Using channeling properties for studying the impact parameter dependence of electron capture by 20 MeV/u uranium ions in a silicon crystal

Abstract

The impact parameter dependence of electron capture by 20 MeV/u U$^{91+}$ ions has been studied by means of channeling in a 11 $\mu$m thick silicon crystal. Such ions are far from their equilibrium charge state in matter, and channeling offers a unique opportunity to study electron capture in conditions going from the extreme case of a single capture event (for the best channeled ions) to the case of multiple charge exchange events leading to the charge state equilibrium (for unchanneled ions). For each incident ion, the charge state at emergence, energy loss, electron emission and X-ray yields are measured. The correlations between these quantities are studied. The data are reproduced by simulations based on the ion flux distribution. We show that the Mechanical Electron Capture (MEC) dominates at impact parameters smaller than 0.5 Angstrom, whereas Radiative Electron Capture (REC) is the only process occurring beyond. Specific features associated to highly charged heavy ions at intermediate velocities are discussed, in particular ionization following capture into highly excited states, and local electron density enhancement due to the electron gas polarization. The measured impact parameter dependence of capture probabilities is compared to CDW-EIS (continuum distorted waves – eikonal initial state) calculations, extrapolated to n>5 final states.

Full text

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Using channeling properties for studying the impact parameter dependence
of electron capture by 20 MeV/u uranium ions in a silicon crystal
E. Testa1, P.N. Abufager2, F. Bosch3, A. Braüning-Demian3, H. Bräuning4, M. Chevallier1, C. Cohen5, D.
Dauvergne1, A. Gumberidze3, A. L’Hoir5, R. Kirsch1, C. Kozhuharov3, D. Liesen3, P.H. Mokler4,5, J.C. Poizat1,
C. Ray1, R.D. Rivarola2, J.P. Rozet6, Th. Stöhlker3, S. Toleikis3, M. Toulemonde7, D. Vernhet6 and P. Verma3
1Institut de Physique Nucléaire de Lyon, Université de Lyon, F-69003 Lyon; Université Lyon 1 and
IN2P3/CNRS, UMR 5822, F-69622 Villeurbanne, France
2Instituto de Fisica Rosario (CONICET-UNR), Av. Pellegrini 250, 2000 Rosario, Argentina
3Gesellschaft für Schwerionenforschung (GSI), D-64291 Darmstadt, Germany
4Institut für Atom und Molekülphysik, Justus Liebig Universität, D-35392 Giessen, Germany
5Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany
6Institut des Nano-Sciences de Paris, CNRS-UMR75-88, Universités Paris VI et Paris VII,
75251 Paris Cedex 05, France
7Centre Interdisciplinaire de Recherche Ions-Lasers, UMR 11 CEA-CNRS, 14040 Caen Cedex, France
Abstract
The impact parameter dependence of electron capture by 20 MeV/u U91+ ions has been
studied by means of channeling in a 11 µm thick silicon crystal. Such ions are far from their
equilibrium charge state in matter, and channeling offers a unique opportunity to study
electron capture in conditions going from the extreme case of a single capture event (for the
best channeled ions) to the case of multiple charge exchange events leading to the charge state
equilibrium (for unchanneled ions). For each incident ion, the charge state at emergence,
energy loss, electron emission and X-ray yields are measured. The correlations between these
quantities are studied. The data are reproduced by simulations based on the ion flux
distribution. We show that the Mechanical Electron Capture (MEC) dominates at impact
parameters smaller than 0.5 Å, whereas Radiative Electron Capture (REC) is the only process
occurring beyond. Specific features associated to highly charged heavy ions at intermediate
velocities are discussed, in particular ionization following capture into highly excited states,
and local electron density enhancement due to the electron gas polarization. The measured
impact parameter dependence of capture probabilities is compared to CDW-EIS (continuum
distorted waves – eikonal initial state) calculations, extrapolated to n>5 final states.
I.
Fast heavy ions traveling through matter may carry bound electrons if their velocity v
is not large compared to the orbital velocity vK of their K-shell electrons. They suffer atomic
collisions that may result in charge exchange processes. For a given projectile velocity the
electron capture and loss cross sections vary in opposite ways with the instantaneous ion
charge state: the higher the charge state, the higher (lower) the capture (loss) cross section. As
a consequence the mean charge state of projectiles penetrating matter tends toward a value for
which electron capture and loss cross sections are equal and that is called the equilibrium
charge state. The measurement of charge state distributions of ions transmitted through thin
Introduction

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targets [1] in combination with calculation codes [2] has been used since long in order to
determine charge exchange cross sections.
In ordinary matter, electron loss is essentially due to Nuclear Impact Ionization (NII),
but to some extend also to Electron Impact Ionization (EII); for non-relativistic ions, electron
capture is essentially due to the Mechanical Electron Capture (MEC), that involves bound
target electrons, whereas the Radiative Electron Capture (REC), that may involve also quasi-
free electrons, has much smaller cross sections. The present study is devoted to the impact
parameter dependence of electron capture (MEC and REC). The Mechanical Electron Capture
is a three-body process of bound electron capture in which energy and momentum are
conserved by means of the target atom recoil. For this reason, and also because initial and
final wave functions of the transferred electron must overlap both in spatial and momentum
spaces, MEC probabilities are expected to be peaked at rather small impact parameters
relative to target atoms, and then to decrease according to the size of electronic shells on
which electrons are captured.
Contrary to total cross sections, the impact parameter dependence of MEC
probabilities cannot be determined directly, because impact parameters of atomic collisions
are not observable quantities. Recently the Recoil Ion Momentum Spectroscopy (RIMS)
technique [3, 4] has been used successfully for studying those processes in fast ion-atom
collision experiments with gas targets. However the use of RIMS for measuring impact
parameter distributions is not straightforward because it is necessary to establish the relation
between the impact parameter of the collision and the transverse momentum of the recoiling
atom and therefore to evaluate the screening of the target nuclear potential. A few
experiments have been performed with light ion-target systems ([5] and references therein)
for which the above relation is not obscured by the presence of too many electrons.
Another way to control impact parameter distributions is to use channeling conditions
in a single crystal [6]. This opportunity has been first demonstrated in the pioneering work of
Datz and co-workers presented in ref [7] where the energy loss rate of ions in planar
channeling conditions was studied as a function of their distance from atomic planes.
Discovered more than forty years ago, ion channeling has been widely studied for decades in
all its aspects (for a review, see for instance references [8, 9]). When fast ions enter a single
crystal along a major planar or axial direction, they experience strongly correlated binary
collisions with target atoms, that repel them from atomic planes or strings. Then the uniform
flux of the incident beam becomes rapidly non-uniform as the ions penetrate into the crystal
bulk. As a result their interaction with the solid is deeply modified: close nuclear encounters
are essentially suppressed and the rate of energy loss is reduced. If the projectiles are heavy
ions traversing a thin crystal, their electron capture and loss probabilities are strongly lowered,
which completely modifies the charge state distribution of the transmitted projectiles. Our
collaboration has already used channeling to study the impact parameter dependence of
ionization processes (and thus the competition between NII and EII). In these experiments, the
incident ions chosen had a strong electron excess compared to their mean equilibrium charge
state in matter [10].
In the present experiment, we use channeling for studying the impact parameter
dependence of electron capture. For this purpose, we have chosen nearly bare very heavy
incident ions (20 MeV/u H-like U91+) with a very low adiabaticity parameter η=v/vK, thus
extending previous studies of capture processes with much lighter ions [11, 12]. In our
experiment the incident ions have a charge much higher than their mean equilibrium charge
state and, as long as the latter is not reached, they mostly experience electron capture. This is
the case for channeled projectiles all along their path through thin single crystals: these ions

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travel far from atomic cores and their charge exchange probability is highly reduced. The
trajectories of these ions in the crystal can be obtained by Monte-Carlo simulations and the
corresponding impact parameter distributions with respect to target nuclei can be deduced.
Then, by measuring charge state distributions at the crystal exit, we have been able to access
with precision to MEC probabilities corresponding to collisions occurring in the 0.2-0.5 Å
impact parameter range. In this range the MEC probabilities decrease steeply for increasing
impact parameters and we have been able to study precisely this variation. Actually,
measuring such MEC probabilities at large impact parameter is essential in order to check
whether bare H-like very heavy ions can be transported inside a crystal without charge
exchange, which would lead to potential applications at lower energies.
In section II, we describe our experiment and present the measurement of the charge
state Q and energy of the projectiles transmitted through the crystal, in coincidence with the
detection of X-rays and electrons emitted in the ion-crystal interaction. The section III is
devoted to the simulations performed to reproduce our experimental data and provide
quantitative impact parameter information on the electron capture. In section IV, we discuss
our results and compare them to CDW-EIS (Continuum Distorted Waves - Eikonal Initial
State) calculations.
II.
Our experimental set-up and part of our data have already been presented in a previous
paper [13]. Briefly, 20 MeV/u H-like U91+ ions were extracted from the storage ring ESR at
GSI (Darmstadt) and sent onto a 9.6 µm thick (111) silicon crystal fixed on by a three-axis
goniometer. The projectiles emerging from the crystal were charge- and energy-analyzed by
means of a magnetic spectrometer and collected in a 2D position sensitive detector located at
the focal plane of the magnet. The crystal was tilted by about 35.2° to the beam direction (its
effective thickness being 11.7 µm), which allowed the crystal alignment along the <110>
direction. The crystal was biased at –10 kV and was faced on both sides by two grounded
silicon detectors (referred to as Si-in and Si-out). These detectors attracted and collected low-
energy electrons emitted under ion impact by the entrance and exit surfaces. The signal they
delivered had an amplitude proportional to the electron multiplicity.
Experiment
A.
measurements
Charge state, energy loss and electron multiplicity
We have shown in a recent paper [14] that the electron multiplicity for channeled
projectiles is smaller than for unchanneled projectiles or than in random conditions, because it
reflects the reduction of the energy loss rate of channeled projectiles. Moreover the electron
multiplicity has been shown to depend on the transverse energy of channeled projectiles.
Hence, electron multiplicity can be used for discriminating between channeled projectiles of
different transverse energies, and, of course, between channeled and unchanneled projectiles.
In fig 1, the position X of transmitted ions in the magnet focal plane is associated to
the normalized electron multiplicity yielded by the detector Si-out, successively for a random
orientation and for alignment along the (111) and (110) planar directions as well as along the
<110> axial direction. This two-dimensional representation provides substantial information
in addition to the simple spatial distributions at the focal plane. It must be noted that each
picture results from the juxtaposition of spectra recorded for different values of the magnetic
field of the spectrometer and that there is no dose normalization between adjacent spectra so
that most of the rare charge states can be seen.

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First, one sees that individual charge states may show up, except in the random
orientation. In this orientation, transmitted ions are distributed rather uniformly in a large
ellipse-shaped spot. The charge state distribution corresponds to charge equilibrium and is
then gaussian-like. The mean charge is close to 74, in good agreement with the semi-
empirical formula of Leon et al. [15]. In fact the charge state distribution at emergence
reflects the one inside the target. As the ion energy loss has nearly a Q2 dependence (for
20 MeV/u uranium ions at charge state equilibrium, the bound electrons are localized very
near the nuclei and the ions are nearly point charges), the charge state fluctuations inside the
target induces energy straggling. The very heavy ions at intermediate velocity used in our
work experience very large charge-changing cross sections and, in this case, the energy
straggling in random geometry is by far dominated by charge state fluctuations [16]. Of
course, energy straggling smears out the position X distribution. However, even if not
resolved, the X distribution reflects the charge state distribution: the higher the charge state,
the stronger the deflection (hence X). We can notice that the electron emission yield, that
reflects the energy loss rate of projectiles when they are about to leave the target, is seen to
increase slowly with the charge state at emergence (from the arbitrary chosen value 1 to about
1.1), which is due to the fact that forward emitted electrons originate from a depth in the
target that is not much larger than the mean free path for ion charge changing.
For (111) and (110) planar alignments, two components are clearly seen. The first one
is random-like, however with a slightly higher mean value for the X distribution, and
corresponds to the unchanneled part of the aligned beam (~34% and 42%, respectively). The
second one is made of a succession of spots corresponding to individual charge states, from
the best channeled projectiles of Qout=91 at the extreme right to the poorly channeled
projectiles of Qout=70 at the extreme left. One may wonder why individual charge states can
show up, especially for the lowest ones (70 to ~78) that are mixed up in random conditions
and for the unchanneled component. As already indicated above, the mixing observed in these
two latter cases is related to the strong energy straggling induced by charge changing events
during the traversal of the target, electron capture, mainly by MEC and electron loss, mainly
by NII. The emergent ions have reached charge equilibrium since long: according to the code
ETACHA [2], this equilibrium is reached after 1 or 2 µm. On the contrary, for channeled
projectiles the MEC probability is suppressed, or considerably reduced and the lower cross-
section REC takes over. On the other hand electron loss by NII is suppressed and only the low
cross section EII can occur. As a result the channeled component does not reach any charge
equilibrium at emergence. The projectiles emerging in a given charge state have essentially
captured electrons, and at about the same pace because they had nearly the same transverse
energy and experienced nearly the same mean electron density. Then they suffered about the
same mean energy loss. Moreover, as the charge exchange straggling is much weaker than
that for ions in random conditions, the individual charge states can be clearly identified. We
can take benefit from this feature to turn the position X scale into a Qout scale, for ions
emerging with a given energy, that can be used to obtain the Qout distribution in random
conditions.
For <110> axial alignment, the picture looks like what is observed in planar
channeling except that the unchanneled component is too weak to be clearly visible. Note
that, for the best channeled projectiles along both planar and axial directions, the individual
spots do not appear vertical, but obliquely oriented. As the position X of a transmitted ion of a
given charge state depends on its energy loss in the crystal, this shows that energy loss and
forward electron emission are correlated, which is not surprising. This feature clearly shows
up for the best channeled projectiles in the case of the <110> direction. In fig 2(a), the scatter
plot of the highest charge states is shown, along with oblique lines that mark the separation

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between charge states. In fig 2(b) we show the corresponding X distribution of the individual
charge states, and also the X distribution of the direct incident U91+ beam. Then we can
determine the mean energy loss of the frozen component of Qout = 91, here 4.5% in units of
the incident energy, which has to be compared to the tabulated mean energy loss in random
conditions, 5.0 %. One learns more on energy loss in fig 2(c) where the two X distributions
correspond to transmitted projectiles with electron multiplicities NSi-out respectively below 0.4
and above 0.9, and thus with, respectively, a relatively small or a large transverse energy. In
particular, one can observe on fig 2(c) that for the frozen component, the energy loss ranges
from 3.8% to more than 5.3%. This shows that the available area of the transverse space for
the frozen component is rather large, and different frozen ions may experience different target
electron densities, from a few 10-2 to a few 10-1 electron per Å3 (the map of the electronic
densities averaged along the silicon <110> axis has been calculated in the reference [17]).
This explains why the NSi-out distribution corresponding to the frozen component is broad.
Then, the frozen U91+ projectiles associated to high NSi-out, may lose more energy than
projectiles traversing the crystal in random conditions (with a mean charge state of 74). Of
course, this is due to the quasi Q2-dependence of the energy loss rate. However, even the
frozen U91+ ions that suffer the highest energy loss rate are still well-channeled projectiles that
experience reduced electron densities: their energy loss rate, when the charge dependence has
been removed, is 0.7 times the random value. For the low NSi-out component, the non-
monotonous decrease of the charge state fraction with decreasing charges is attributed to
multiple electron capture events suffered by very well channeled projectiles in the thin
amorphous surface layers on each side of the crystal. The simulations presented below have
shown that the best agreement with experimental charge distributions is obtained for a total
thickness of amorphous layers of 60 Å.
The experimental charge state distributions extracted from the results shown in fig 1
are given in fig 3: for a random orientation and for <110> axial alignment in fig 3(a), and for
(111) and (110) planar alignments in fig 3(b). One of the main features we can notice in fig 3
is the mean charge state at emergence of poorly-channeled ions in axial and planar
orientation;
In fig 3(a) we show also the results of Monte-Carlo simulations [18]. These
simulations, that reproduce the general trends of the charge distributions, were based on the
full description of ion trajectories and charge exchange in the crystal.
In fig 4 we show, for the <110> direction, how the measured charge fractions F(Qout),
for Qout values from 91 to 86, vary with the angle ψ0 between the incident beam direction and
the <110> direction. These charge states, that disappear when ψ0 increases (because they do
not show up in random conditions), represent the best channeled part of the aligned beam
Note that trying to reproduce these variations is a very severe test for the simulations that we
have performed (see section III).
B.
X-ray measurements
X-ray measurements provide additional information on the nature and probabilities of
the charge exchange processes [19]. The Ge detector we used allowed us to observe the filling
of the K- and L-shells of the H-like incident ions. In fig 5 we show two X-ray spectra
obtained with a tightly collimated detector located at 90° to the beam direction, for a random
incidence (fig 5(a)) and for alignment along the <110> axial direction (fig 5(b)). These
spectra are normalized to the same number of non frozen ions. This normalization is based on
the fact that the filling of the K-shell vacancy occurs at most once (excitation or ionization
from the K-shell of uranium ions can be neglected).

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The spectrum of fig 5(a) is dominated by L- and K- photons that are mainly due to
decay cascades following MEC events. For 20 MeV/u uranium projectiles in a silicon target,
these cascades are due essentially to the capture of silicon K-shell electrons into (n≥4) shells
of uranium ions (this is required by the necessary matching of the initial and final velocities of
the captured electron). Moreover the REC lines are essentially absent because the K- and L-
REC cross sections are smaller than MEC cross sections by orders of magnitude. K- and L-
shells are very rapidly filled by the cascading processes following MEC events and thus K-
and L-REC events cannot occur. In contrast, in the spectrum of fig 5(b), the L-lines are
strongly reduced and L- and K-REC lines do show up because the MEC probability is
drastically reduced for channeled projectiles. K-lines still appear. They originate from the
radiative decay following capture events either by MEC, into high n-shells, or by REC,
mainly on the L-shell. On the other hand, the sharp components of the L-lines observed in
fig.5(a), which correspond to cascade events taking place after charge state equilibrium has
been reached (i.e. cascade between rather well defined energy levels), has disappeared in the
spectrum obtained in the <110> orientation (fig.5(b)). The main reason of this effect is
obvious: most of the ions are well channeled, they experience reduced capture probabilities
and exit the crystal with few electrons on their L-shell. However the effect also concerns less
well channeled ions that do experience significant electron capture. For instance, ions with
“intermediate” transverse energies may approach atomic strings close enough to experience
many MEC events, filling progressively their L-shell. However they are too far to undergo
ionization in this shell, as such an event requires atomic collisions at very small impact
parameters of the order of 10-2 Å. Finally we have shown in ref.[18] that even the ions with
very high transverse energy, do not show up the sharp components of the L-lines. As these
ions approach enough atomic strings to suffer L-shell ionization, this result proves that the
filling of the L-shell takes place much less rapidly than for unchanneled ions. We have
attributed this fact to a so-called “superdensity effect”: a capture in an excited state is often
followed by re-ionization before that the captured electron could be stabilized in an inner
shell.
For <110> alignment, we show in fig 6 the dependence on Qout of the various REC and
MEC rates, for Qout values ranging from Qout = 90 (that correspond to the best channeled
projectiles producing X-rays), to Qout = 76 (that correspond to rather poorly channeled
projectiles). These rates have been determined by assuming that radiative decay emission is
isotropic in the projectile frame, and that REC photons are emitted according to a sin2θ law
[20, 21], where θ is the angle between the photon and the beam directions in the laboratory
frame. In fig 6(a) we give the Qout dependences of K-REC, L-REC and L-line (Balmer)
photon rates per ion transmitted with the charge Qout. The decrease of the Balmer rate when
Qout increases is due to the progressive suppression of MEC events when the transverse
energy decreases. Then, as already stated, the K- and L-shells vacancies of ions with low
transverse energy are not filled by MEC and these projectiles may experience large radiative
capture rates. In fig 6(b) we give the relative contributions of REC and MEC processes to
electron capture. The main feature of this figure is that mechanical capture is the dominant
process as soon as ions capture more than one electron whereas its contribution for ions
transmitted with the charge state Qout = 90 is about 30%. This is a qualitative evidence for a
rapid increase of the MEC probability with the ion transverse energy, i.e. when projectiles can
approach closer to atomic strings or planes.
The experimental results presented in fig 4 and fig 6 provide very detailed information
on the rate of the various electron capture processes as a function of the ions transverse
energy. This information could be reached because we used very heavy H-like ions at
intermediate velocities (beams extracted after deceleration from the storage ring at GSI).

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Then: i) the charge state at emergence is strongly connected to the transverse energy. ii) the
very high electron capture probability allowed us to accumulate sufficiently high statistics X-
ray spectra to perform a detailed analysis of these spectra for each charge state Qout at
emergence.
The transverse energy of an ion defines its available transverse space, and thus its
impact parameter distribution with respect to target nuclei. We have then attempted to fit our
experimental results in order to reach information on the competition between REC and MEC
events as a function of the impact parameter, and thus on the impact parameter dependence of
the MEC probability per target atom. For this purpose, we have performed simulations that
are presented in the next section.
III. Simulations
Simulations are necessary to get quantitative information out of our experiments. We
show in what follows that for simulating the charge exchange of rather well channeled
projectiles, that keep far from atomic strings or planes, typically at distances larger than
0.25 Å, a full Monte Carlo calculation is not mandatory. We have used a much less time
consuming approach, based on the following strategy: first, impact parameter distributions are
determined by trajectory calculations, neglecting charge exchange. Second, electron capture
rates are adjusted as a function of the ion transverse energy in order to reproduce the
experimental data. Third, we extract the impact parameter dependence of MEC and REC
capture per target atom by unfolding the preceding values..
A.
energy
Impact parameter distribution as a function of transverse
Simulating charge exchange events requires essentially to determine the impact
parameter distribution of channeled projectiles and the mean electron density they encounter.
We have therefore performed calculations of channeled ion trajectories all along the crystal
For each crystal orientation (axial and planar), this procedure allowed us to set the impact
parameter distribution and the mean electron density encountered as a function of the
transverse energy.
Impact parameter distributions were calculated considering the successive collisions
suffered by projectiles. For this, we used the Moliere analytical approximation of the Thomas-
Fermi screening function to calculate the ion-atom potential. The distribution of thermal
displacements of target atoms from lattice sites was considered to follow a gaussian law with
a variance calculated from the Debye theory. Correlation between atomic displacements was
neglected. Then we used the transverse continuum potential approximation to provide the
transverse energy value E⊥ (this approximation was not used for trajectory calculations).
The transverse energy distribution at the crystal entrance is somewhat modified along
the ion path. This is due, on the one hand, to transverse energy changes related to charge
exchange and, on the other hand, to multiple scattering. Let us first evaluate the influence of
charge exchange, which has been studied in detail by Grüner et al. [22]. The variation
(
∆
⊥
∆ε
of the reduced transverse energy
ε
)Q
Q
E⊥
⊥=
of a channeled projectile due to a charge
change Q
(
ε∆
⊥
mainly at the closest approach of the atomic strings or planes, i.e. when ψ has the smallest
∆
=
occurring when its trajectory makes the angle ψ with the channeling direction is
(
EQ
ψ∆−
. As the dominant charge exchange (the mechanical capture) occurs
))
22Q
Q
∆

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values, (
angle) that is of the order of 1 eV for Q
along the axial and planar directions we have studied are respectively about 120 eV and
20 eV, the influence of charge changing on the channeled projectile trajectories has been
neglected in our simulations.
)Q
∆
⊥
∆ε
is small compared to
()
22Q
c
EQ
ψ
∆−
(where ψc is the Lindhard critical
=1 in the <110> direction. As the potential heights
∆
As for transverse energy changes induced by multiple scattering, the effect of elastic
collisions is already taken into account in the trajectories calculations that are achieved by
considering all the consecutive binary collisions with target atoms. But multiple scattering on
target electrons must also be considered; it is in fact the main source of transverse energy
change for well channeled ions. According to Bonderup et al. [23], one may neglect the
contribution to multiple scattering of non local distant interactions and the mean rate of
transverse energy increase induced by this process is connected to the energy transfer by close
collisions on target electrons. Thus we calculated the mean increase of the transverse energy
using the measured energy loss values and considering only the fraction of this loss that can
be attributed to close collisions (this procedure is described in ref [10]). The increase in
transverse energy is small but not negligible and was considered in the simulations.
There is a one-to-one relationship between the transverse energy of an ion and its
minimum distance of approach to atomic strings or planes rmin. In the following paragraph, we
will mostly consider rmin distributions. They give access to the impact parameter distribution
with respect to target atoms, which has to be determined in order to extract information on the
impact parameter dependence of charge exchange processes from our experimental data.
B. Charge exchange
For poorly channeled ions, the charge state at emergence arises from a competition
between capture by MEC and ionization by NII, both types of events occurring very
frequently. For these ions the rmin distribution varies strongly with the penetration depth. In
such a situation the information that can be reached from our experiments on the impact
parameter dependence of MEC probability would be questionable at small impact parameters.
We will thus focus on electron capture events of channeled ions that remain at distances rmin
larger than about 0.25 Å from atomic strings or planes. At such distances, the projectile
ionization by nuclear impact (NII) is strongly reduced and was neglected in our simulations
(this approximation is justified in section IV B). The projectile electron loss can therefore
occur only by EII of electrons captured in outer shells (that had not enough time to cascade
into inner shells). In order to limit the number of parameters in our simulations, we have
introduced an effective capture probability defined as the probability of capturing an electron
that is not lost afterwards (of course, this probability results both from capture events and
from electron loss through EII).
Our simulations of charge exchange are based on the determination of the minimum
distance
beam angular divergence and on the crystal orientation. We consider their evolution induced
by electron multiple scattering, as a function of penetration depth. To each
associate the mean numbers of effective mechanical and radiative capture events, NMEC and
NREC, and a mean transverse energy increase per unit path,
energy loss per unit path
dzdE/
[23]. The ε⊥ increase results in an increase with depth of the
transverse accessible space, i.e. a rmin decrease and a capture probability increase.
The influence of the projectile charge state on the capture probabilities is taken into account:
whereas the mechanical capture does not depend much on the projectile charge state (since it
irmin distributions near the entrance crystal surface. These distributions depend on the
irmin value, we
dzd
/
⊥
ε
, proportional to the mean

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occurs in highly excited states), the REC probability for an U90+ ion (that has no K-vacancy)
is for example 30 % lower than for a H-like U91+ ion. The electron capture events occurring in
the amorphous layers of the crystal surfaces are also taken into account in the simulations.
The adjustable parameters introduced in the simulations are the following: i) the mean
numbers
(
N
directions; ii) a coefficient CREC (this coefficient is justified below) applied to the REC cross
section (this cross-section is readily obtained in the frame of the non-relativistic dipole
approximation [21]); iii) the beam angular divergence (that could not be measured
separately); iv) the electron capture in the amorphous layers. The main experimental data that
have been used to constrain these parameters are (see section I): the charge state distributions,
the K-, L-REC probabilities and the angular scans of the emergent charge state distribution
across the <110> axis (and across the (110) and (111) planes, which were also obtained even
if not presented here). In order to fit these scans, we had to assume a two-component beam
profile: 65 % of the beam described by a narrow gaussian distribution with 1D standard
deviations σX=σY=0.14 mrad and the remaining 35 % by wider Gaussian wings with
σX=σY=0.43 mrad, to be compared to the critical angle of 1.4 mrad for the <110> axis, and of
0.50 and 0.55 mrad respectively for the (110) and (111) planes.
)
i
MECr
min of mechanical capture events for each of <110>, (110) and (111)
The overall best agreement between the simulations and the measurements, presented
in fig. 4, 6.b and 7.b, leads to the mean effective capture numbers
given in fig. 7.a. In all cases, the agreement between the numerous experimental data and the
simulations is remarkable. In figure 7.b we compare the simulated and measured charge state
distributions for the <110>, (110) and (111) orientations. Having limited our simulations to
ions of relatively low transverse energy, that undergo less than nine effective captures, the
ions that capture more than ten electrons are gathered in the fraction called F(Qout<82). The
mean effective captures NMEC presented in fig 7.a for
simulations providing the best overall agreement with the data. NMEC increases strongly for
ions that approach the atomic strings or planes at distances
distance that roughly corresponds to the spatial extension of the silicon core orbitals. On the
contrary, the mean number NREC is nearly constant although the mean density
is about twice the mean density
e
ρ (
for poorly channeled ions of small
of their inner shells by MEC followed by electron cascades.
()
i
MECrN
min and
()
i
RECrN
min
irmin > 0.25 Å correspond to the
irmin smaller than about 0.4 Å, a
e
ρ (
irmin=0.3 Å)
irmin=0.8 Å) sampled by well channeled ions. This is due,
irmin values, to the fast filling (close to the crystal entrance)
C. Impact parameter dependence of MEC probabilities
Our experimental results and the above simulations provide the effective MEC
probability
( ) bPeff
MEC
per target atom at a given impact parameter b. This probability is deduced
from the mean numbers
(
N
distributions
( ) b
Φ
by assuming independent single charge exchange events for well
(
N
Φ
are then linked, for a given crystal direction, by the
following relation:
(
bPrN
MECMEC
min
Φ
)
i
MECr
min of MEC events and from the impact parameter
irmin
channeled ions.
)
i
MECr
min and
( ) b
irmin
)
( )( )dbb
ir
min
effi
=∫
.
( 1 )
Within the experimental uncertainties, the probability
for the three crystal directions studied (<110>, (110) and (111)), a strong indication of the
( ) bPeff
MEC
is found to be the same

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10
self-consistency of our simulations. The larger
(see the fig 7.a) is explained by the fact that the impact parameter distribution
()
i
MECrN
min values obtained for the (110) plane
( ) b
irmin
Φ
in this
plane is slightly shifted towards the small impact parameters compared to the impact
parameter distributions obtained for the <110> and (111) directions.
IV. Discussion and comparison with theoretical calculations
A.
REC cross sections
Detailed information on the radiative capture (namely the influence of the electron gas
polarization) has been reported and discussed already in [24]. In this paper, we observed that
theoretical REC cross sections have to be multiplied by a factor CREC=1.5 to reproduce our
results. This is explained by a ion-induced polarization of the target electron gas resulting in a
local electron density enhancement in the vicinity of these slow, highly charged projectiles. In
order to complement this point, we present in fig 8 the measured evolution of K-REC cross
sections as a function of the adiabaticity parameter. The gas and amorphous solid target data
correspond to a compilation by Th. Stöhlker et al. [25]. Channeling data were obtained by our
collaboration at GANIL [26, 27] and in the present work. The theoretical values correspond to
the calculations of ref. [21] performed in the frame of the non-relativistic dipole
approximation and they correspond to non-perturbed electron densities in the target. At low
energies, measurements on solid state targets tend to exhibit enhanced cross sections when
compared either to measurements on gas targets or to theoretical estimates. In contrast, at
higher energies, i-e in situations for which η>1, a good overall agreement is observed.
These results for REC cross sections suggest some remarks: as discussed in ref. [24], a very
strong electron gas polarization is induced by the relatively slow and very highly charged ions
used in the present experiment. In such a situation, first order perturbation calculations predict
a very strong enhancement of the electron density at the ion site that should induce an
enhancement of the REC yield much higher than the 50% effect that we observe
experimentally. In fact, higher order–or non-perturbative - calculations would be required to
get a quantitative estimate of the density enhancement. Besides, although channeling
conditions are the most efficient way to measure REC cross sections in solids at such low
values of the adiabaticity parameter, gas target measurements would certainly provide
complementary information on the observed electron density enhancement observed in
channeling conditions and on the validity of the dipole approximation at low η values.
B.
Effective MEC probabilities per target atom
In fig 9, we show the b dependence of the probability
experimental results, via eq.1, along with CDW-EIS calculated values of MEC probabilities
( ) bP
MEC
. The error bars become important for b values smaller than 0.2 Å, because the
assumptions subtending our treatment are only valid for relatively well-channeled ions; they
also become important for b values above 0.55 Å because, at large b values, the electron
capture events are rare and mainly due to REC.
( ) bPeff
MEC
extracted from our
ntheory
−
The CDW-EIS calculations were limited to capture into shells with n-value ≤ 5
because of the enormous complexity of such analytical methods. The sum of these theoretical
probabilities is larger than the probability
Peff
MEC
This could be expected as, for small b values, NII becomes important and thus
strongly affected by electron loss events. On the contrary, this sum is much smaller than
( ) b
for impact parameters b below 0.15 Å.
( ) bPeff
MEC
is

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11
( ) bPeff
MEC
the integrated CDW cross sections [28] presented in fig 10: the mechanical capture
probability is maximum for final states n equal to 5 and 6 (which are not considered in the
CDW-EIS calculations) and then decreases for larger n-shells. The decrease predicted by ref
[28] and presented in fig 10 is much slower than the n-3 scaling law usually assumed [29].
Indeed, the mechanical capture requires the overlap of the initial and final wave functions of
the captured electron, which leads, in our experimental situation, to a high probability of
electron capture from the silicon K-shell into the projectile outer shells. Besides, since the
silicon K-shell extension is very small (of the order of 0.05 Å), the MEC probability is
expected to depend mainly on the projectile orbitals on which a K-shell target electron is
captured; this has been verified by comparing
bP
( ) b
n
ϕ
is the radial wave function of the H-like uranium). These two functions appear
to be nearly similar and we used this similarity to extrapolate the theoretical probability
( ) bP
MEC
for n>5. The overall electron capture probabilities on all the projectile shells
(indexed by n) and the corresponding sum for n-values running from n =1 to n=10 are
presented on fig 9. The fact that these theoretical probabilities are much larger than the
probability
( ) bPeff
MEC
of effective capture for impact parameters b above 0.25 Å demonstrates
the importance of EII on outer shells (let us recall that the values
when fitting our experimental data by simulations result from both capture and ionization
processes).
at large impact parameter. This confirms the dependence on the shell number, n, of
( ) b
ntheory
MEC
−
functions up to n=5 and
( )
b
2
b
n
ϕ

(where
ntheory
−
( ) bPeff
MEC
that were obtained
In order to really compare the experimental and theoretical results for impact
parameters b > 0.25 Å, we have evaluated the projectile ionization probability to deduce a
theoretical probability for effective capture
P
MEC
is negligible for impact parameters b > 0.25 Å: the outer shell electrons of the projectile with
n-value close to 6 or 7 present binding energies of the order of a few keV and orbital
extensions of about 0.2 Å. As the ionization of these electrons requires impact parameter
smaller than 0.05 Å between projectile electrons and silicon atoms, the NII process is much
less important than the EII process for impact parameters b above 0.25 Å. The EII probability
has been estimated by coupling the cross sections
projectile n-shell to the radiative decay times and to the decay branching ratios that
correspond to the probability for an electron initially on a n-shell to reach a n’-shell with
n’<n. The cross sections
( )
n
EII
σ
have been deduced from the cross section
determined for 29 MeV/u Pb56+ ions by L’Hoir et al. [18] and a scaling law based on the Lotz
formula [30]. The radiative decay time and the branching ratio have been obtained from the
calculations of Omidvar for H-like ions (with a Zp-4 scaling law for the decay times) [31]. The
EII process leads to a complete ionization of the shells above n= 9 whereas it does not much
affect the probability for electron capture on the other shells.
( ) bP
MEC
for effective capture is compared, in fig 11, to
the probability
( ) bPeff
MEC
deduced from our measurements. These two probabilities are in good
agreement. This is a good indication that: i) the CDW-EIS calculations provide a realistic
impact parameter dependence for MEC at medium and large distance; ii) our extrapolated
estimates of MEC probabilities on high n>5 shells (total cross sections, impact parameter
dependence) are valid and thus MEC into highly excited states plays a major role. We are
currently undertaking CTMC calculations to provide complementary estimates of the capture
probability and confirm our extrapolation procedure.
( ) b
efftheory
−
. The nuclear impact ionization (NII)
( )
n
EII
σ
for electron impact ionization of the
() 3
=
n
EII
σ

The theoretical probability
efftheory
−

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12
Conclusion
Whereas full Monte Carlo calculations are required to follow the evolution of charge
states and electronic configuration of high transverse energy channeled ions approaching
atomic strings [14], we have shown that one may use a simpler calculation procedure based
on the ion flux distributions when only relatively larger distances from target atoms are
considered. This has allowed us to determine the dependence of MEC and REC probabilities
by fast heavy ions as a function of impact parameters. REC probabilities are increased by the
local electron density enhancement induced by the very heavy and highly-charged ions at
intermediate velocities. Experimental MEC probabilities are consistent with the results of
CDW-EIS calculations performed up to n=5 and extrapolated for larger n-shells. Our
experiments have shown that the mechanical capture at large impact parameter arises mainly
from capture into highly excited states, up to n=9. Besides, we have seen that frozen U91+
projectiles channeled in a 11.7 µm crystal may lose more energy than projectiles traversing
the crystal in random conditions. This charge state effect has been used in a recent experiment
to study the feasibility of strongly slowing down very highly charged ions by transmission
through a relatively thick crystal in channeling conditions. The capture cross sections of the
best channeled ions is so strongly reduced that they may have a significant probability to
emerge with their initial charge state. The results of this study are to be published in a
forthcoming paper.
In view of the future FAIR accelerator in Darmstadt, our work stresses the interest of
channeling for the study of recombination and energy loss of the slow, highly-charged ions
that will be available at the FLAIR facility.
Acknowledgments
This work was partly supported by the French-German GSI-IN2P3 collaboration
agreement 97-35.

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fig 1. Detection of projectiles transmitted for various beam incidence conditions of U91+ ions: (a) random
incidence, (b-c) (111) and (110) planar alignments, (d) <110> axial alignment. Scatter plots of forward
electron emission multiplicities (NSi-out) as a function of X, the position in the focal plane of the analyzing
magnet. NSi-out values have been normalized in such a way that the mean value of the left part of plot (a) is
equal to unity.

Qout=74
(arb. units)
(arb. units)

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14


fig 2. (a) Scatter plot of forward electron emission multiplicities vs position X for the best channeled
projectiles incident along the <110> axial direction. The oblique lines are used to delimit charge states and
(b) generate, by projection, the X distribution of each charge state; the X distribution of the direct 91+
incident beam is also shown, which yields the energy loss of the frozen fraction.(c) Here two X
distributions are shown, for transmitted ions of NSi-out above 0.9 and below 0.4, respectively (these
distributions have been smoothed).

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15

fig 3. Experimental charge distributions, F(Qout), of incident 20 MeV/u U91+ ions transmitted (a) in random
conditions and for <110> axial alignment (along with Monte-Carlo simulations). Full symbols:
experiment, open symbols: simulations by L’Hoir et al. (see text); (b) for (111) and (110) planar
alignments.

0.1
1
10
simulation
(b)
<110> axis
F(Qout) (%)
Random
orientation
(a)
68 70 72 74 76 78 80 82 84 86 88 90
Charge state Qout
0.1
1
10
experiment
Incident
charge state
(110) plane
(111) plane
F(Qout) (%)