Stopping of molecules and clusters - Argentina.gob.arStopping of molecules and clusters N estor R. Arista Divisi on Colisiones At omicas, Centro At omico Bariloche and Instituto Balseiro,

Stopping of molecules and clusters

N�estor R. Arista

Divisi�on Colisiones At�omicas, Centro At�omico Bariloche and Instituto Balseiro, Comisi�on Nacional de Energ�õa At�omica, 8400 S.C.

Bariloche, Argentina

Abstract

The interaction of swift cluster and molecular ions with solids has been studied with increasing interest in the past

decades. Experiments have produced valuable information on cluster energy losses, screening e�ects, charge states, and

dynamical interactions among correlated ions, which in many cases have not been obtained using single atomic ions.

These studies show di�erences (vicinage e�ects) in the energy loss per nucleon and also in charge equilibrium conditions

both for small and large clusters.

Theoretical studies dealing with vicinage e�ects in ion cluster interactions, using the most basic models, have pro-

vided a semi-quantitative view of cluster energy losses which partially agree with experiments, but still leave several

questions open. In particular, the dielectric models predict a diminished stopping power in the low-velocity range, as a

result of negative interferences, whereas at high velocities a signi®cant enhancement of the energy loss is expected. The

behavior at high velocities has been con®rmed experimentally using small molecular ions. In addition, there is clear

experimental evidence of diminished stopping values at low velocities, but still the possible in¯uence of various

mechanisms should be clari®ed, like the role of charge state equilibrium, alignment, and non-linear quantum e�ects.

Modi®cations in charge state equilibrium may be specially important in the case of large clusters.

The purpose of this work is, on one side, to review the current knowledge and recent progress in this ®eld, and, on the

other, to reformulate the theory of cluster energy loss in order to incorporate the e�ects of charge equilibrium according

to the most recent experimental evidences. Ó 2000 Elsevier Science B.V. All rights reserved.

PACS: 34.50.Bw; 36.40.+d

1. Introduction

Most of our knowledge on the interactionprocesses that govern the behavior of swift ions insolids derives from experiments using beams ofatomic projectiles. In these experiments, the dis-tances among the beam particles are so large that

the solid returns to equilibrium long before a newparticle enters the region explored by a previousone (except in cases where radiation damage oc-curs). In these conditions, there is no interferencein the interactions of di�erent particles with themedium.

In contrast, when molecules or ion clusters areused, the distances among the particles in thecluster are comparable to the interatomic distancesin the solid, and therefore, the particles interact in

Nuclear Instruments and Methods in Physics Research B 164±165 (2000) 108±138

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E-mail address: [email protected] (N.R. Arista).

0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 0 6 9 - 1

a coherent way with the medium. Hence, it may beexpected that new physical phenomena will resultfrom these interactions. In this way, the use of ionclusters provides a powerful tool to investigate thedynamical interactions of atomic particles withmatter.

The interaction of swift ion clusters with solidshas been a subject of intense activity during thepast decades, starting with the observation byPoizat and Remillieux [1], of the transmission ofH�2 ions in carbon foils [1]. The ®rst experimentalevidence of vicinage e�ects in the energy loss of ionclusters was obtained shortly after by Brandt et al.[2], by observing the energy loss of transmittedprotons produced by the incidence of H�2 and H�3ions on thin carbon foils, and it was followed bysimilar experiments performed in various labora-tories [3±7].

This vicinage e�ect was theoretically explainedas a result of interferences in the excitation oftarget electrons by the simultaneous interactionswith two or more ions moving in a correlated wayand with short internuclear distances through thesolid [2,8±10]. One of the interesting aspects is theconnection between the energy loss of swift ionclusters and the so-called wake phenomenon alsostudied intensively in those years, both experi-mentally [11±17] and theoretically [18±20].

Numerous experiments with molecular ions,dealing mainly with the observation of molecularexplosion, dynamical screening and wake e�ects,transmission of molecular ions, and related phe-nomena, have been reported since, and severalreview papers have also been published [21±24].Various aspects related to elastic collision cascades[25,26] and cluster energy losses [27] have also beencovered in previous papers.

Moreover, recent advances in experimentaltechniques allow the possibility of acceleratinglarge molecular clusters containing up to severalthousand molecules. Clusters of hydrogen, car-bon and water molecules are among the mostfrequently studied cases [28±32]. Studies withsmall and large clusters have shown that theconditions of charge state equilibrium are sig-ni®cantly modi®ed by the presence of close ions[33±36]. This e�ect has not been taken into ac-count so far in the theoretical descriptions, and

it is one of the main points addressed in thispaper.

The studies of the interaction of large ionclusters with matter have also attracted the atten-tion of several investigators in relation with pos-sible applications to plasma physics [37,38] andmaterials research [39,40]. In particular, due to thelarge concentration of deposited energy that theycan provide, swift ion clusters are also consideredas possible drivers in inertial fusion studies [41±47].

After almost three decades of research, the areaof cluster±solid interaction has become a very rich®eld; the present paper will refer only to thoseaspects of the interaction processes that bear adirect connection with the energy loss problem.Further information on related aspects may beobtained from the previously mentioned articles.

The purpose of this paper is, on one side, toreview the present knowledge of the energy loss ofmolecules and swift ion clusters in solids, and, onthe other, to present new developments on thetheory of the energy loss of ion clusters, incorpo-rating in the description the e�ects of chargeequilibrium.

First, the theoretical formulation will be re-viewed, and several known problems and new as-pects will be discussed; in particular, those relatedto random and aligned-cluster di�erences, non-linear e�ects for slow clusters, and modi®cations incharge state equilibrium conditions for small andlarge clusters, including the existence of vicinagee�ects in the population of bound states. Thestandard formulation of cluster stopping powerwill be extended and updated, in the light of newexperimental evidences, to include the possibilityof multiple charge states for each of the individualions in the cluster.

2. Cluster stopping power

The stopping power due to valence electronswill be discussed here within the framework of thedielectric formalism, and the connection withother methods will be indicated. In the presentapproach the properties of the target are describedby its dielectric function ��q;x�, where q and xrepresent the momentum and energy transferred in

N.R. Arista / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 108±138 109

an inelastic process to the system. This approachhas the possibility of describing in a self-consistentway the screening of the intruder ions as well as theexcitations of valence electrons in the solid, in-cluding both collective and single-particle (orelectron±hole) excitations.

Since the internal motion of the ions within thecluster is usually very slow in terms of character-istic electronic times (due to the large mass di�er-ence between electrons and ions), the excitation ofelectrons may be considered decoupled from therelative motion of the ions. Within this scheme, theinstantaneous stopping power of a swift molecularion or cluster of N atomic ions, moving with ve-locity v; is given by the average of the energy lossfor the whole cluster as follows [9,10,20]:

Scl�v� � dEdx

� �� 1

2p2v

Zd3q Fcl�~q�� 2

� ~q �~vq2

Imÿ1

��q;~q �~v�

" #; �1�

where Fcl�~q� is the form factor of the incidentcluster, de®ned in terms of its total charge distri-bution qcl�~r�, by

Fcl�~q� �Z

d3rqcl�~r� exp�i~q �~r�: �2�

In the case of molecular ions, the total chargedensity is qcl�~r� � qcore�~r� � qmol�~r�; where qcore�~r�is the charge density of the ionic cores (includingthe nuclei), and qmol�~r� is the electronic chargedensity corresponding to the molecular state, whichmay be calculated in terms of the molecular wavefunction W�~rn� � W�~r1;~r2; . . . ;~rNe

� (for a moleculecontaining Ne valence electrons) as follows:

qmol�~r� �XNe

n�1

Z�d3rn�d�~r ÿ~rn� W�~rn�

�� 2; �3�

where �d3rn� � d3r1d3r2 � � � d3rNe:

Most frequently, the molecule dissociates afterpenetrating the solid. In this case one can expressthe charge density, for a cluster of correlated ions,as

qcl�~r� �XN

i�1

qi�~r ÿ~ri�; �4�

where qi�~r ÿ~ri� is the total (electronic and nuclear)charge density of an ion located at position ~ri;relative to the cluster center of mass.

From Eqs. (2) and (4) one obtains

Fcl�~q� �XN

i�1

fi�~q� exp�i~q �~ri�; �5�

where fi�~q� is the atomic form factor of ion i,namely

fi�~q� �Z

d3rqi�~r� exp�i~q �~r�: �6�

In particular, for a bare nucleus of charge Zie,qi�~r� � Zied�~r�; and so fi�~q� � Zie:

Using Eq. (5) in Eq. (1) the following expansionfor Scl is obtained:

Scl�v� �XN

i�1

Si�v� �Xi6�j

Iij�~rij;~v�; �7�

where

Si�v� � 1

2p2v

Zd3q fi�~q�� 2

� ~q �~vq2

Imÿ1

��q;~q �~v�

" #�8�

and

Iij�~rij;~v� � 1

2p2v

Zd3qfi�~q�f �j �~q�

�~q �~vq2

Imÿ1

��q;~q �~v�

" #exp�i~q �~rij�: �9�

Si�v� represents the stopping power of each in-dividual ion in the cluster due to its self-interactionin the medium, whereas Iij � Iij�~rij;~v� are interfer-ence terms which depend on the relative positions~rij of each pair of ions �i; j�. One should note aminor di�erence in the de®nition of the interfer-ence terms with respect to some previous papers

110 N.R. Arista / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 108±138

[9,27], in that the magnitudes used here have di-mensions of stopping power.

From Eq. (9) it may be shown that the inter-ference terms satisfy a reciprocity relation:

Iij�~rij;~v� � Iji�~rji;~v�: �10�This can be shown using the property that

Im ÿ1=��q;x�� is an odd function of x, and thatfi�ÿ~q� � f �i �~q�; as seen from Eq. (6). Eq. (10) is ageneralization of a similar property shown in [9]for the simplest case of point-like particles.

The quantity of central interest to measure thevicinage e�ect in the energy loss of a cluster ofparticles is the stopping power ratio, de®ned by

R � Scl�v�PNi�1 Si�v�

��ion-beam

: �11�

The subscript ion-beam in this ratio indicatesthat the stopping power values Si�v� in the de-nominator correspond to measurements usingatomic-ion beams. At this level of the formulation,there is no di�erence between these Si values andthose in Eq. (7), but this distinction will becomeimportant for the discussion in Section 4, wherethe charge state problem will be addressed.

Before going on with this description, it isuseful to establish the relation between the di-electric formalism and other perturbative modelsdealing with atomic excitations [10,20,48]. Thesemodels are usually expressed in terms of the gen-eralized oscillator strength F0n�~q�, de®ned by thetransition-matrix elements for electron excitationsj 0i !j ni, as follows [49±51]:

F0n�~q� � 2m�En ÿ E0��h2q2

jhnj exp�i~q �~r�j0ij2: �12�

The connection with the dielectric function can bemade, in the present notation, with the replace-mentZ

d3q~q �~vq2

Imÿ1

��q;~q �~v�

" #. . .

) 4p2e2na

m

Xn

Zd3qq2

F0n�~q�d ~q �~v�

ÿ DE0n

�h

�. . . ;

�13�

where na is the atomic density, and DE0n ��En ÿ E0� are the excitation energies.

With this correspondence, Eq. (1) may betransformed into

Scl�v� � 2e2na

mv

Xn

Zd3qq2

Fcl�~q�� 2F0n�~q�

� d ~q �~v�

ÿ DE0n

�h

��14�

and in the same way other equations may betransformed to the atomic perturbation theoryform.

This brings the connection between the dielec-tric formalism and other developments using theoscillator strength representation. While bothformulations are based on perturbative approxi-mations, each one has its own properties and ad-vantages. The dielectric formalism is particularlyuseful to evaluate the contribution of valenceelectrons since dynamical screening and plasmone�ects are included, whereas the atomic oscillatorstrength description may be more useful to incor-porate the e�ect of the various atomic shells. Al-ternatively, the dielectric formalism may beextended to incorporate also the atomic propertiesof the target material, using for instance an ex-tended description in terms of Mermin dielectricfunctions, adjusted with optical or electron energyloss data [52,53].

If the atomic form factors are isotropic: fi�~q� �fi�q�; an average over the orientation of the in-ternuclear axis ~rij in Eq. (9) can be easily per-formed, and one gets a simple expression for theangular average of the interference term:

Iij�r; v� � 2

pv2

Z 1

0

dqq

sin�qr�qr

fi�q�f �j �q�

�Z qv

0

dxx Imÿ1

��q;x��

: �15�

In particular, for bare ions with charges Zi; Zj awell-known expression is retrieved [9]:

Iij�r; v� � ZiZjI0�r; v�; �16�


where

I0�r; v� � 2e2

pv2

Z 1

0

dqq

sin�qr�qr

�Z qv

0

dxx Imÿ1

��q;x��

; �17�

and similarly for the individual stopping terms,

Si�v� � Z2i S0�v�

� 2Z2i e2

pv2

Z 1

0

dqq

Z qv

0

dxx Imÿ1

��q;x��

:

�18�

In a simpli®ed e�ective charge description [54], thecharges Zi may be replaced by the correspondinge�ective charges Z�i : One should note however thata more consistent description will be obtained byintroducing a full representation of the di�erentcharge state fractions and corresponding ionicform factors as it will be discussed later in thispaper.

In the following, the application of this modeland the available experimental information will bediscussed, ®rst for the case of diatomic clusters,and then for more complicated structures.

3. Small clusters

3.1. Diclusters

The simplest case to be considered here is thatof a pair of ions Z1; Z2 in correlated motion; astructure that may be obtained by the incidence ofdiatomic molecules. The study of this case is basicin order to understand more complicated clusterstructures, since the formulation for an arbitrarycluster may be expressed in terms of pair interac-tion terms. So it is convenient to start by reviewingthis case.

The main electron gas parameters to be usedhere are the following: Fermi velocity vF, plasmafrequency xP, and Wigner±Seitz radius rs (withrelations: vF � 1:919=rs, xP �

��3=r3

s

p; in atomic

units). For simplicity, this discussion will referonly to the high velocity range, v� vF; a discus-

sion of the case v < vF can be found in [9]. Then,two characteristic ranges of interaction are dis-tinguished: ®rst, a range of close or binary ion-electron interactions, given by distances r in therange �h=2mv < r < vF=xP; whose main e�ect is theexcitation of electron±hole pairs in the electrongas; and second, a domain of distant or collectiveinteractions, with typical distances vF=xP < r <v=xP; where resonant or plasmon excitation is therelevant process.

The importance of plasmon excitations in thisenergy range was one of the main questions ana-lyzed in the earliest publications [2,8,9,18]. It maybe noted however, that the quantal properties ofthe plasma waves are not directly involved in thisprocess (the quantum parameter �h=2mv appearinghere pertains to the description of individualelectron excitations), so that a similar behaviormay be obtained using for instance a harmonicoscillator model with a resonance frequency xP

[27].In terms of these parameters, the following

cases arise, depending on the internuclear distancer12:

(a) r12 < �h=2mv: In this case, the two ions willbehave as a single united ion (with charge Z1 � Z2 )with respect to both collective and single-particleexcitations. This may be called the united atomlimit.

(b) �h=2mv < r12 < v=xP: Here the two ions be-have as separate ions with respect to the closestinteractions (r � �h=2mv) with target electrons,while at the same time they behave as united ionswith respect to the most distant collective inter-actions (resonant or plasmon excitations) withadiabatic distance v=xP:

(c) v=xP < r12 < 5v=xP: Here the ions start toseparate; however, there is still an interesting rangeof negative interferences, for r12 � 2v=xP ÿ 5v=xP.

(d) r12 > 5v=xP: The ions become independent.For realistic internuclear distances, the condi-

tion r12 < �h=2mv is never achieved (in the low ve-locity limit [9] the relevant velocity is not v but vF,so that this condition becomes r12 < �h=2mvF;which again cannot be ful®lled with the usual r12

and vF values). Therefore, the cases of actual in-terest for important vicinage e�ects in cluster±solidinteractions are those of cases (b) and (c).


The occurrence of positive and negative inter-ferences becomes particularly notorious in the caseof aligned diclusters, where these e�ects are max-imal. A simple interpretation may be given byconsidering the interference process as a time-correlation problem [8], namely: two perturbationsshifted in time (Dt � r12=v) acting on a harmonicoscillator with frequency xP. In this case, positiveor negative interferences may be obtained, de-pending on the value of the phase xPDt � xPr12=v:

3.1.1. Analytical approximationsThe value of the interference term I0�r; v�; Eq.

(17), can be calculated analytically in the high-velocity limit (v� vF), using e.g. the plasmon poleapproximation. This yields the simple result [9]

I0�r; v� � exP

v

� �2 sin�x�x

�ÿ Ci�x�

��x2

x1

; �19�

where x1 � 2mvr=�h; x2 � rxP=v; and Ci�x� denotesthe cosine-integral function.

Under the same conditions, the individualstopping term, Eq. (18), becomes

S0�v� � exP

v

� �2

ln2mv2

�hxP

� �: �20�

Moreover, when condition (b) is satis®ed, one getsa further approximation [9]:

I0�r; v� � exP

v

� �2

ln1:526v

rxP

� �: �21�

These simple results serve to illustrate the mostbasic properties of the pair-interference e�ect. InFig. 1 the ratio I0�r12; v�=S0�v�; Eqs. (19) and (20),is shown as a function of the internuclear distancer12, for velocities v � 1, 2, 5 and 10 a.u. It may beobserved that there is a range of distances wherethe interference is negative, and this region movesto larger values of r12 with increasing v: Therefore,positive or negative interferences arise, dependingon the internuclear distance and cluster velocity.The main range of close positive interferences isgiven by r12 < 2v=xP. In the case of large clustersat high velocities, the interference among closeneighboring ions is positive but it turns negativefor more distant ions, so that, in general, both

e�ects compete (it may be seen that the amplitudeof the oscillations of I0�r; v� decays with 1=r2, butthis decay is exactly compensated by the increasingnumber of distant ions in a large cluster).

However, it may be expected that with in-creasing velocities the range of positive interfer-ences (�2v=xP� will continuously grow, reachingeventually to all the ions in the cluster (except ofcourse for relativistic limitations if the size ofcluster is very large). The consequences of thisbehavior will become very important for the caseof large clusters as it will be discussed in Section 6.

3.1.2. Classical and quantum descriptionsTo complete this review of basic properties, the

relationship between the quantum and the classicalpictures should be mentioned. The di�erence be-tween both pictures arises from the treatment ofshort-range excitations of individual electrons, justas in the usual stopping power theory for singleions [57]. The parameter that regulates the transi-tion between both regimes is the usual Bloch pa-rameter [58], g � Ze2=�hv; so that when g < 1 thequantum mechanical (perturbation theory) ap-plies, and then the value of bqm � �h=mv (corre-sponding to qmax � 2mv=�h) determines the limitingbehavior pertaining to individual electron excita-tions; whereas when g > 1 a classical picture

Fig. 1. Ratio I0�r12; v�=S0�v�; from Eqs. (19) and (20), as a

function of the internuclear distance r12; for velocities v � 1; 2, 5

and 10 a.u., for an electron gas with xP � 0:5 a.u.


becomes more appropriate, and in this case theproperties of close collisions are best described interms of the classical collision radius bcl �Ze2=mv2: Thus, classical corrections would benecessary in the case of highly charged ions, orwhenever the condition Ze2=�hv > 1 is satis®ed, likefor slow ions in plasmas (the case of slow ions insolids, however, requires a di�erent approach totake into account the strong quantum e�ects to bediscussed in Section 3.5).

The relationship between classical and quantumdescriptions in the interaction of ion clusters withharmonic oscillators has been reviewed in [27]. Inaddition, the interaction of correlated ions withclassical (dilute) plasmas has been studied in[55,56], using classical and semiclassical concepts.

The description of distant interactions in thesepictures remains essentially the same, and ischaracterized by the adiabatic parameter bad �v=xP. Therefore, in the cases discussed here (in-ternuclear distances of a few atomic units), theinterference term I0�r12; v� is insensitive to thechoice of either a classical or a quantum mechan-ical (perturbation) model, in as much as the con-ditions r12 � bqm; r12 � bcl; are always ful®lled[56]. On the other hand, the individual stoppingterm S0�v� has a logarithmic dependence on theminimum distance parameter bmin (being either bqm

or bcl ), S0�v� � ln�bad=bmin�, and the interpolationformula derived by Bloch [57,58] correspondsroughly to taking the maximum between these twovalues of bmin, in such a way that S0�v� may beapproximated by the smallest of the two corre-sponding limiting values [57]. As a consequence,the stopping ratio, depending on I0�r12; v�=S0�v�;increases when classical corrections are taken intoaccount [27,56].

3.2. Statistical models and Thomas±Fermi scaling

In the case of carbon targets to be considered inthe following sections (the case where most ex-perimental information on cluster stopping exists),the dominant contribution to the stopping powerat low and intermediate energies comes from thefour valence electrons of carbon. But for heavierelements and higher velocities the contributionfrom inner shells becomes increasingly more

important. The integration of the former resultsfor a non-uniform electron density distribution,trying to represent in a more realistic way theproperties of real solids, has been one of themethods used to take into account the electronicstructure of the medium [59±61].

A simple approach to illustrates general prop-erties of the stopping power at large velocities wasproposed earlier by Lindhard and Schar� [62],based on the statistical properties of the Thomas±Fermi (TF) atom. Since this approach provides ageneral view, it is useful to cast the present for-mulation of interference e�ects in the scheme ofthe statistical description.

In order to simplify expressions, atomic unitswill be used in the following.

Let us consider the case of a simple dicluster ofpoint-like particles with charges Z1 and Z2, andinternuclear distance r12; and integrate the clusterstopping power over the electronic distribution ofthe target atom, following [62]. This yields

�Scl � 4pv2

Ztna �Z21

h� Z2

2��L� 2Z1Z2�Ji; �22�

where Zt and na are the target atomic number andatomic density, respectively, and �L and �J are thestopping and interference numbers averaged overthe whole atom. Using the high-velocity approxi-mations of Eqs. (19) and (20) one gets

�L � 1

Zt

Zd3r n�r� ln 2v2

x0�r��

; �23�

�J � 1

Zt

Zd3r n�r�g r12x0�r�

v

� �; �24�

being

g�x� � sin�x�xÿ Ci�x�: �25�

In Eq. (24) the function g�x� is evaluated atx2 � r12x0=v; after using the limit 2mvr12=�h� 1 inEq. (19). The local resonance frequency in Eq. (24)is given by x0�r� � cxP�r� � c

��4pn�r�p

; withc � ��

2p

[62]. The factor c � ��2p

has been intro-duced in [62] as a correction to take into account inan approximate way the fact that the atomicelectrons are not free but subject to atomic binding


forces. This correction has become of common useeven though it has not been fully justi®ed (from apractical point of view, the introduction of thisfactor yields values of the mean excitation energy,I � ZtI0; in better agreement with experiments).For reasons of internal consistency, the samecorrection is applied here to the resonance fre-quency arising in Eqs. (23) and (24).

The Thomas±Fermi scaling is introduced viathe expression for the electron density, n�r� �Z2

t f �x�; with

f �x� � 32

9p3

u�x�x

� �3=2

; �26�

being u�x� the TF screening function in terms ofthe reduced distance x � rZ1=3

t =b, with b � 0:8853:Therefore, the resonance frequency scales with Zt

according to x0�r� � cZt

��4pf �x�p

:Introducing these expressions in Eqs. (23) and

(24) one gets

�L�u� � b3

Zd3x f �x� ln

u

c��pf �x�p !

; �27�

�J�q12� � b3

Zd3x f �x�g c

��4pf �x�

pq12

� �; �28�

in terms of reduced variables u; q12 representingvelocity and internuclear distance,

u � v2

Zt

; q12 �r12

vZt: �29�

Using the normalization condition for the densityfunction f �x�; the stopping number may be writtenin the well-known asymptotic form [62]

�L�u� � ln2uI0

� �� ln

2v2

ZtI0

� �; �30�

and I0 may be calculated from

ln�I0� � b3

Zd3x f �x� ln c

��4pf �x�

p� �: �31�

The numerical result for I0 depends slightly onthe model for the screening function used toevaluate the integral. In particular: I0 � 8:9 eV forthe TF model, I0 � 10:7 eV for the Lenz±Jensen

model, and I0 � 11:7 eV using a Moli�ere approxi-mation to the screening function.

On the other hand, if the approximation of Eq.(21) is used for the g-function in Eq. (28), corre-sponding to the case r12 � v=x0 (high velocitylimit), a similar logarithmic approximation for the�J -function is obtained:

�J�q12� � ln1:526

I0q12

� �: �32�

The function �J�q12� integrated from Eq. (28) forthe TF model is shown in Fig. 2, as well as theanalytical high velocity approximation of Eq. (32).

These results yield a di�erent scaling for theinterference term: while the stopping number �Lscales with v2=Zt; the interference number scaleswith q12 � r12Zt=v: Therefore, a simple scaling ofthe ratio �J=�L in terms of a unique reduced variableis not possible.

A comparison of the behavior of the functions�L�v2=Zt� and �J�r12Zt=v�, for di�erent values of Zt

and v; shows that the interference e�ect, measuredby the ratio �J=�L, will generally be larger for lowerZt values and for larger velocities. The decrease ofthe interference e�ects with increasing Zt is physi-cally explained since the inner shells should beexcited in a rather incoherent way by the twoparticles [20].

Fig. 2. Average interference term �J�q12� integrated using

Eq. (28) for the TF model, as a function of the reduced pa-

rameter, qÿ112 � v=r12Zt; Eq. (29). The dashed line is the analyt-

ical high-velocity approximation of Eq. (32).


In conclusion, the use of low-Z materials ispreferred to maximize the interference e�ects in theenergy loss of ion clusters. In particular, carbontargets are mostly favorable since the contributionof the inner K-shell is very small [63,64].

3.3. Beam-foil experiments

In order to discuss the behavior of the inter-ference e�ects for proton diclusters, experimentalresults of the stopping power ratio R in Eq. (11),from various authors [2±7,65±67], have been col-lected in Fig. 3. All these experiments were madeusing thin carbon foils, with thicknesses between 2and 5 lg=cm2. The ®gure also shows results ofcalculations of the stopping ratio for pairs ofcorrelated protons obtained with the present for-malism and using Lindhard's dielectric function[69]. The calculations correspond to a set of in-ternuclear distances in the range of those expectedin thin-foil experiments [2,70]. In addition, the®gure shows the stopping ratio corresponding toan undissociated H�2 molecule, calculated directlyfrom Eqs. (2) and (3), using a form factor Fcl�~q� forthe molecular ion corresponding to a variationalwave function proposed in [71]. The result of thiscalculation is indicated by the curve denoted H�2 inFig. 3.

This simple representation provides some clearindications. First, there is a signi®cant increase ofthe interference e�ects for energies larger thanabout 40 keV/amu. This increase in the vicinagee�ects is due to the appearance of characteristicelectronic excitations for velocities larger thanabout 1 a.u., followed by an enlargement of theinteraction (or adiabatic) distances relevant forthese excitations. Secondly, in the low-energyrange a decreased stopping ratio (R < 1) is ob-served. This diminished stopping power of the di-cluster, as compared with separated protons, hasreceived the attention of various authors, and itwill be considered in more detail below.

The open data points in Fig. 3 correspond totransmitted molecules; these are the cases wherethe lowest energy losses are obtained. However, theloci of these data lie su�ciently above the H�2 curveso that the possibility of an explanation in terms ofundissociated molecules may be discarded. The

processes leading to transmitted molecules havebeen studied extensively [34,72±75], and the fol-lowing two cases can be distinguished: (i) moleculesfrom the initial beam passing through the foilwithout breaking up (direct transmission regime,predominant for small dwell times, s < 1 fs), and(ii) reconstituted molecules originated at the exitsurface from ions that emerge with appropriatedistances and relative velocities (recombination re-gime, dominant for large dwell times, s > 1 fs). Thecases shown in Fig. 3 correspond to the recombi-nation regime.

Fig. 3. Calculations and experimental values of stopping power

ratios for H�2 on carbon foils. The data points represent the

experimental values from various sources, as indicated in the

inset. The solid points show the average energy loss of emerging

protons from dissociated molecules; the open data points cor-

respond to the energy loss of transmitted (recombined) mole-

cules. The curves show the calculations of the stopping power

ratio R; with Eqs. (15)±(18), using Lindhard's dielectric function

for an electron gas with rs � 1:6 a.u., and for internuclear dis-

tances r12 between 1.2 and 2.5 �A (in the range expected for these

experiments). The foil thickness in these experiments range

between 2 and 5 lg=cm2.


In order to establish the range of internucleardistances of interest in these experiments, one mustconsider the e�ects of the screened Coulomb ex-plosion of the molecular fragments and the e�ectsof multiple scattering in the foil.

The screened interaction between two protonsmay be represented by a screened potential of theform exp�ÿar�=r, with a ' xP=

��v2 � v3

F=3p

; inter-polating the limits of low and high velocities [76].Results of these calculations are shown in Fig. 4,including values of the mean internuclear distance

at the exit of the foil, rexit part (a), and of the av-erage internuclear distance hri inside the foil part(b), which is de®ned by

hri � 1

s

Z r�s�

r0

rvrel�r� dr; �33�

where s � D=v is the dwell time in the foil andvrel�r� is the relative radial velocity between bothions. Also shown in the ®gure are the results cor-responding to the pure Coulomb explosion andthose for dynamically screened interactions, forvarious initial internuclear distances r0: The valuesare plotted as a function of the beam energy perunit mass. These calculations correspond to acarbon foil of thickness D � 100 �A (�2 lg=cm2�:

The phenomenon of multiple scattering pro-duces a dispersion of the internuclear distancesaround the mean values calculated before. Themagnitude of this e�ect may be estimated using theformulation for the lateral dispersion derived frommultiple scattering theory [77]. This assumes thatthe multiple scattering of the particles is incoherent,which may not be the case for short internucleardistances. The e�ects of coherent interactions in themultiple scattering process for swift diclusters havebeen studied theoretically by several authors [78±81] and will be further considered in Section 3.4. Inaddition, it should be noted that the straggling ofenergy induces a longitudinal spread of the inter-nuclear distances; however it may be shown that,for slow ions, this e�ect is much smaller than thelateral spread produced by multiple scattering.

The e�ects of multiple scattering in the smear-ing of the internuclear distances are represented byvertical bars in Fig. 4. The ®gure clearly shows thatboth e�ects, Coulomb explosion and multiplescattering, become important for thin foils and forenergies below about 100 keV/amu (dwell timeslarger than 2 fs). In particular, multiple scatteringis considered to be very important in the process ofmolecular transmission via recombination, since itenhances the probability of recombination at theexit surface by providing a mechanism that in-creases the probability that two protons emergewith internuclear distances and relative velocitiesadequate to form a bound state when emerg-ing from the foil [74,83]. In fact, molecular

Fig. 4. Internuclear distances corresponding to the dissociation

of H�2 molecules and screened explosion of the pair of protons

in a carbon foil of thickness D � 100 �A, as a function of the

impact energy, and for three values of the initial internuclear

distance, r0 � 1:2; 1:3 and 1:4 �A. The dotted lines with square

symbols are the results for unscreened Coulomb interaction.

Part (a): calculations of the mean internuclear distance at the

exit of the foil, rexit. Part (b): average internuclear distance

within the foil, rav � hri, de®ned by Eq. (33). The vertical bars

in these ®gures represent the smearing of the internuclear dis-

tances due to multiple scattering in the foil estimated using the

multiple scattering theory for the lateral spread [77].


recombination may be viewed as a kind of ``sur-vival of the ®ttest'' phenomenon; those moleculeswith smaller internuclear distances and relative ve-locities will have more chances of recombination.

To obtain more realistic representations of theprocesses of molecular break up or recombination,one should resort to numerical simulations. Severalstudies along this line have been produced, most ofthem at high energies [23,82±85]. Among the mainingredients to be included in accurate simulationswe should mention: description of the screened in-teractions and wake forces, using for instance aMermin-type representation of the dielectricproperties [85±88], multiple scattering with andwithout coherent interaction e�ects [78±81], andthe inclusion of energy loss straggling and charge-state ¯uctuations. So far, no calculations incorpo-rating all these aspects have been produced.

3.4. Angular orientation e�ects

The earlier experiments using H�2 ; H�3 and Oÿ2ions [2,3,5] provided stopping power values forrandomly oriented molecules for intermediate andhigh energies. In all these experiments an en-hancement of the energy loss (R > 1� due to thevicinage e�ect was observed. The ®rst observationof diminished stopping power values (R < 1) fortransmitted molecules was made by Eckardt et al.[4], and the possibility of partial orientation e�ectswas suggested, based on the comparison withtheoretical calculations. Further measurements byseveral authors con®rmed the diminished stoppingratios (or negative vicinage e�ects) for transmittedmolecules at low energies; the e�ect is clearlyshown by the open data points in Fig. 3.

The angular distributions of emerging (recom-bined) H�2 molecules at low energies (3±30 keV/amu) in thin carbon foils [89] show angular widthsreduced by a factor

��2p

relative to equal-velocityH0 atoms, indicating that the two hydrogen ionsscatter in a rather independent way during theirtransit through the foil. However, the completeangle-energy distributions (the ring patterns)which could provide direct information of possiblealignment e�ects, have not been obtained.

Other experiments by Steuer et al. [67,68] werespecially designed to measure the stopping ratios

for the dissociation fragments of molecular ions(N�2 ; O�2 ; He�2 and H�2 ) with internuclear axesoriented parallel to the beam direction (one mayexpect that multiple scattering e�ects will producea dispersion of the angular orientations, but thesee�ects may be estimated to be very small in theseexperiments). The experiments provided furtherevidence of diminished stopping values for lowvelocity diclusters (R � 0:85±0.95), and were the-oretically analyzed in [90].

An estimation of the orientation e�ects usingthe dielectric formulation is given in Fig. 5,showing the stopping ratio R; calculated for thecases of random (a) and aligned (b) diprotons. Asigni®cant decrease is obtained for velocities below2 a.u., where the stopping for aligned diclustersdrops to values around 0.7 or below. These valuesare lower than those reported experimentally, butthe results may be consistent with the possibility ofa partial alignment of the transmitted particles.

Fig. 5 includes two types of calculations: thesolid lines are the results corresponding to twocorrelated protons, whereas the dashed lines cor-

Fig. 5. Stopping ratios for random and aligned hydrogen dicl-

usters in carbon foils. Curves (a) correspond to the angular

average of the interference term (random orientation of the

internuclear axis); curves (b) correspond to pairs of particles

travelling with the internuclear axis oriented parallel to the

beam velocity. The solid curves are the results for pairs of

protons, whereas the dashed lines include the equilibrium

fractions of protons and neutral atoms in the beam according to

Eqs. (42)±(46). The equilibrium fractions in this velocity range

have been represented by empirical ®ts to the data from

[35,36,91] .


respond to charge equilibrium (i.e., including thee�ect of neutral hydrogen atoms in the beamproduced by capture and loss processes) [35,36,91].This is an e�ect that has not been consideredpreviously and will be discussed in Section 4.

The limitations in the possibility of obtainingwell-aligned diclusters due to the process of (in-coherent) multiple scattering have already beenmentioned. On the other hand, if the process ofcoherent multiple scattering is considered (i.e., thesimultaneous elastic interactions of both movingions with target atoms), one ®nds a possiblemechanism to induce an alignment e�ect on thedicluster. Previous calculations by Kononets [79]at higher energies show that coherent elastic in-teractions may collaborate with electronic wakeforces in producing an alignment e�ect. In fact, thee�ects of wake forces on the angular distributionsof the emerging ions have been studied extensivelyfor swift diclusters [11±17]. However, at low en-ergies wake forces are replaced by nearly spheri-cally symmetric screened potentials, producingthen a symmetric repulsion. So it seems that theonly mechanism capable of producing an align-ment e�ect at low energies would be the one pro-vided by coherent multiple scattering. Calculationsand simulations for low energy diclusters incor-porating this e�ect will be quite useful to furtherelucidate the e�ective mechanisms leading to neg-ative vicinage e�ects and to H�2 recombination.

In addition, in the case of transmitted H�2molecules, it was noted that the e�ects of foilroughness may lead to a phenomenon of prefer-ential transmission of particles traversing thinnerparts of the foil, and therefore su�ering lower en-ergy losses, which may also contribute to the ob-served e�ect of reduced energy losses [6,92]. Sincethe yield of transmitted molecules has a criticaldependence on the thickness of the foil and on theincident velocity [18,74], in order to estimate thecontribution of this e�ect it would be necessary tohave experimental information on the behavior ofthe yield of transmitted molecules for the range oflow energies of interest here.

In summary, several mechanisms may contrib-ute to diminish the stopping power ratios in thelow-energy range. The largest e�ects appear to bethose produced by the alignment of the internu-

clear axis. However, the process leading to suchalignment at low energies has not been yet clearlyidenti®ed.

But there is still a further question to be ana-lyzed, which is the role of non-linear e�ects arisingfrom quantum scattering theory, which are knownto be very important for slow ions. The magnitudeof these corrections will be considered next.

3.5. Non-linear quantum e�ects

It is well-known, from extensive studies of theenergy loss of slow atomic ions in solids, thattheoretical results based on perturbative or di-electric models fail for velocities below the stop-ping power maximum, even for the lightestprojectiles [93±97]. The most evolved theoreticalmodels in the low-energy range are those based onquantum scattering theory and transport cross-section calculations, where the screening of theintruder ion and the scattering of electrons are si-multaneously described in a self-consistent way[96±99]. The results of these treatments show fullynon-linear features, like a complete departure fromthe Z2 scaling of the stopping power (characteristicof all perturbative models), showing instead anoscillatory behavior which agrees very well [99]with the experimentally observed Z1-oscillations[100±103].

The application of similar methods to the caseof molecular ions poses however a more di�culttask, due essentially to the breakdown of thespherical symmetry that makes the problem easyto treat only in the case of single slow ions.

In a previous approach to this problem [104],the total scattering amplitude of electrons by thedicluster was expressed as a linear combination ofthe non-linear amplitudes (LCNLA), fi�h�; pro-duced by each scattering center, and the di�eren-tial cross-section was calculated using aperturbative approximation for the interferencebetween two scattering centers [105] given by

drdX

� �LCNLA

� f1�h�j j2 � f2�h�j j2 � f1�h�f �2 �h��

� f �1 �h�f2�h�� sin�qr12�

qr12

; �34�


with q � 2vF sin�h=2�: With this approximation,the stopping power of the composite may be cal-culated from

S � nvvF

ZdX 1� ÿ cos�h�� dr

dX

� �LCNLA

: �35�

Calculations with this model are straightfor-ward using the phase shift values and scatteringamplitudes for single protons derived from densi-ty-functional theory (DFT) for the jellium model[98]. These calculations are shown in Fig. 6 (with asolid line) for proton diclusters in an electron gaswith rs � 2. The results are compared with calcu-lations using dielectric theory (dashed line).

Here two important features are noted: (i) theinterference e�ect becomes negative (i.e., R < 1)for internuclear distances between 2 and 4 a.u.,

showing in general a behavior quite similar to thatobtained from linear dielectric theory, (ii) the valuein the united-atom limit (r12 � 0) becomes R � 2.

The ®rst feature seems to con®rm the valuescalculated with dielectric theory (not for the ab-solute stopping values but for its ratio). However,it could be argued that this behavior derives simplyfrom the special form of amplitude interference,Eq. (34), borrowed from linear theory.

The second property re¯ects a basic failure ofthis scheme: the value R � 2 in the united-atomlimit is a consequence of the linear superpositionof amplitudes, i.e., a typical perturbation charac-teristic (in fact, Eq. (34) pertains to the frameworkof the ®rst-order Born approximation [105]).

Fortunately, the right value in the united-atomlimit can be calculated in an exact way from thetransport cross-section for helium ions, using alsothe density functional theory [98]. Thus, the exactDFT value for the stopping ratio in the united-atom limit is found to be R � 0:79; which is shownwith an open-square symbol in Fig. 6.

To complete this comparison, Fig. 6 includessome relevant experimental data. The solid-dia-mond symbol in the ®gure, Rexp � 1:02 (at r12 � 0),is the value of the stopping ratio in the real united-atom limit, obtained from the measured stoppingpower values for low-energy helium and hydrogenin aluminum [106] (corresponding to rs � 2:07). Inaddition, Fig. 6 shows also the experimental valuesof the stopping ratio for dissociation fragments(Rexp � 1:01) and transmitted molecules (Rexp �0:86) in thin aluminum foils, from [4] (the esti-mated internuclear distances for these experimentsare in the range �2±2.8 a.u.).

These results clearly show the failure of thementioned approximations in the range of smallinternuclear distances, and the need for more ac-curate calculations in the relevant range 0 <r12 < 3 a.u. The results also illustrate the impor-tance of non-linear e�ects in the low energy range,even for the lightest ions.

So far the only fully consistent non-linear cal-culation of dicluster stopping powers, taking intoaccount the asymmetry of the scattering potential,was made by Urbassek et al. [107], by solving thescattering problem in ellipsoidal coordinates andcalculating the transport cross-section for di�erent

Fig. 6. Non-linear e�ects in the stopping power ratio for hy-

drogen diclusters at low velocities in aluminum (corresponding

to rs � 2:07�. The dashed line shows the prediction from the

dielectric theory using Lindhard's dielectric function for rs � 2:

The solid line is the result obtained from a linear combination

of non-linear amplitudes (LCNLA), Eq. (34), using the scat-

tering amplitudes for individual protons calculated from the

density-functional theory. The open diamond at r12 � 0 shows

the exact non-linear result from density-functional theory, in

the limit of united protons (i.e., the stopping power ratio be-

tween a helium nucleus and two protons). The solid square at

r12 � 0 shows the experimental value from [106] for the united

atom limit (i.e., the measured stopping power ratio between

helium and two hydrogen ions at low velocities in aluminum).

The data points at r12 � 2:4� 0:4 show the experimental values

of the stopping ratios for dissociation fragments (R � 1:01) and

for transmitted H�2 molecules (R � 0:86) in aluminum foils,

from [4].


orientations of the dicluster. The self-consistencycondition was the Friedel sum rule for the scat-tering phase shifts. The authors used this sum ruleas a criterion to adjust the value of the screeningconstant a; which is the only parameter in thecalculation. Their results are expressed in terms ofa vicinage function g, whose relation with thestopping ratio R used here is simply: R � 1� g:

This calculation has the correct united-atomlimit. In particular, the results shown in Fig. 6 ofthis reference yield, for r12 � 0; the values:R � 1:05 for rs � 2; and R � 0:5 for rs � 3 (thevalue of R depends critically on rs within thisrange). In addition, one can obtain other useful R-values for random ( �R) and parallel (Rk) orientationof the internuclear axis, using the relationRk � �rk=�r��R; and using the values of the corre-sponding transport cross-sections, �r and rk; andthe g-values given in [107]. In this way one obtainsthe following indicative values for an internucleardistance r12 � 2 (with rs �2): �R � 1:18; Rk � 0:85:

These values show again a diminished stoppingpower for parallel orientation of the internuclearaxis.

Unfortunately, the potential used in these cal-culations does not have the proper separated-atomlimit for large r12: In fact, the model potential al-ready fails for internuclear distances of a fewatomic units [107], so that even the values indi-cated before are inaccurate. To diminish theseuncertainties, similar calculations using more re-alistic screened potentials would be necessary.

In conclusion, the non-linear calculations yieldR-values lower than 1 for aligned diclusters for therange of internuclear distances in molecular ions.Further calculations would be necessary in orderto get a more de®nitive test of the non-linear vic-inage e�ects.

4. Charge state e�ects

4.1. Preliminary considerations

In the previous sections, the simplest cases ofproton diclusters produced by the incidence of H�2molecules have been discussed in detail. Earlierstudies of the energy loss of these diclusters have

given almost no attention to the role of captureand loss of electrons during transit through thesolid, although the importance of these e�ects wasalready pointed out by Laubert and Chen [108]from the analysis of coincidence experiments be-tween the emergent H0 and H� fragments. In ad-dition, the existence of bound states aroundprotons moving in solids was also concluded fromtheoretical many-body calculations [109±111].

Further calculations of stopping powers of hy-drogen beams in solids have taken into account theprocesses of electron capture and loss at low andintermediate velocity, using also the concept ofpartial stopping powers to obtain the average en-ergy loss [112,113]. There is however some limita-tion in modeling the whole process due to the lackof direct experimental information on the popu-lation of charge states when the ion is moving in-side the solid (since the observable charge states ofemerging ions may be a�ected by the interactionwith the exit surface).

The relevance of charge state e�ects may be il-lustrated for clusters containing a fraction ofneutral atoms as follows: the dressed interactionbetween two separate ions with charges qi; qj in thecluster is, to a ®rst approximation, proportional toqiqj: So that if any of these ions is neutral the in-teraction becomes zero or very small. On the otherhand, the self-induced stopping power for a neu-tral atom is not negligible (at low energies, thestopping power for a neutral hydrogen atom issimilar to that for a proton with the same velocity,whereas at high energies it is about half that value[114,115], as expected from the partition rule).Hence, the e�ect of a signi®cant neutral fraction inthe stopping power ratio R, Eq. (11), is expected tobe important. Similar (perhaps less critical) con-siderations may be applied to other charge statesof di�erent ions.

The in¯uence of charge states in the stopping ofheavier molecular ions was ®rst discussed bySteuer and Ritchie [90], who reformulated the ef-fective charge model developed by Brandt andKitagawa [116] to describe molecular ions, con-sidering in particular the case of aligned N�2 ions.They found a diminished stopping power at lowvelocities, with a minimum value of about 0.7for internuclear distances of 2 a.u. (close to the


equilibrium value for these molecules). Thesenegative interference e�ects were attributed mainlyto electron±hole pair excitations. Recently, calcu-lations at higher velocities based on this modelhave also been reported [117].

In a further study [118], Steuer and Ritchie in-troduced the idea that the charge state of closelyspaced ions would be further reduced, below thenormal equilibrium value, due to an increase ofthe electron capture probability produced by theproximity of the partner ion. A model to describethese modi®cations in charge state equilibriumconditions was formulated by an extension of theBohr±Lindhard model [119].

Signi®cant changes in the charge states of hy-drogen clusters emerging from carbon foils were®rst observed by Meggitt et al. [33] from the dis-sociation of incident H�2 molecules, and also byGaillard et al. [34] using H�2 and H�3 molecules;later on, similar results were obtained using H�Nclusters with N values up to 23, and with energiesbetween 30 and 120 keV/amu [35,36]. In this rangeof energies, the electron capture process involvesmostly carbon valence electrons [120], so that thee�ects discussed by Steuer and Ritchie may beapplicable. Similar results have been obtained forN�2 ions [121,122] and C�5 clusters [123] in the MeVrange of energies. In all cases a shift toward lowercharge states has been observed, denoting an en-hancement in the beam neutralization.

These modi®cations in the equilibrium chargestates of cluster ions due to the proximity of otherions will be called in the following the ``vicinagee�ect in charge states''.

Recent experiments for transmitted H�2 and H�3molecules by Susuki et al. [124±126], and calcula-tions by Kaneko [71], reveal the critical impor-tance of bound electrons in reducing the stoppingpower and corresponding e�ective charges. Let usconsider brie¯y these cases. The stopping ratios Rmeasured in these experiments are shown in Fig. 7.Part (a) corresponds to measurements for trans-mitted H�2 molecules, while part (b) corresponds totransmitted H�3 molecules. We show for compari-son the calculations using Eq. (1), with the ap-propriate form factors [125,126] to describe boundmolecular ions (solid curves denoted by H�2 andH�3 , respectively) as well as similar calculations for

diprotons and triprotons resulting from brokenmolecules (curves denoted 2p and 3p).

The experiments show the di�erent regimes ofmolecular behavior: measurements at 10 MeV/amu and small dwell times (0.2±1.1 fs) [124,126]give e�ective charges of 1.19 and 1.45 for H�2 andH�3 , respectively, corresponding to diminishedstopping ratios R ' 0:7 in both cases, and in goodagreement with theoretical calculations for the

Fig. 7. Stopping power ratio R for transmitted H�2 and H�3molecules in carbon foils according to [124±126]. Part (a) cor-

responds to measurements for H�2 molecules, while part (b)

corresponds to H�3 molecules. Each data point is an average of

energy loss measurements performed with various foils. The

curves show the theoretical values calculated from Eq. (1) with

the appropriate form factors to describe two possible situations:

bound molecular ions (solid curves denoted H�2 and H�3 , re-

spectively), and dissociated clusters (diprotons or triprotons)

resulting from broken molecules (dashed curves denoted 2p and

3p, respectively). Part (a) illustrates two di�erent mechanisms

for H�2 : direct transmission at high energies (small dwell times)

and recombination at low energies (large dwell times). Part (b)

shows only the direct transmission mechanism in the case of H�3molecules.


transmission of unbroken H�2 and H�3 ions (directtransmission regime). Similar measurements forH�2 at lower energies (0.5 MeV/amu) and largerdwell times (1.3±6 fs) [125] yield a stopping ratioR � 1:15 for the emergent H�2 ion, which agreeswell with the theoretical estimation for two cor-related protons (recombination regime). Theseresults for H�3 molecules are also in agreementwith previous experiments where no recombina-tion regime was observed for this molecule[72,73].

The results of Fig. 7 pose also a puzzlingquestion: contrary to the case of dissociationfragments (cf. Fig. 3), the two experimental valuesof stopping ratio R for transmitted H�2 moleculesin Fig. 7(a) show a decrease of R with energy,which was explained as an evidence of the transi-tion between the two mentioned channels of mo-lecular transmission. Hence, it should be expectedthat further measurements for intermediate ener-gies (i.e., between 1 and 4 MeV) will show theexistence of both channels, presumably in the formof a two-peak structure (taking into account thatthe straggling is comparatively small in these ex-periments). The observation of a double peakstructure at intermediate energies would be a clearcon®rmation of the transition between both re-gimes of molecular transmission (previous evi-dences of this transition arise from the change inthe slope of the molecular yield as a function ofdwell time [74]).

The importance of collective or vicinage e�ectsin the equilibrium charge state of large ion clusterswas also noted by Sigmund et al. [26,27], indicat-ing the possibility of increased binding energiesand capture cross-sections due to the proximity ofother moving ions [27]. They proposed a qualita-tive estimation of the cluster ionization by apply-ing the Thomas±Fermi scaling properties and aBetz-type formula [127] to estimate the averagecharge of the cluster under the assumption that allparticles may be supposed to be concentrated in apoint. In this united-atom limit, the expectedcharge of the whole cluster Zeff would be

Zeff � Zclqeff � Zcl 1

"ÿ exp

ÿ v

Z2=3cl

!#; �36�

with Zcl � NZ1 for a homonuclear cluster consist-ing of N ions of nuclear charge Z1; here qeff is anormalized e�ective charge (0 < qeff < 1). Thisrelation was used to estimate, for di�erent types ofion clusters and energies, when an enhancement orde-enhancement of the stopping power might beexpected [26].

Fig. 8 shows a set of experimental data ofemerging proton fractions for hydrogen clustersfrom [35,36], which were produced by the inci-dence of H�N molecules (with N � 1±21) in carbonfoils with thicknesses of 2.1 lg/cm2. The ®gureshows the fraction of emerging protons,~/1 � 1ÿ ~/0 (the fraction of Hÿ at these energies isgenerally very small [36] and will not be consid-ered), and compares it with the prediction ofEq. (36) (solid lines). The dotted lines indicate theexperimental values for incident protons. The ef-fect mentioned above of an increased fraction ofneutrals (or decreased fraction of protons), ascompared with the case of incident proton beams,may be clearly observed. The comparison with Eq.(36) shows a moderate agreement for some inter-mediate values of N : The estimation fails for sim-ple protons or diprotons (except, perhapsaccidentally, for the E � 120 keV case). In addi-tion, the data show some indications of saturationof the charge state fractions with increasing N :

These experiments give the populations ofcharge states of the emerging ions, which may bedi�erent from the actual values inside the solid dueto the e�ects that may take place at the exit sur-face. However, to a ®rst approximation thesevalues may be considered as representatives of thesituation inside the solid [91], except possibly forvery low energies.

An additional complication in the analysis ofthe experiments is the question of charge equilib-rium, which has to be reformulated. In fact, thereis here an additional target-thickness dependenceof the charge state fractions due to the fact that thedistances among the various particles increase withdwell time (and so the vicinage e�ect in chargeequilibrium decreases). Therefore, one should ex-pect that in the limit of long dwell times theequilibrium fractions will coincide with those ofsimple proton beams. This limit has not been ob-served in these experiments with hydrogen clusters,


since they have been carried out using very thinfoils; however, within these limitations, a decreaseof the enhancement of neutrals using slightlythicker foils has been observed [35].

Very recent experiments by Brunelle et al. [128]using MeV clusters of carbon, C�N ; with N valuesbetween 3 and 10, in carbon foils, show also sig-ni®cant reductions in the charge states of theemerging C ions as compared with single carbonprojectiles at the same velocity. The averagecharge per atom decreases continuously with thenumber of constituents. In addition, the vicinagee�ect in the charge state tends to disappear whenthe foil thickness increases.

Ben-Hamu et al. [129] have applied the united-atom limit to estimate the stopping powers for

various ion clusters, showing a qualitative agree-ment with experiments for several cases. Thecomparison, however, is in all cases restricted toindicate the extrapolated value for zero internu-clear distance, since due to the nature of the unitedatom limit no calculations for ®nite internucleardistances can be made. The authors also providephysical arguments to justify the di�erences in thecharge state populations for cluster ions as com-pared to individual ions. This is explained by thedi�erent dependences of the electron capture (rc)and loss (rl) cross-sections on the projectile atomicnumber Zp according to Bohr's analytical expres-sions [130], namely: rc / Z5

p, rl / Zÿ1p , so that

rc=rl / Z6p. Therefore, in this picture the lower

average charge state for cluster ions may also be

Fig. 8. Set of experimental data of emerging proton fractions for hydrogen clusters, produced by the incidence of H�N molecules (with

N �1±21) in carbon foils, with energies of 30, 40, 60 and 120 keV/amu. The data points are the values of 1ÿ ~/0; where ~/0 is the

fraction of neutrals emerging from the foil, according to [35,36]. The solid lines show the e�ective charge estimations, by applying the

united-atom limit to the whole cluster, from Eq. (36). The dashed lines show in each case the emerging proton fraction corresponding

to incident protons. The di�erence between the dashed lines and the data points correspond to the enhancement in the yield of neutrals

(vicinage e�ect in charge states) for clusters of closely spaced ions.


explained by the strong increase of the electroncapture cross-section.

All these experimental evidences and theoreticalarguments point to the necessity of reformulatingthe cluster stopping power model, in order tomake possible the description of di�erent charge-state equilibrium conditions, including the addi-tional ``vicinage e�ects'' in the population ofbound states for closely spaced ions.

An extension of the early theory, in a way thatmay be useful to represent this situation, is pre-sented here.

4.2. Extended formulation of cluster stopping power

To start with a simple system, it is enough toconsider a hydrogen dicluster with two chargestate fractions (H0 and H�) and random orienta-tion of the internuclear axis. The extension of theformulation to other ion clusters and the inclusionof multiple charge states will be rather straight-forward.

In the present approach, the ions will be as-sumed to reach charge equilibrium in times muchshorter than the dwell time. In the case of very thinfoils, the e�ects of the initial approach to equilib-rium (starting from the incident molecular state)may give place to observable di�erences whichwould require special consideration. In fact, asimilar problem is found when dealing with singleatomic ions, so that the net e�ect of the initialtransition to equilibrium on the stopping powerratios will be attenuated.

The equilibrium fractions of H0 and H� will bedenoted by ~/0 and ~/1; where the tildes are used tonote that these fractions, for correlated ions, maybe di�erent from the corresponding values /0;/1,for atomic-ion beams (i.e., the previously men-tioned vicinage e�ect in the charge states).

For each possible combination of frozen chargestates, a combined stopping power may be de®nedas follows:

S�00� � 2S�0�H � I �00�12 � I �00�

21 ; �37�

S�01� � S�0�H � S�1�H � I �01�12 � I �10�

21 ; �38�

S�11� � 2S�1�H � I �11�12 � I �11�

21 ; �39�which correspond to the cases of having twoneutrals, one neutral and one proton, and twoprotons, respectively. The values of S�0�H and S�1�H

are the stopping powers for neutral hydrogen andfor protons in frozen charge states (partial stop-ping powers), which may be calculated by Eq. (8)with the corresponding form factors.

The notation of the interference terms indicatesthat one has now to consider both the ion pair(lower indexes) as well as the possible charge states(upper indexes). Hence, instead of Eq. (9), oneshould consider the following set of interferenceterms:

I �ab�ij � 1

2p2v

Zd3qf �a�i �q�f �b��j �q�

� ~q �~vq2

Imÿ1

��q;~q �~v�

" #exp�i~q �~rij�; �40�

where f �a�i �q� denotes the form factor of an ion i ina given charge state a:

Then, these ``frozen'' stopping terms may becomposed, using the corresponding probabilityfactors (of having two neutrals, one neutral andone proton, or two protons), to obtain the statis-tical average of the stopping power for thedicluster, as follows:

hSicl � ~/20S�00� � 2 ~/0

~/1S�01� � ~/21S�11�: �41�

Substituting Eqs. (37)±(39) in Eq. (41) and usingthe condition ~/0 � ~/1 � 1, one gets the clusterstopping in an equivalent form:

hSicl � 2h~SiH � 2hIi; �42�where

h~SiH � ~/0S�0�H � ~/1S�1�H : �43�The ®rst term 2h~SiH in Eq. (42) may be interpretedas the statistical average of the stopping power fortwo equivalent ions, having each the same averagestopping ~/0S�0�H � ~/1S�1�H , but with modi®ed charge

state fractions ~/0;~/1 (because of the in¯uence of

the other ion). The average interference term hIi inEq. (42) is given by


2hIi � ~/20 I �00�

12

h� I �00�

21

i� 2 ~/0

~/1 I �01�12

h� I �10�

21

i� ~/2

1 I �11�12

h� I �11�

21

i: �44�

The reciprocity relation (10) applies also to theseinterference terms, so that the reciprocal terms inEq. (44) are in fact equal.

On the other hand, the stopping power corre-sponding to a hydrogen beam with the samevelocity is

hSiH � /0S�0�H � /1S�1�H �45�(note that the values of /0 and /1 for isolated ions,i.e., the normal values, are used here).

Therefore, the stopping ratio,

R � hSicl

2hSiH� h

~SiH � hIihSiH

�46�

contains now two terms. A ®rst term, h~SiH=hSiH �� ~/0S�0�H � ~/1S�1�H �=�/0S�0�H � /1S�1�H �, which may beexpected to be slightly less than one, due to thementioned vicinage e�ect in the charge state pop-ulations. And a second term, hIi=hSiH; whichcould be larger or smaller than 1, containing thevicinage e�ects in electron excitations due to theinterferences between di�erent ions.

In the approximation ~/0 ' /0, ~/1 ' /1, oneretrieves a previous and widely used form

R � 1� hIihSiH: �47�

The previous description can now be easily gen-eralized to larger clusters of N ions of type A andwith multiple charge states. Using ~/a to denote thecharge state fractions for each charge state a, thecluster stopping power becomes

hSicl � Nh~SiA � NhIi; �48�where

h~SiA �X

a

~/aS�a�A �49�

is the modi®ed atomic stopping power, in terms ofthe partial stopping powers S�a�A ; and hIi is an av-erage interference term per particle, given by

NhIi �Xa;b

~/a~/b

Xi<j

I �ab�ij

h� I �ba�

ji

i� 2

Xa;b

~/a~/b

Xi<j

I �ab�ij ; �50�

where the reciprocity relation, I �ab�ij � I �ba�

ji ; hasbeen used to reduce the number of terms.

This result corresponds to homonuclear clus-ters, but it may be generalized to heteronuclearclusters with only a slight complication in the no-tation.

Let us consider a cluster consisting of twoatomic species, with NA total ions of type A and NB

total ions of type B, and let ~/a and ~ub be thecorresponding charge state fractions. The clusterstopping power for this case may be written asfollows:

hSicl � NAh~SiA � NBh~SiB � �NA � NB�hIiAB

� NA

Xa

~/aS�a�A � NB

Xb

~ubS�b�B

� �NA � NB�hIiAB; �51�where S�a�A (S�b�B � is the partial stopping power of anion of type A (B) in charge state a (b); and h~SiA;Bare the modi®ed stopping powers for each type ofion. The average interference term hIiAB is nowgiven by

�NA � NB�hIiAB �2Xa;a0

~/a~/a0XNA

i<j

I �aa0�ij

� 2Xb;b0

~ub ~ub0XNB

i<j

I �bb0�ij

� 2Xa;b

~/a ~ub

XNA

i�1

XNB

j�1

I �ab�:ij �52�

(note that the case i � j is included in the last sum,since in this case Iij refers to interactions betweenions of di�erent sets).

This value of hSicl, Eq. (51), should be com-pared with the stopping power for NA independentions of type A and NB ions of type B (with normalstate fractions /a;ub) namely,

hSiion-beams � NAhSiA � NBhSiB� NA

Xa

/aS�a�A � NB

Xb

ubS�b�B : �53�


Applications of some of these results to H�Nclusters and to very large clusters will be consid-ered below.

5. Clusters of intermediate size: N-clusters

In this section, calculations of stopping powersfor clusters of intermediate sizes, and the expectedmodi®cations due to charge state equilibrium inthe cluster, will be considered.

To begin with the simplest case, consider againthe case of hydrogen diclusters. The e�ect of theneutral fraction for these diclusters is shown in theprevious Fig. 5 with dashed lines. In these calcu-lations the experimental values of the charge statefractions of hydrogen in carbon [35,36,91] havebeen used. The e�ect shown in Fig. 5 is a decreaseof about 5% in the case of random cluster orien-tation, and a similar increase for aligned diclustersat low energies. The fact that the correction is onlyof a few percent at low energies is an indicationthat a strongly screened proton in a solid does notdi�er too much from a neutral hydrogen atom. Athigh energies this di�erence is more important, butsince the fraction of neutrals drops with energy theglobal e�ect still remains low.

In order to evaluate the stopping power for agiven N-cluster, the ®rst question to be consideredis the available information on cluster structureand population of charge states. For hydrogenclusters of intermediate sizes, H�N (with N � 3±21),there is enough information on the distribution ofcharge states provided by previous experiments[35,36] (possible changes in the charge state frac-tions at the exit surface will be neglected in thisanalysis).

With regard to cluster structures, it is knownthat for N > 2 mostly odd values of N are exper-imentally observed. The explanation for this is thathydrogen clusters grow by accretion of H2 mole-cules around a central core formed by a H�3 mol-ecule. Information on the structures of theseclusters for N values from 5 to 13 may be obtainedfrom previous studies [131,132]. The present for-mulation has been applied to several of these H�Nclusters using the information on cluster structuresfrom these references, and considering the cases of

(a) full ionization (proton clusters), and (b) chargeequilibrium conditions, according to the experi-mental charge state distributions shown in Fig. 8.The results of these calculations are shown inFig. 9, and they are compared with experimentalresults of the stopping ratios obtained by Farizonet al. [133], for cluster energies of 60 keV/amu.

The open circles correspond to calculationsassuming full ionization of the hydrogen atoms,whereas the open squares are similar calculationsfor conditions of charge equilibrium. The point ofinterest here is the important decrease in thestopping ratios R due to charge neutralization.The enhancement e�ect drops with increasingnumber of particles so that the e�ect of partialneutralization in this energy range becomes in-creasingly important for larger clusters, and it mayeven lead to de-enhancement e�ects (i.e., valuesR < 1).

The comparison with the experiments is quali-tatively good. It should be noted that the targethas been represented here by a simple Lindharddielectric function and that the dynamics of thecluster particles moving inside the solid has notbeen considered. For more quantitative estima-tions one should consider computer simula-tions using molecular dynamics or alternative

Fig. 9. Stopping power ratios for hydrogen clusters in a

2 lg=cm2 ± carbon foil, at energies of 60 keV/amu. The solid

points are the experimental values from [133]. The open circles

are the values calculated for each cluster using the information

on the molecular structure from [131,132], and assuming con-

ditions of full ionization (proton clusters). The open squares are

similar calculations but including the equilibrium fractions of

neutral hydrogen atoms, as discussed in the text.


techniques, and the use of more accurate dielectricfunctions.

6. Large clusters

6.1. General formulation

The application of the discrete formulation sofar used to very large clusters becomes unpracticalas the number of particles increases. Moreover,due to straggling and multiple scattering e�ects therelative positions of the ions in the cluster spreadand tend to randomize as the cluster penetrates ina solid. In these conditions, a full statistical for-mulation of cluster structure and energy loss maybecome more useful.

The statistical formulation of cluster stoppingpower was developed in previous papers, both forsolid [134±139] and plasma targets [140,141], onthe assumption of either a single charge state or ane�ective-charge representation for all ions. Anextension of the formulation is introduced here inorder to provide a more general representation interms of charge states, including the correspondingform factors for each ion, and the vicinage e�ectsin the equilibrium charge-state fractions. As be-fore, and for the sake of clarity, the case of hy-drogen clusters will be considered ®rst, includingonly the two most relevant charge states (H0 andH�). The generalization to other cases will bestraightforward.

Following the formulation of [136,138,139,141],the stopping power for a cluster of N hydrogenions may be written as

h�Sicl � Nh~SiH � Nh�Iicl; �54�where h~SiH is the average self-stopping term forany of the identical ions in the cluster (using themodi®ed charge state fractions),

h~SiH � ~/0S�0�H � ~/1S�1�H : �55�The cluster-averaged interference term is now ex-pressed in terms of the pair correlation function forthe ions in the cluster gcl�r�; as follows [134±136]:

h�Iicl � nZ

d3r gcl�r�hI�r�i; �56�

where n is the density of ions in the cluster and

hI�r�i � 2

pv2

Z 1

0

dqq

sin�qr�qr

~/20 f0�q�j j2

h� 2 ~/0

~/1�f0�q�f �1 �q�� ~/21 f1�q�j j2

i�Z qv

0

dx x Imÿ1

��q;x��

: �57�

This expression for hI�r�i may be derived usingprobability factors and considerations similar tothose leading before to Eq. (41).

The values h�Sicl and h�Iicl contain now a doublestatistical average: over charge state distributions(denoted by hi) and over the distribution of in-teratomic distances within the cluster (denoted bythe upper bars in �S and �I). The pair correlationfunction gcl�r� in Eq. (56) carries full statisticalinformation on the distribution of close and dis-tant neighbors around a given reference ion. Inparticular, for hydrogen clusters containing H2

molecules, the pair correlation function may beseparated in intramolecular and intermolecularterms: gcl�r� � gintra�r� � ginter�r� [134±136].hI�r�i is the dicluster interference function av-

eraged over angular orientations and over chargestate populations. The functions f0�q� and f1�q�are the form factors for neutral and ionizedhydrogen (f1�q� � 1 for bare protons) de®ned byEq. (6).

Inserting Eq. (57) in Eq. (56) and changing theorder of integration, one gets

h�Iicl �2

pv2

Z 1

0

dqq

Gcl�q� ~/20 f0�q�j j2

h� 2 ~/0

~/1�f0�q�f �1 �q�� ~/21 f1�q�j j2

i�Z qv

0

dxx Imÿ1

��q;x��

; �58�

where

Gcl�q� � nZ

d3r gcl�r� sin�qr�qr

: �59�

Explicit analytical expressions for the cluster formfunction Gcl�q� have been given in [136,141], for aspherical random cluster model.


Since ~/0 and ~/1 are constant factors in the in-tegrals of Eq. (58), h�Iicl may be written as follows:

h�Iicl � ~/20�I �00� � 2 ~/0

~/1�I �01� � ~/2

1�I �11�; �60�

where the �I �ab� are given by

�I �ab� � 2

pv2

Z 1

0

dqq

Gcl�q� fa�q�f �b �q�h i

�Z qv

0

dxx Imÿ1

��q;x��

: �61�

These expressions provide the solution to the hy-drogen cluster problem.

Eq. (60) provides also the clue to a more gen-eral formula for homonuclear clusters of any typeof ions and including multiple charge states.

The obvious generalization to large homonu-clear clusters is the following:

h�Sicl � Nh~SiA � Nh�Iicl; �62�

where

h~SiA �X

a

~/aS�a�A ; �63�

h�Iicl �Xa;b

~/a~/b

�I �ab�: �64�

Note that the indexes here refer only to thecharge states, since the summations over pairs ofions i; j (used in the previous section) are statisti-cally accounted for in the integrated �I �ab� terms.

A last ®nal extension of these expressions pro-vides the generalization to heteronuclear clusterscontaining two types of ions, A and B. The stop-ping power in this case may be written as

h�Sicl�AB� � NAh~SiA � NBh~SiB � NA h�Ii�A�cl

h� h�Ii�AB�

cl

i�NB h�Ii�B�cl

h� h�Ii�BA�

cl

i;

�65�

where h�Ii�A�cl and h�Ii�B�cl are the average interferenceterms for the separate sub-clusters A and B, givenby

h�Ii�A�cl � nA

Zd3r g�A�cl �r�hI�r�iA

�Xa;a0

~/a~/a0

�I �aa0�A ; �66�

h�Ii�B�cl � nB

Zd3r g�B�cl �r�hI�r�iB

�Xb;b0

~ub ~ub0�I �bb0�B �67�

and the cross-interference terms h�Ii�AB�cl and h�Ii�BA�

cl

in Eq. (65) are given by

h�Ii�AB�cl � nB

Zd3r g�AB�

cl �r�hI�r�iAB

�Xa;b

~/a ~ub�I �ab�AB ; �68�

h�Ii�BA�cl � nA

Zd3r g�BA�

cl �r�hI�r�iBA

�Xb;a

~ub~/a

�I �ba�BA : �69�

The terms �I �aa0�A , �I �bb0�

B , �I �ab�AB and �I �ba�

BA on the right-hand side of these expressions are given as inEq. (61), with the corresponding Gcl-functions

G�A�cl �q� � nA

Zd3r g�A�cl �r�

sin�qr�qr

; �70�

G�B�cl �q� � nB

Zd3r g�B�cl �r�

sin�qr�qr

; �71�

G�AB�cl �q� � nB

Zd3r g�AB�

cl �r�sin�qr�

qr; �72�

G�BA�cl �q� � nA

Zd3r g�BA�

cl �r�sin�qr�

qr: �73�

Note here that one should take into account, in thecalculation of the interference terms, three di�er-ent pair-correlation functions: g�A�cl �r� for ions oftype A, g�B�cl �r� for ions of type B, and g�AB�

cl �r� �g�BA�

cl �r� for crossed interactions between ions Aand B.


These functions should be normalized as fol-lows:

nA

Zd3r g�A�cl �r� � NA ÿ 1; �74�

nB

Zd3r g�B�cl �r� � NB ÿ 1; �75�

nB

Zd3r g�AB�

cl �r� � NB; �76�

nA

Zd3r g�BA�

cl �r� � NA: �77�

An example of interest would be the case of largeclusters of water molecules. It was considered in[134,135] using a simpli®ed description in terms ofe�ective charges.

6.2. Application to hydrogen clusters

Experiments using hydrogen clusters [66,133]give the possibility to test the vicinage e�ects pre-dicted by the energy loss models. The stopping ofhydrogen clusters in solid targets has been investi-gated theoretically using the dielectric formalism[133,136,138,139], as well as the harmonic oscillatormodel [66]. The results are qualitatively similar,depending particularly on the parameters used torepresent either the dielectric properties of the me-dium or the distribution of atomic oscillator fre-quencies in each case. Calculations based on thesemodels are in good agreement with the experiments[66,133] for clusters with up to N � 25 particles.

Previous calculations were done on the as-sumption that the particles in the cluster are fullyionized (proton clusters). In order to analyze theadditional e�ects due to charge state equilibrium itbecomes of interest to review some of these cal-culations.

Fig. 10 shows the integrated values of the in-terference terms as well as the normal stoppingpower of individual protons SP using the presentformalism. These calculations correspond to acluster of 100 H2 molecules (N � 200; cluster ra-dius rcl ' 20 a.u., with a hydrogen atom density of4:24� 1022 cmÿ3), under the assumption of full

ionization of the beam (proton clusters) in amor-phous carbon (with rs � 1:6 , xP � 0:853 a.u.).

The model used for large clusters contains theintramolecular term for the H2 molecule,gintra

cl �r� � Ad�r ÿ r12�; and approximates the in-termolecular distribution by a ``hole'' in the cor-relation function, ginter

cl �r�, for r < rcorr; with acorrelation (or exclusion) radius rcorr determinedfrom the nearest-neighbor distance in the actualhydrogen cluster [136,137]. The applicability ofthis spherical random model was justi®ed in [136]by comparing calculations for an ordered structurerepresenting an arrangement of 13 H2 moleculeswith the hcp structure of solid hydrogen [142].

The contributions to the interference termcoming from intramolecular (Iintra) and intermo-lecular (Iinter) interactions have been separated.They bear di�erent characteristics: Iintra shows abehavior associated to the proton stopping powerSP; with a maximum at v ' 2 a.u., whereas Iinter

shows a wide negative trough for v � 2±6; fol-lowed by a very wide maximum at high velocities.The reason for this behavior is the following: theinterference between close ions (Iintra) is positive, inagreement with the enhancement due to vicinagee�ects for unexploded diclusters in this energyrange (cf. Fig. 1), but it turns negative for largerinternuclear distances (also as in Fig. 1). With in-creasing velocities the interference e�ect extends to

Fig. 10. Proton stopping power, SP � S�1�H ; and interference

terms due to intramolecular (Iintra) and intermolecular (Iinter)

interactions, calculated from Eqs. (55)±(61). The calculations

correspond to a cluster of 100 H2 molecules (N � 200), in an

electron gas characterized by rs � 1:6; under the assumption of

full ionization of the beam (proton clusters), ~/1 � 1:


larger distances incorporating a larger numbers ofions and producing the wide negative trough.However, for large enough velocities the positivephase of the interference ± corresponding to theso-called case (b) in Section 3, Eq. ( 21) ± starts todominate. This occurs when the condition forpositive collective interferences, v � rxP, is satis-®ed for most of the particles in the cluster. Takinga range of values r � rcl=2ÿ rcl, one gets v � rxP

� 8ÿ 16; in agreement with the wide-maximumbehavior of Iinter shown in Fig. 10. The ®nal declineof Iinter for very large velocities is produced by theusual Bohr factor �xP=v�2 in Eqs. (19)±(21).

The contribution from carbon K-shell electrons(not included here) is an important correction athigh velocities. This will increase the value of theindividual stopping term by about 20% for a ve-locity of 10 a.u., but will not change in a signi®cantway the values of the interference terms Iintra andIinter. Hence, the values of the stopping ratio willdecrease in about the same proportion.

Let us now consider the e�ect of neutral atomsin the cluster. Due to the lack of information oncharge state fractions for these large clusters, aqualitative description of the expected behaviorwill be considered here, assuming a set of ®xedfractions of neutrals in the cluster.

Fig. 11 shows the values of the stopping powerratios as a function of the cluster velocity. Thesolid line denoted as a is the calculation for fullionization (proton cluster), while the curves b, c, d,e, f, correspond to neutral contents ranging from10% to 50%. The inset shows the low-velocity be-havior. It may be noted that for low velocities thestopping ratio R is only slightly larger than 1, andit drops below 1 for v � 1±2 a.u. when the neutralfraction is increased. The interference e�ects re-main negative (R < 1) for v � 5 in all cases (i.e.,independently of the neutral fraction).

On the other hand, there is a signi®cant increaseof the interference e�ect at large velocities, which isdue to the fact that the interferences in this rangeof velocities are dominated by long-range collec-tive excitations (plasmons) produced by chargedparticles. The decrease of R with increasing neutralfractions (curves b±f) is a consequence of theshort-range interactions of neutral atoms (ex-pressed through the corresponding form factors)

giving place to a signi®cant reduction of plasmonexcitations.

Hence, a simpli®ed but essentially correct de-scription of the high-energy behavior may be ob-tained by referring to the properties of the pairinterference function depicted in Fig. 1, and con-sidering only long-range interactions between pairsof protons (i.e., neglecting the contribution ofneutral particles).

6.3. Resonant e�ect

An interesting prediction of the cluster stoppingtheory is the existence of a maximum in the stop-ping power ratio for a given radius of the cluster,or in practice, for a given number of particlescomposing the cluster. This e�ect was found forclusters interacting both with solids [138,139] andwith plasmas [140,141].

In order to explain the origin of the resonante�ect, let us consider ®rst the case of a large clusterof protons with radius rcl. Then, the interferenceterm h�Iicl, Eq. (56), may be written in the simpleform

h�Iicl � 4pnZ

dr r2gcl�r�I0�r; v�; �78�

where I0�r; v� is given by Eq. (17).

Fig. 11. Values of the stopping power ratios as a function of

cluster velocity for a cluster of 100 H2 molecules (N � 200), in

an electron gas characterized by rs � 1:6. The solid line denoted

a is the calculation for full ionization (proton cluster), while the

curves b, c, d, e and f correspond to neutral fractions of 10%,

20%, 30%, 40% and 50 % respectively. The inset shows the low-

velocity behavior.


The shape of the function r2gintercl �r�I0�r; v� in the

integral of Eq. (78) is shown in Fig. 12. To illus-trate the behavior, two cases are shown, for clusterradii rcl � 2v=xP and rcl � 10v=xP; and both for avelocity v � 10 a.u.. The results have been nor-malized and parameterized in terms of the reducedvariable r=rcl to allow a direct comparison (forsimplicity the correlation hole was not depicted inthis ®gure). The modulation of the interferences isproduced by the phase of the collective resonances(plasmons) excited in a coherent way by the wholecluster. This resonant behavior is produced by thepole in the response function Im ÿ1=��q;x�� when the interference term I0�r; v� is integrated. Asthe ®gure clearly shows, one can obtain the max-imum coherent behavior when the three parame-ters rcl; v and xP are tuned, such that the conditionrcl ' 2v=xP is satis®ed (more precisely, the maxi-mum is located at rcl � 1:8v=xP; see below). In thiscase, most of the particles in the cluster interfere ina constructive way with the ®eld induced by thewhole cluster, and then the e�ect is maximized.

The resonant e�ect is illustrated by consideringthe stopping ratio, which in this case may bewritten as

R � h�Sicl

hSiH� h

~SiH � hIiH2� h�Iiinter

SP

; �79�

and studying the variation of R a function of thecluster size, for a ®xed value of velocity, in thiscase v � 10. A constant cluster density is assumed,so that the number of particles increases with thesize of the cluster (the result would of course bedi�erent if the cluster were to be expanded keepingthe number of particles constant).

Note that in the denominator of Eq. (79) thevalue of hSiH has been replaced by simply theproton stopping power SP; since at the velocityconsidered the fraction of neutrals for a beam ofhydrogen ions is negligible. The fraction of neu-trals in the cluster will be here a parameter, in-cluded in the calculation of h~SiH; hIiH2

and h�Iiinter.Fig. 13 shows the dependence of this stopping

ratio on the cluster radius rcl; or equivalently, onthe total number of particles in the cluster Ncl: Forthe chosen parameters one ®nds a prominentmaximum at rcl ' 21 a.u., or Ncl ' 250. Thismaximum corresponds approximately to the con-dition rcl ' 2v=xP illustrated in Fig. 12. This isfollowed by a minimum at rcl ' 47 a.u., and asecond maximum at rcl ' 60 a.u. This behavior isfully in agreement with the more general descrip-tion of the resonant e�ect based on scaling func-tions developed in [141]. Similar results have been

Fig. 12. Integration of the interference term h�Iicl for large

clusters, leading to the resonant e�ect. The curves show the

function r2gcl�r�I0�r; v� in the integrand of Eq. (78), showing one

case of important coherent behavior (rcl � 2v=xP ) and one case

of near-cancellation (rcl � 10v=xP). The calculations corre-

spond to v � 10 and two di�erent cluster radii rcl; but have been

re-scaled in terms of r=rcl (so that 06 r=rcl6 2�.

Fig. 13. Resonant e�ect for large hydrogen clusters. The ®gure

shows the stopping ratio R, from Eq. (79), as a function of: (a)

cluster radius rcl, and (b) number of particles in the cluster Ncl;

for a ®xed velocity v � 10 a.u. The curves a, b, c, d, e and f

correspond to increasing neutral fractions of 0%, 10%, 20%,

30%, 40% and 50 %.


obtained for large hydrogen clusters in C, Al, andSi [138,139], as well as in fusion plasmas [140,141].

Fig. 13 also shows the e�ect of di�erent frac-tions of neutrals in the hydrogen beam, corre-sponding to ~/0 � 0:1; 0:2; . . . ; 0:5: The e�ect of theneutral component is to decrease the R-values butthe characteristics of the resonance phenomenonremain. Note that the stopping ratio decreasesfaster than linearly with the neutral fraction. Inaddition, for large rcl the value of R does not go tounity, but lies slightly below. This is the e�ect ofresidual interferences for very large clusters de-scribed in [141] (they arise from short-range cor-relations between neighboring ions).

6.4. Experimental observations and computer sim-ulations

The resonant e�ect is a theoretical predictionthat requires accelerated clusters with high en-ergies, and it has not been experimentally testedyet. However, it may be noted that the reportedexperiments [66,133] with hydrogen clusters ofintermediate sizes (with N values up to 25) showa maximum or a ``saturation'' e�ect in the energyloss as a function of the cluster size. Althoughthese clusters are not large enough to justify thepresent description (and the energies are in anintermediate range not properly covered here)the experiments could be interpreted qualitativelyas an indication of the resonance condition,which, for velocities v � 1±2 a.u., yields a softmaximum for N � 3±10, in fair agreement withthese experiments. The stopping enhancement inthis case is relatively small, and depends criti-cally on the existence of neutral particles in thecluster.

Predictions of large enhancement e�ects havealso been made by Nardi et al. [45,46] from com-puter simulations of the interaction of C60 clusterswith both hydrogen plasmas [45] and solid-stateplasmas [46]. The expected enhancements aremuch larger in plasmas because of the higher val-ues of v=xP; due to the usually lower plasma fre-quencies. In the case of solids, the calculations fora C60 cluster with energy of 200 keV/amu showinitially a large enhancement (R � 5� in the ®rst100 �A of penetration, followed by a decrease to

values R � 1 (or even below) after 600 �A, which isdue to cluster disintegration and equilibriumcharge state e�ects. For thicker foils the overallintegrated e�ect becomes small.

On the other hand, the only reported energyloss experiment with C60 clusters in solids wasmade at energies below 40 keV/amu, and no de-tectable di�erences (within �5%) in the stoppingpowers were observed [31]. In this energy range thee�ects of charge equilibrium have a strong in¯u-ence on the outcome of the experiments.

In conclusion, the observation of large en-hancement e�ects for solid targets will requireenergies higher than those so far used, both toachieve larger cluster ionization fractions as well aslarger v=xP values (i.e., large coherence lengths).

6.5. Scaling laws

To end up the description of the enhancemente�ects for large clusters and high velocities, let uscomment brie¯y on some scaling properties of theinterference term, which allows to predict the po-sition and magnitude of the resonant e�ect. Theseproperties have been derived and analyzed in de-tail in [141] for the case of plasma targets. How-ever, in the high-velocity limit the results obtainedtherein using the plasmon-pole approximationapply also to the case of solid targets. The mainresults are summarized here.

The cluster radius for which the stopping ratioreaches a maximum value is more precisely given,for v� vF, by [141]

rcljmax � 1:8v

xP

; �80�

and the value of the interference ratio I=S at themaximum becomes

�Icl

S0

��max

� 4:3

L�v� ncl

vxP

� �3

; �81�

where L�v� is the stopping number for individualions, given, in the high velocity limit, by the Bethe±Lindhard form:

L�v� � ln2v2

xP

� �: �82�


These scaling laws apply to the case of clusters ofbare ions (e.g., proton clusters) and are in excellentagreement with the results in Fig. 13. The e�ect ofcharge states will lower the values of the stoppingratios, but should not a�ect very much the positionof the maximum, as shown in Fig. 13.

6.6. The Z3 e�ect

The large enhancement e�ects just describedbring the question of whether these e�ects may bedescribed adequately by the present formulation,without exceeding the limitations of a perturbativeapproach. In fact, the dielectric formalism for theenergy loss of a particle with charge Z gives astopping power proportional to Z2; which may beconsidered as the ®rst term of a perturbative ex-pansion in powers of Z, namely

S�v� � 4pne

v2Z2L0�v�� Z3L1�v� � � � �

�: �83�

The next term in this expansion, of order Z3; is theso-called Barkas correction.

The theory of the Z3 e�ect was ®rst formulatedby Ashley et al. [143±145], by extending Bohr'smodel of atomic oscillators. Further developmentswere made by various authors [147,148], includingexact calculations for the quantum harmonic os-cillator [146] and for the electron gas model [149].

According to these formulations, the ratio be-tween the Z3 and the Z2 terms in Eq. (83) is of theorder of

B � ZL1�v�L0�v� '

3p4

ZxP

v3: �84�

All these descriptions apply to the case of simpleions, so in order to estimate the magnitude of thenon-linear corrections expected in the case of alarge cluster, very simple assumptions will bemade. The previous case of a cluster with Zcl � 200particles and velocity v � 10 will be taken as areference.

If full ionization of the cluster is assumed, thenZ � 200, and one gets from Eq. (84) a maximumvalue B � 0:4: However, since Eq. (84) corre-sponds to a point nuclear charge, this value shouldbe considered only as a gross overestimation of the

magnitude of the actual correction for a distrib-uted charge.

If, on the other hand, the e�ective charge of thecluster is estimated using the statistical approxi-mation of Eq. (36), one obtains (after expandingthe exponential, where v=Z2=3

cl � 0:3):

Zeff ' Zcl 1

"ÿ exp

ÿ v

Z2=3cl

!#� Z1=3

cl v: �85�

This yields an e�ective charge Zeff � 60; and so theBarkas factor becomes B � 0:12:

These two estimations of B are based on theassumption of the united atom limit, thereforethey provide only crude upper bond limits. Hence,one may expect that the Z3-corrections will nota�ect signi®cantly the predictions of large stoppingvalues given before in the range of high velocities,although it seems that a quantitative considerationof non-linear corrections would be justi®ed.

Another way to interpret these results is thefollowing: by comparing the values of the inter-ference term Iinter with the proton stopping term SP

in Fig. 10, one sees that the maximum value of Iinter

is lower that the maximum of SP: So that, althoughIinter�v� is much larger than SP�v� at high velocities(v � 10), it remains lower than the usual maximumstopping power for single protons. Then, it may beexpected that the non-linear corrections would notbe larger than those applicable to single protonsnear the stopping maximum.

In conclusion, the most important questionwith regard to the stopping of large clusters seemsto be the problem of partial neutralization of thebeam, which may provide the largest correctionwith respect to calculations for full ionization ofthe beam. This e�ect may be taken into account ina complete way with the present formalism.

7. Concluding remarks

From the present study of experimental evi-dences and theoretical developments, several con-clusions and comments on remaining openquestions can be formulated.

There is a wealth of information on the energyloss of diatomic clusters and molecular ions at low,


intermediate, and high energies, but there is notyet a complete understanding of the most e�ectivephysical mechanisms in all cases.

In the case of dissociated fragments at interme-diate and high energies, the energy loss values arereasonably well accounted for by the theoreticalestimations. This is the energy range where per-turbative and dielectric models work best. It is alsothe range of important collective e�ects and wakephenomena. So, the use of small ion clusters withintermediate or large energies provides a powerfultool for studying collective and wake phenomena.

In the low energy range, there is clear experi-mental evidence of diminished stopping powerratios for diatomic molecules of various elements.This e�ect is particularly notorious in the case ofemerging (recombined) molecules. Among thevarious processes that may contribute to this e�ectwe summarize the following:· The e�ect of partial alignment is potentially the

largest one. It may account for the observed di-minished stopping ratios, but the mechanismleading to such alignment in this energy range(below the threshold for wake excitation) hasnot been identi®ed. The role of coherent or cor-related multiple scattering at low energiesshould be further studied.

· The e�ects of charge equilibrium, including asigni®cant fraction of neutralized atoms, mayalso explain a lower energy loss. An inconve-nience here is our poor knowledge of chargestates of ions inside the solid, so that in mostcases one has to resort to theoretical estimationsof charge state fractions.

· Non-linear quantum e�ects become very impor-tant for slow molecular ions, even for diatomichydrogen molecules or diclusters. There is cur-rently a need for fully non-linear transportcross-section calculations following the lines ofa self-consistent approach [107]. These calcula-tions may provide a test to the use of dielectricor perturbative models in the case of slow ionclusters.

· The e�ects of foil roughness may partially con-tribute to the observed e�ect; in order to estimatethis contribution, experimental information onthe molecular recombination yields at low ener-gies would be required.

On the other hand, coincidence experiments at lowand intermediate energies with detection of neutralhydrogen atoms, following the pioneering ideas ofLaubert [6,108], may provide a valuable tool toinvestigate the role of neutrals within the solid.These experiments may also shed some light on therelevant question of inside charge states. In thissense, ion clusters provide a unique system tostudy the interactions and charge states of ionsinside the solid, which cannot be obtained withatomic ions.

The question of charge states becomes crucialto the study of small and large clusters, althoughthe importance of this question clearly increaseswith the number and concentration of ions in thecluster. There is experimental evidence indicatingthat the conditions of charge equilibrium are sig-ni®cantly modi®ed in ion clusters, with respect tothe conditions for atomic-ion beams with the samevelocity. The presence of neighboring ions pro-duces an increase in the capture cross-section, andthus a larger fraction of neutrals or lower-charge-state ions. Further experimental information oncharge state populations for various types of ionclusters is necessary, both in order to improve ourknowledge of the less explored vicinage e�ects incharge equilibrium, as to be able to calculate therelated vicinage e�ects in the energy loss.

The coexistence of multiple charge states of theions within a cluster is a question that a�ects in alarge way the energy loss process. Here the theoryof cluster stopping power has been extended andupdated in order to incorporate the e�ects relatedto the interactions between ions with di�erentcharge states. Being based on dielectric or pertur-bative approximations, the theory is expected toapply in the case of swift ion clusters (or even slowion clusters in plasma targets).

The prediction of a large enhancement of thestopping power due to a collective resonance is aninteresting new e�ect to be yet experimentallytested using high-energy clusters. It may have im-portant applications in beam-plasma interactionsand fusion research. The magnitude of the e�ectwould depend on the degree of ionization of thebeam, and so the use of hydrogen or deuterium (tominimize spread) clusters may be more convenientfor this observation.


Still several other questions beyond the scope ofthis paper remain to be considered, like the ap-proach to the charge-state equilibrium as the ionsenter the solid, and the e�ects of energy exchangerelated to electron capture and loss processes.Other relevant questions for cluster±solid interac-tions include the aspects related to elastic±collisioncascades [26], and conditions for stability or frag-mentation of large clusters [150,151], as well asapplications to materials research [39,40] and nu-clear fusion studies [41±47]. A better knowledge ofthe energy loss will also contribute to furtherdevelopments in these areas.

The physics of small, intermediate and largeclusters is a wide ®eld of research, and a continu-ous source of new physical problems on correlatedatomic collisions in solids and many practicalapplications.

Acknowledgements

This work was supported in part by ConsejoNacional de Investigaciones Cient�õ®cas y T�ecnicas,and Agencia National de Promoci�on Cient�õ®ca yTecnol�ogica of Argentina. The author wishes tothank J.C. Eckardt, G.H. Lantschner, I. Abril andR. Garc�õa Molina for many stimulating discus-sions on molecular and cluster e�ects over theyears. Useful comments and suggestions on themanuscript by P. Sigmund are also gratefullyacknowledged.

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