Charge-State Equilibration of Fast Ions
Passing Through a Carbon Foil and a Single C60 Molecule
Itoh Akio
Quantum Science and Engineering Center,
Kyoto University, Kyoto 606-8501, Japan
1. Introduction
Spherical cage structure of a C60 fullerene, composed of sixty carbon atoms on the surface of radius 6.6
a.u., readily reminds us of an intuitive picture that the molecule may act as a thin foil target for an incoming
projectile particle. The target thickness of single C60 molecule is approximately 60 πa 2  1.6 1016
(molecules/cm2), being equivalent to e.g. a dense gaseous target of pressure 0.1-Torr and 5-cm length at
20 C . Thus, it is suggested reasonably that the charge-state equilibration may be achieved for such
projectile ions that penetrate through the C60 cage, if the projectile atomic number is not so high. It is,
however, an open question whether the C60 can be regarded as a thin carbon foil, since no such experimental
study has been carried out so far. The present report is concentrated on this subject, and a similarity between
the two materials is discussed. For this purpose equilibrium charge fractions of fast Li beams have been
measured for real carbon foils [1], and those for a single C60 molecule were estimated from cross sections
for C60 fragmentation accompanying charge-changing collisions [2].
In general, two different mechanisms are relevant to the charge equilibration of outgoing projectile ions;
i) small-angle multiple collisions and ii) large-angle single collisions. The former mechanism is the most
popular and predominant process, and a large number of experimental works have been reported [3]. The rate
equation of charge fractions Fk as a function of the target
100
thickness x, is expressed by
F3
dFk
  (Fq σ qk Fk σ kq ) ,
dx
q
The charge equilibration is attained asymptotically for
infinite target thickness ( x   ). By contrast, the
charge-equilibration in large-angle single collisions, memory
of initial electronic states may be lost during a small-impact
collision, resulting in the same charge distributions
irrespective of the initial charge state. In other words,
difference of initial wave functions due to the initial charge
state is smeared out during passing through a dense electronic
cloud of the target atom. A similar situation is expected for a
C60 target, in which the electronic cloud of 60 π -electrons
spreads spherically at radii r=2~5Å and 180 σ -electrons
at about 3.6 Å . Thus, the charge equilibration may be
achieved in penetrating particles without accompanying large
deflection. This is the key scenario of the present work.
Equilibtrium Fractions
with charge-changing cross sections σ ij .
F2
10-1
F1
10-2
10-3
10-4
present
Stocker
celluloid
F0(Pivovar)
ZMY
10-1
100
7
2. Equilibrium fractions in carbon foils
F0
101
Li Energy (MeV)
Fig. 1. Equilibrium charge fractions.
Equilibrium charge fractions of 1~6 MeV lithium ions
after passing through carbon foils were measured with a
traditional method as described in [1] at the Tandem accelerator facility of Nara Women’s University.
Equilibration was confirmed by measurements for various combinations between incident charge state and
foil thickness. The incident beam flux was kept below 300 cps to avoid counting loss of the
position-sensitive detector. The results are shown in Fig. 1 together with other experimental [4-6] and
theoretical vales [7]. To our best knowledge, present data are the first and only ones in this energy range.
Overall expressions for the fractions of F0 and F1 were obtained as a function of the incident 7Li energy E
(in MeV) above 1 MeV range;
F0  (1.2  0.06)  10 2 E -(3.70.04)、　F1  (0.33  0.005)E -(2.670.02)
The present results are in remarkably good agreement with semiempirical calculations (denoted by ZMY) by
Zaidins [7]. He employed the independent electron model (IEM) based on the assumption that the probability
for the removal of a projectile electron is solely determined by the ratio between the electron orbital velocity
and the projectile velocity.
Until now various semiempirical formulae for average charge Q have been derived to obtain the most
reliable universal expression as functions of projectile atomic number Z, impact energy E and target atomic
number, as summarized in [8]. In the following, we use three representative formulae, which may be
applicable in the present energy range. Note that all the formulae were derived from available
100
He
Li
Li
Be
B
SIM
1-Q/Z
Average Charge Q
3
10-1
present
Stocker
celluloid
ND
TD
SIM
TRIM
ZMY
presnt
2
1
100
7
101
Li Energy (MeV)
Fig. 2. Comparison with semiempirical formulae
10-2
10-3
0
1
2
3
4
5
Reduced Velocity X
Fig. 3. Comparison with other projectile ions
experimental data, and consequently, the range of validity is limited as shown below. For ions with energy E
(MeV) and mass M (amu), the value of Q in carbon foils is given in the following way:
ND : Q Z  (1  X 5 / 3 ) 3 / 5 , for 0.3  Q Z  0.7, 16  Z
TD : Q Z  1  exp(  X ) , for 0.2  Q Z  1.6, 5  Z  18
SIM : Q Z  1  exp( 1.251X  0.32 X 2  0.11X 3 ) ,　fo r X　 2.4, 8  Z
The quantity X is the reduced velocity X defined by, X  3.86Z 0.45 E M . Abbreviated letters ND, TD
and SIM correspond to refs [9-11], respectively. The present data of average charges are compared with these
formulae in Fig. 2. Obviously, any of these formulae cannot reproduce the experimental values in the whole
range of incident energy depicted. It should be emphasized that these formulae may be applicable only for
relatively high Z projectiles (Z>5). Thus, large discrepancies observed indicate that the quantity Q/Z should
contain Z dependent terms explicitly as discussed again in more detail below. In this figure, the experimental
data are also compared with effective charges Zeff calculated with the TRIM code which is widely used in
ion-solid collision experiments. Compared to other formulae given above the TRIM data are in much better
agreement with experimental values. However, somewhat large deviations are observed outside 2 MeV
region. We should bear in mind the fact that the Zeff is essentially a different quantity from the equilibrium
average charge. This is because the projectile electrons also take part in the energy loss (stopping) processes
via electronic excitation of the target atoms, and consequently, the Zeff would be higher than Q.
On the other hand, the IEM calculation by Zaidins is found to give perfectly equivalent Q to the
experimental values. This finding is rather surprising, because in this IEM model, the probability of electron
loss is dependent only on the velocity ratios mentioned above, and any physical quantity related to the target
atom is not taken into consideration.
In order to obtain a semiempirical formula for the present case, the experimental data are plotted in Fig. 3
as a function of the reduced velocity X, where the ordinate represents 1－Q/Z to emphasize the variation in
high velocity region. One can see clearly that the log(1－Q/Z) is not a linear function of X, instead it
contains higher order terms as in the SIM formula. The best fit obtained over the entire velocity range is
Q / Z  1  exp( 0.706  1.98 X  0.0883 X 2 )　, for 0.8  X  5
This formula seems to be applicable in a wide range of incident energy from 0.8 to 30 MeV.
In Fig. 3 are also depicted the experimental data for light ions of He [12], Be and B [8] and
SIM-calculations. There exists obviously a systematic deviation depending on Z. Namely, the values increase
with increasing Z. The data for B ions coincide fairly well with the SIM formula, which is successfully
applicable for heavy projectiles, implying that the data for heavy ions of Z >5 can be approximated by the
SIM formula. The deviation seems to be largest in the X range investigated here. This is due to the fact that
the Q/Z must approach 0 at low velocity limit, while at high velocities (X>4) the Q/Z shows no longer Z
dependence since the charge fractions are dominated by fully striped ions and hydrogen-like ions. Thus, the
asymptotic deviation for low Z ions postulates a requirement of explicit Z-dependent terms in the Q/Z
formula.
2. Charge fractions in single C60 molecule
The time-of-flight experiments of C60 fragmentation was performed at the QSEC tandem accelerator
facility of Kyoto University. A gas-phase C60 target
was produced by heating 99.9% pure powder at
500 C in a temperature controlled quartz oven.
The molecular C60 beam was effused through a
small aperture of the oven, and the “target” gas
pressure at the beam line was about 9  10 7 Torr
under the condition of a base pressure of
3  10 7 Torr. It should be noted that, in such low gas
pressure, almost all the outgoing projectiles pass
through
the
target
region
without
any
charge-changing collisions. It implies that the
traditional method of charge fraction measurements
cannot be used unless the gas pressure is varied
widely. This is the case in the present work.
From a series of our previous experiments on
ion-induced C60 fragmentation, it is known that the
fragmentation always occur in close collisions,
namely, in cage-penetrating collisions. Since the
desired quantities are the number and the charge state
of particles penetrating a C60 cage, we measured
fragmentation cross sections of C60 in coincidence
with outgoing charge state of Li ions including
neutral components. The detailed description of the
experimental method is given in [2]. The total
fragmentation cross sections σ(qk) obtained for
charge-changing collisions of q→k can reasonably be
assumed to reflect the number of cage-penetrating
particles with charge k. The charge distribution
Fqk for these particles can then be calculated by
Fig. 4. Fragment ion spectra from C60 obtained in
coincidence with final charge state.
Fqk  σ(qk)
3
 σ(qk)
k 0
Fractions
for the incident charge state q. The experiment was carried out for all charge states (q=0~3) at 2 MeV
incident energy. Fig. 4 demonstrates the time-of-flight spectra of fragment ions measured for various
combinations between q and k. The fragment pattern varies strongly depending on the different q→k
collisions. For instance, in one electron loss collisions (1→2), ionized parent ions are also observed in
addition to small fragment ions produced via multifragmentation processes. On the other hand, in double
electron capture (3→1), only the small fragment ions are created, implying the predominance of violent
collisions.
The charge fractions obtained by the
2 MeV Li on C60
above method are shown in Fig. 5 as a
0
10
function of k for all q → k collisions.
Obviously the overall shapes of the
distributions reveal remarkably similar
distributions to each other, particularly for q=
2 and 3. In fact the average charge of
outgoing articles was found to differ only
10-1
slightly for different incident charges. This
finding indicates strongly that the
cage-penetrating particles can attain nearly
N2gas
equilibrium charge distributions.
C-foil
It is worthwhile to compare the present
results with equilibrium fractions measured
10-2
C60
for other target materials such as N2 gas and
0
Li
carbon foils mentioned in the above section.
1+
The results for C60 are, in overall, in fairly
Li
2+
good agreement with these targets. In
Li
particular, the values for q=2 and 3 coincide
Li3+
fairly well with those for N2 gas. This
10-3
experimental finding implies clearly that the
single C60 molecule is almost equivalent to
0
1
2
3
the thick gaseous target. Comparison with
carbon foils shows a somewhat large
Outgoing Charge State k
discrepancy at q=3. It may indicate the
difference of so-called density effect between Fig. 5. Charge fractions of C60 cage-penetrating projectiles.
C60 and real carbon foils.
In conclusion, we have certainly observed the thin foil property of C60 using a rather sophisticated
experimental procedure as introduced in this report. It is concluded that the single C60 molecule acts as, at
least, a condensed gaseous target. More systematic and detailed investigations are presently planed.
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