T. Kaneko et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 76–80

We can grasp the characteristic points in this figure as follows: (1)S(k) has an oscillatory behavior, (2) in small k, the value of S(k) in aring structure yields larger than in a linear-chain structure, (3) inthe region of k > 2 (a.u.), S(k) fluctuates around the value ofS(k) = n. This oscillatory structure originates from the pair correla-tion of constituent atoms in the cluster.

2.3. Electronic stopping power

Let us assume that the cluster projectile moves in a target mate-rial with velocity V

!, where the cluster is composed of homo-atoms

and contains n atoms of atomic number Z. We regard the cluster asan ensemble of partially stripped ions, and assume that the elec-tron cloud of each ion is described by the Thomas–Fermi–Molierestatistical distribution. Then the charge density of jth ion located atthe origin is given by qjð r!Þ ¼ Zedð r!Þ � eqjeð r!Þ, with

qjeðrÞ ¼Nj

4pr

X3

s¼1

asbs

Kj

� �2

exp � bsrKj

� �: ð9Þ

Here Nj and Kj are the number of bound electrons and the sizeparameter of the electron cloud, respectively. The values of a1, a2,

Fig. 2. The speed dependence of the average charge of a linear-chained Cn cluster with(R = 3a0), dot-dot-dash line (R = 4a0), dot-dash line(R = 6a0), short dashed line (R = 10a0). Sshown in (c): long dashed line (a central ion of a C6), dot-dash line (an edge ion of a C6), ddiamond and the open square refer to the middle ion and an edge ion of a C3, respectiv

a3 and b1, b2, b3 are 0.10, 0.55, 0.35 and 6.0, 1.20, 0.30, respectively,and Kj ¼ 0:6269N2=3

j a0=ðZ � 17 NjÞ.

Let us find the stopping power of a dielectric medium for a mov-

ing cluster. First, the force F!

j, acting on the jth partially stripped ion

with charge density qjð r!; tÞ ¼ Zedð r!� V!

tÞ � eqjeð r!� V!

tÞmoving

at velocity V!

in a dielectric media with the dynamical dielectric

function eð k!;xÞ is given by F

!j ¼ �

Rd3rqjð r!; tÞruindð r!; tÞ. Here

the induced scalar potential uindð r!; tÞ is connected with the in-

duced charge density qindð k!;xÞ through

qindð k!;xÞ ¼ 1

eð k!;xÞ� 1

" #qextð k!;xÞ: ð10Þ

The external charge density for the incident cluster ions in Fou-rier space is given by

qextð k!;xÞ ¼ 2p

Xj

qjð� k!Þexpð�i k

!R!

jÞdðx� k!

V!Þ; ð11Þ

where qjð k!Þ denotes the charge density in Fourier space of the jth

ion located at position vector R!

s with respect to the center-of-mass(CM) of the cluster, moving at velocity V

!in a dielectric media. By

summing up the total force acting on every ions in a cluster, one fi-nally obtain the expression of the electronic stopping power as

dEdx¼ 2

pV2

Z þ1

0dk

1kjqextð k

!Þj2

D EZ kv

0dxxJm

�1eðk;xÞ

� �: ð12Þ

In the above equation, the square of the external charge densityis averaged over the cluster orientation, becomes

jqextð k!Þj2

D E¼Xn

j¼1

qjðkÞh i2

þXn

j

Xn

‘ð–jÞqjðkÞq‘ðkÞ

sinðkRj‘ÞkRj‘

; ð13Þ

and the jth charge density has a form of qjðkÞ ¼ e½Z � qjeðkÞ�, with

qjeðkÞ ¼ Nj

X3

s¼1

asb2s

b2s þ k2K2

j

: ð14Þ

It is noted here that if the charge density in Eq. (13) are set to beunity, this equation give rise to the quantity S(k) in Eq. (8). In thisrespect, the external charge density reflects the spatial structureof the cluster in Fourier space.

equal separation R for (a) n = 4 and (b) n = 6: dashed line (R = 2.4a0), dotted lineolid line indicates for a single C ion. The position effect for C3 and C6 with R = 2.4a0 isotted line (the middle ion of a C3), short dashed line (an edge ion of a C3). The filled

ely [9].

Fig. 3. The speed dependence of the average charge of a Cn cluster in a ring structure with equal separation R: (a) R = 2.4a0, and (b) R = 4.0a0: dashed line (n = 20), dotted line(n = 10), dot-dot-dash line (n = 6), dot-dash line (n = 4), short dashed line (n = 3), long dashed line (n = 2). For reference, solid line indicates a single carbon ion.

Fig. 4. The ratio of the electronic stopping power, as a function of speed, of (a) Al for a linear-chained Cn, (b) Al for a ring Cn, (c) Si for a linear-chained Cn, and (d) Si for a ring Cn,with R = 2.4a0 to that for a single C ion at equivalent speed: solid line(n = 2), short dashed line(n = 3), dot-dash line (n = 4), dot-dot-dash line (n = 6), dashed line (n = 10).

T. Kaneko et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 76–80 79

3. Numerical results and discussion

First we show the dependence of the cluster averages on thespeed, ranging from m0 ð1 a:u:Þ to 10m0. Fig. 2(a) and (b) showsthe calculated average charge per ion, defined asQ ¼ ð1=nÞ

Pnj¼1Qj, of swift C4 and C6 clusters in a linear-chain struc-

ture, respectively. In each figure, the dashed line, the dotted line,the dot-dot-dash line, the dot-dash line, and the short dashed linesindicate the cases of the separation of R = 2.4, 3, 4, 6 and 10a0,respectively. For reference, the average charge of a single carbon

ion is drawn by the solid line. From these figures, we find that(1) over the whole speed the average charges of the cluster ion issmaller than that of a single C ion at equivalent speed, (2) for eachCn, the cluster average charge approaches the corresponding aver-age charge value of a C ion when the cluster speed becomes high orthe inter-atomic separation becomes larger, (3) at a given speed,the ratio of the cluster average charge to that of a C ion reducesmore with increasing the number of constituent atoms in the clus-ter. Fig. 2(c) displays the position effect of the cluster averagecharge. The short dashed line and dotted line refer to an edge ion

80 T. Kaneko et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 76–80

and the middle ion of a linear-chained C3, respectively, togetherwith the corresponding data [9]. The dot-dash line and the longdashed line indicate an edge ion and a middle ion of a linear-chained C6, respectively. This clearly shows the ion in a middle po-sition has a smaller average charge than an edge ion.

Fig. 3(a) and (b) shows the calculated average charge per ion ofa Cn (n = 2–20) in a ring structure with equal separation of R = 2.4a0

and 4a0, respectively. In both figures, the dashed line, the dottedline, the dot-dot-dash line, the dot-dash line, the short dash line re-fer to n = 20, 10, 6, 4, 3, respectively. For reference, we also drawthe curves for C (the solid line) and for C2 (the long dashed line).As an important result, we remark that comparing Fig. 2 withFig. 3, the average charge of Cn (n = 3, 4, 6) clusters with R = 2.4a0

and 4a0 in the ring structure is larger than that of the correspond-ing clusters in the linear-chain structure with an equivalent sepa-ration. As for the 1 MeV/atom C3 cluster, this was confirmed by theexperimental data by Chiba et al. [9].

Fig. 4 shows the ratio of the stopping power, S(n), of aluminumand silicon targets for swift linear-chained and ring Cn; clusterswith inter-atomic separation of R = 2.4a0, to the stopping power,S(1), for a single carbon ion with equivalent speed. In our calcula-tion, the stopping power was evaluated in a shell-by-shell way,based on Eqs. (12)–(14). These targets contain the conduction elec-trons, the number of which is assumed to be three and four peratom, and characterized by the so-called rs values. We adoptrs = 2.070 and 2.006 for Al and Si, respectively. The excitation ofthese conduction electrons is dealt with the dielectric functionmethod [26]. These targets also hold the inner-shell electrons in1s, 2s, and 2p states. The excitation of those electrons is treatedin the wave-packet model [27]. In these figures, the solid line,the dashed line, the dot-dash line, the dot-dot-dash line, and thelong dashed line refer to n = 2, 3, 4, 6, and 10, respectively. As aglance, one finds that S(n) represents a remarkable super-lineardependence for each target over a wide range of speed. In thelow speed region, on the other hand, the stopping power ratioyields the sub-linear dependence on the atom number withincreasing the number of constituent atoms, especially in the ringstructure. In addition, the stopping ratio S(n)/(n � S(1)) has a pla-teau up to m � 5m0 and has a rapid increase beyond this region. Thistendency is attributed to excitation of the inner-shell electrons anda strong charge correlation. This character is common to aluminumand silicon, regardless of linear-chain or ring structures, while it isbit different from carbon target. We mention that the present esti-mation of the stopping power contains the reduction of the clusteraverage charge.

In conclusion, we studied the cluster effect in the averagecharge and the stopping power for MeV/atom carbon clusters.

Regarding the cluster average charge, it shows the sub-linear clus-ter effect regardless of low or high speed. We studied in detail thedependence on the speed, the number of constituent atoms, the in-ter-atomic distance, and the spatial structures of a linear-chain anda ring. Asymptotic formula for a ring cluster was obtained. Theelectronic stopping-power ratios for linear-chain and ring carbonclusters moving in Al and Si targets have a plateau picture, and astrong super-linear-dependent increase with increasing both thespeed and the number of constituent atoms.

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