ARTICLES
PUBLISHED ONLINE: 9 FEBRUARY 2014 | DOI: 10.1038/NCHEM.1860
State-resolved diffraction oscillations imaged
for inelastic collisions of NO radicals with
He, Ne and Ar
Alexander von Zastrow1†, Jolijn Onvlee1†, Sjoerd N. Vogels1, Gerrit C. Groenenboom1,
Ad van der Avoird1 and Sebastiaan Y. T. van de Meerakker1 *
Just as light scattering from an object results in diffraction patterns, the quantum mechanical nature of molecules can lead
to the diffraction of matter waves during molecular collisions. This behaviour manifests itself as rapid oscillatory
structures in measured differential cross-sections, and such observable features are sensitive probes of molecular
interaction potentials. However, these structures have proved challenging to resolve experimentally. Here, we use a Stark
decelerator to form a beam of state-selected and velocity-controlled NO radicals and measure state-to-state differential
cross-sections for inelastic collisions of NO with He, Ne and Ar atoms using velocity map imaging. The monochromatic
velocity distribution of the NO beam produced scattering images with unprecedented sharpness and angular resolution,
thereby fully resolving quantum diffraction oscillations. We found excellent agreement with quantum close-coupling
scattering calculations for these benchmark systems.
T
he outcome of a collision between molecules is determined by
the paths they take over the underlying potential energy
surfaces (PESs). The accurate measurement of the deflection
of the molecules as a result of their encounter, that is, of the
differential cross-section (DCS), is one of the most rigorous tests
for ab initio computed PESs1. Since the 1950s, experimentalists
have engineered ever more complex crossed molecular beam
machines to measure DCSs in more and more detail. In the early
experiments, product recoil velocities and angular distributions
were measured by time-of-flight analysis of the scattered molecules
using electron impact ionization detectors that rotated around the
beam crossing area. Enhanced detection efficiencies and resolutions
were realized with the advent of tunable lasers. For light atoms
such as H and D, the extremely high resolution afforded by
modern ‘Rydberg tagging’ and Doppler time-of-flight techniques
has made a significant impact on the understanding of elementary
chemical reactions2–4.
The ability to record DCSs took a major leap forward with the
development of ion imaging and velocity map imaging (VMI) techniques5,6. The power of these methods is derived from the ability to
image velocity vectors onto a position-sensitive detector, thereby
probing all recoil angles simultaneously7. In combination with
laser-based ionization methods, this offers the revolutionary capability to generate images of recoiling molecules that directly reflect
the state-resolved DCSs8,9. In the last decade, VMI has opened new
avenues in molecular collision research and has already proven indispensable in unravelling the mechanisms that govern half-collisions10,11, inelastic energy transfer12,13 and chemical reactivity14–18.
Although VMI and time-of-flight techniques have comparable
inherent resolutions, the image resolution attainable with VMI is
limited by the velocity spread(s) of the molecular beam(s) used7.
For uni-molecular dissociation processes—where the VMI spectrometer can be directed along the beam flight direction—the
narrow transverse velocity spread of a well-collimated beam
results in sharp images of the recoiling fragments19. In crossed
beam experiments, however, the detector needs to be directed perpendicular to the plane of the intersecting beams and the much
larger forward velocity spreads of both beams significantly blur
the image. The resulting image resolution is often not sufficient to
a
b
NO (X 2Π1/2)
cm–1
fe
11/2
50
25
9/2
fe
7/2
f
e
0
Pulsed
valve
Laser 2
Laser 1
fe
fe
fe
5/2
VMI
system
3/2
1/2
j
Initial state
Stark decelerator
Skimmer
Pulsed valve
Figure 1 | Schematic representation of the experimental set-up. a, Energylevel diagram of NO, where the energy splitting between the L-doublet
components of each rotational level is greatly exaggerated for clarity.
b, A packet of NO (X2P1/2 , v ¼ 0, j ¼ 1/2, f ) radicals with a small velocity
spread is produced by passing a beam through a 2.6-m-long Stark
decelerator. This packet scatters with neat beams of He, Ne or Ar atoms.
The state-resolved angular distribution of the scattered NO radicals is
measured using a VMI detector.
1
Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands; † These authors
contributed equally to this work. * e-mail: [email protected]
216
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|| Velocity (m s–1)
a 360
100%
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DOI: 10.1038/NCHEM.1860
b
100%
NO
Ar
COM
370
REL
380
Results
0%
10
0
–10
them. NO (X2P) þ Rg systems have emerged as the paradigm for
rotational energy transfer and have frequently been the system of
choice to benchmark new theoretical methods or experimental
approaches. Indeed, ion imaging techniques were first applied to
crossed beam experiments using NO þ Ar (refs 8,28).
300 m s–1
0%
Velocity (m s–1)
Figure 2 | Typical velocity-mapped ion images. a, VMI image of the reagent
packet of NO (1/2f ) radicals indicating the velocities parallel () and
perpendicular (⊥) to the NO propagation direction. Each pixel corresponds
to a velocity of 1.2 m s21. b, Newton diagram for the raw experimental
scattering image of NO þ Ar collisions producing the j ¼ 1/2e state,
indicating the initial NO and Ar velocities, as well as the COM velocity.
Scattering images are presented such that the mean relative velocity (REL)
is always oriented horizontally.
resolve the detailed structures in the DCS that are predicted by
theory. It has proven extremely challenging to overcome this fundamental obstacle7.
Exquisite control over the forward velocity of molecular beams
has become possible with the development of the Stark deceleration
technique20,21. By applying a burst of high-voltage pulses to a series
of electric field electrodes at appropriate times, samples of stateselected molecules with computer-controlled velocity and a
narrow velocity distribution can be produced22,23. Here, we report
the first crossed molecular beam scattering experiment that
employs a Stark decelerator to obtain full control over a molecular
beam prior to the collision, in combination with VMI to record
the scattered products. State-to-state rotationally inelastic scattering
between NO (X2P) radicals and the rare gas (Rg) atoms He, Ne and
Ar was studied. The narrow angular and velocity distribution of the
reagent NO beam resulted in scattering images with an unsurpassed
radial and angular resolution. This enabled us to fully resolve
quantum diffraction oscillations in state-to-state scattering crosssections, which are among the most detailed structures that can
occur in any DCS24.
For inelastic scattering processes the structure in DCSs is strongly
affected by two phenomena: rainbows and diffraction oscillations1.
Two types of rainbow occur for molecule–atom scattering: L-type
and rotational rainbows29. L-type rainbows, which also occur in
atom–atom scattering, are attributed classically to the cumulation
of trajectories near the maximum negative deflection angle. They
originate from the attractive part of the interaction potential and
produce a narrow peak in the DCS for smaller scattering angles.
This peak is often washed out in a quantum mechanical description
of the molecule–atom scattering process, however29,30. Rotational
rainbows, which originate from the repulsive part of the potential
and occur at larger scattering angles, correspond to maxima in
the scattered intensity caused by extrema in the angular momentum
of the outgoing molecule as a function of its initial orientation angle.
For favourable systems, rainbow structures in DCSs can be resolved
using VMI. Diffraction oscillations originate from quantum interference of different trajectories of the colliding molecules on the
PES leading to the same final deflection angle. The resulting rapid
oscillatory structure in the DCS occurs in a narrow range of
angles in the vicinity of forward scattering, which makes their experimental observation extremely challenging25–27, and thus far elusive
to ion imaging detection.
We chose the NO (X2P) þ Rg system because it is one of the
most intensely studied systems in molecular collision research,
both experimentally and theoretically. Interest in this system
stems from the open-shell character of NO giving rise to two
Born–Oppenheimer PESs with non-adiabatic couplings between
We used a crossed molecular beam apparatus, which is shown
schematically in Fig. 1b. A 2.6-m-long Stark decelerator was operated such that a packet of NO radicals was produced with a mean
velocity of 370 m s21, a velocity spread of 2.4 m s21 and an
angular spread of 0.18 (all spreads in this manuscript refer to the
width of a fitted Gaussian distribution; HWHM, half-width at
half-maximum). About 99.0% of the NO radicals that exit the
Stark decelerator reside in the X2P1/2 , v ¼ 0, j ¼ 1/2, f, referred to
hereafter as the (1/2f ), state. Labels X2P1/2 , v and j indicate the electronic, vibrational and rotational state of the NO radical. Each
rotational level is further split into two L-doublet components
that are labelled by the spectroscopic symmetry labels e and f (see
energy level scheme of NO in Fig. 1a). The packet of NO radicals
arrived temporally separated from the remainder of the molecular
beam pulse in the interaction region and was scattered with pure
beams of He, Ne and Ar under single-collision conditions at an
angle of 908 and at collision energies of 630 cm21, 485 cm21 and
450 cm21, respectively. The scattered molecules were detected
state-selectively by a (1 þ 1′ ) resonance enhanced multi-photon ionization (REMPI) scheme using two tunable dye lasers that cross each
other at 908 in the plane of the molecular beams. The first laser was
tuned to selected rotational branches of the A2Sþ X2P transition and allowed for the probing of scattered NO in a specific
final rotational ( j) and L-doublet (e or f ) state. The second laser
was tuned to subsequently ionize the NO radicals at their energetic
a
b
11/2e
100%
5/2f
Δθ
Δθ
300 m s–1
0%
c 1.0
11/2e
0.8
Intensity (a.u.)
NATURE CHEMISTRY
ΔErot
5/2f
0.6
Δv
0.4
370 380 390
0.2
0.0
0
100
200
300
400
Velocity (m s–1)
Figure 3 | Velocity-mapped ion image for inelastic collisions of NO (1/2f)
radicals with Ne atoms, probing simultaneously the final states (5/2f ) and
(11/2e ). a, Raw experimental image showing two rings that correspond to
scattering into the (11/2e, inner ring) and (5/2f, outer ring) states. b, Image
resulting from a full simulation of the experiment, using DCSs from ab initio
calculations as input. c, Radial intensity distributions of experimental (black
curve) and simulated (red curve) images for angular sections Du ¼ 408, as
indicated in a and b. The outer rim of the experimental image has a width of
Dv ¼ 3.1 m s21 (at half maximum).
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217
ARTICLES
NATURE CHEMISTRY
DOI: 10.1038/NCHEM.1860
a
NO + He
NO + Ne
NO + Ar
300 m s–1
300 m s–1
300 m s–1
40 m s–1
40 m s–1
40 m s–1
100%
b
0%
c
100%
d
0%
Figure 4 | Experimental and simulated ion images revealing quantum diffraction oscillations. a, Raw experimental ion images for the scattering process NO
(1/2f ) þ Rg NO (3/2e) þ Rg. The diameter of the Newton sphere increases for the series Rg ¼ He, Ne, Ar due to differences in reduced mass and
collision energy. b, Enlarged view of parts of the images, revealing a rapid oscillatory structure in the angular scattering distribution. c, Three-dimensional
representation of the experimental data. d, Three-dimensional representation of the angular scattering distributions derived from simulations of the
experiments, which are based on theoretical predictions for the DCSs.
threshold, thus eliminating blurring of the images due to ion
recoil31. Scattered NO radicals were detected using a standard
VMI spectrometer. Figure 2a shows a VMI image of the reagent
packet of NO (1/2f ), indicating the velocities parallel () and
perpendicular (⊥) to the NO propagation direction. Each pixel
corresponds to a velocity of 1.2 m s21.
For all three collision partners, VMI images were recorded for
NO radicals that scattered into final states ranging from (1/2e) to
(15/2e). Due to conservation of energy and momentum, scattered
NO radicals lie on a sphere with a radius determined by the collision
energy, the reduced mass of the scattering partners and the
rotational energy that is taken up during the collision. This sphere
is then ‘crushed’ onto the plane of the detector7. Figure 2b depicts
such an experimental scattering image of NO þ Ar collisions producing the j ¼ 1/2e state, together with the vector (Newton)
diagram defining the NO, Ar, centre-of-mass (COM) and relative
(REL) velocities.
To illustrate the high image resolution afforded by the Stark
decelerator, Fig. 3a shows the image obtained for scattering into
the (11/2e) level for NO þ Ne as an example. The REMPI transition
218
used to probe the (11/2e) level simultaneously probes the (5/2f )
level, and the image contains the contributions of both scattering
channels. These final states have an energy difference of DErot ¼
45.1 cm21, corresponding to a difference in circle radius of
19.0 m s21, yet the two rings pertaining to both inelastic channels
are fully resolved. The image that results from a simulation of the
experiment, which is based on the temporal, spatial and velocity distributions of both beams, is also shown. The simulations use the
DCSs predicted by high-level quantum mechanical close-coupling
(QM CC) calculations as input, which are based on the most accurate PESs presently available (Supplementary section ‘Theory’).
These ab initio computed PESs were used without any scaling
or modifications.
Diffraction oscillations mainly occur for collisions that induce
a small change in the rotational angular momentum j of NO.
Figure 4a shows the images that were recorded for NO radicals scattering into the (3/2e) state for the three collision partners. Parts of
the images are shown at an enlarged scale in Fig. 4b. A rapid
oscillatory structure in the angular distribution is observed. The
four most prominent diffraction peaks are indicated by arrows.
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NATURE CHEMISTRY
ARTICLES
DOI: 10.1038/NCHEM.1860
1
1
0
20
40
0
0
60
120
180
θ (deg)
1
Intensity (a.u.)
Intensity (a.u.)
NO + He
485 cm–1
Intensity (a.u.)
NO + Ne
600 cm–1
0
20
600 cm–1
40
485 cm–1
0
0
60
120
180
Intensity (a.u.)
NO + Ar
20
40
0
0
60
10
20
30
40
θ (deg)
1
0
0
0
θ (deg)
120
180
θ (deg)
Figure 5 | Diffraction oscillations for the scattering process NO (1/2f )1 Rg
NO (3/2e )1 Rg for Rg 5 He, Ne and Ar. Black curves: experimentally
determined angular scattering distributions. Red curves: angular scattering
distributions that result from the simulated images. Green curves:
theoretically predicted DCSs at the mean collision energies of the
experiment. The experimental angular distributions are in excellent
agreement with the simulations, but show small shifts with respect to the
theoretically predicted DCSs. These subtle effects are due to the kinematics
of the experiment and the projection of Newton spheres in the COM frame
to the detector in the laboratory frame.
The full structure is appraised best by a three-dimensional representation of the data shown in Fig. 4c. Small segments of the distributions around forward scattering are masked by imperfect state
selection of the reagent NO packet. For NO þ Ar, a signal from
trace amounts of NO in the Ar beam is visible near backward scattering angles. Figure 4d shows the corresponding three-dimensional
representations of the angular scattering distributions that result
from simulations of the experiment with the DCSs from QM
CC calculations.
Figure 5 shows the angular scattering intensity distributions
derived from both the experimental (black curves) and simulated
(red curves) images for NO þ He, NO þ Ne and NO þ Ar collisions. We use the convention that u ¼ 08 and u ¼ 1808 correspond
to forward and backward scattering, respectively. The DCSs predicted by the QM CC calculations at the mean collision energies
are shown by the green curves in the insets. The peak positions of
selected diffraction peaks, as predicted by the theoretical DCSs,
are indicated by vertical dashed green lines.
Discussion
The experimentally observed angular scattering distributions, featuring broader envelope structures with superimposed rapid diffraction oscillations, are in very good agreement with the DCSs
predicted by QM CC calculations. The peak positions of the
Figure 6 | Diffraction oscillations for the scattering process NO (1/2f ) 1
Ne NO (3/2e ) 1 Ne at a collision energy of 485 cm21 and 600 cm21.
Black curves: experimentally determined angular scattering distributions. Red
curves: distributions that result from simulated images. The theoretically
predicted DCSs at the two mean collision energies are shown by the lower
curves. The peak positions of selected diffraction peaks, as predicted by the
theoretical DCSs, are indicated by dashed vertical lines to guide the eye. The
upper and middle curves are given a vertical offset for clarity. At higher
collision energies, the diffraction peaks are more closely spaced and shifted
towards forward scattering.
diffraction oscillations, however, appear shifted by up to a few
0.18 in both the experimental and simulated distributions with
respect to the theoretically predicted DCSs (Fig. 5, insets). These
shifts stem from the projection of three-dimensional Newton
spheres onto a two-dimensional plane and from the angular and velocity spreads of the Rg atom beams (Supplementary section ‘Data
analysis and comparison to theory’). Although the appearance of
such distortions is inherent to VMI applied to crossed beam experiments, these effects are usually blurred when conventional beams
are used. The observation of these details is a further testimony of
the high resolution obtained in our experiments.
A qualitative understanding of diffraction oscillations can be
obtained from a semiclassical picture in which a matter wave
scatters on a structureless target. In a simple hard-sphere model
for atom–molecule collisions, the angular spacing Du of the oscillations is given approximately by p/kR, where k is the wavenumber
of the incoming wave and R is the radius of the target25. We applied
this model to our experimental conditions and derived R from the
effective shell radius of the NO–Rg interaction potential at the collision energy of the experiment (Supplementary section ‘Theory’).
The model predicts Du ¼ 6.08 (He), 3.48 (Ne) and 2.68 (Ar), in
qualitative agreement with the values Du ¼ 4.98 (He), 2.68 (Ne)
and 1.98 (Ar) found here.
In our experiment, the collision energy, and therefore k, can be
tuned by either changing the velocity of the reagent NO packet, or
by changing the temperature of the valve producing the collision
partner. Additional measurements on diffraction oscillations were
performed for the (3/2e) channel for NO þ Ne at a higher collision
energy of 600 cm21. The resulting angular scattering distribution is
shown in Fig. 6 and compared to the angular distribution at a collision energy of 485 cm21. At the higher collision energy, the diffraction peaks are closer spaced and shifted towards forward
scattering, in full agreement with the QM CC calculations.
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In conclusion, the resolution achieved here demonstrates that
velocity-mapped images in crossed beam experiments can now be
obtained with a sharpness that has thus far been reserved for
uni-molecular processes. This is illustrated by resolving quantum
diffraction oscillations for inelastic scattering between NO radicals
and rare gas atoms, which is one of the most intensely studied
and well-known systems in the field of molecular collision dynamics.
Near-exact theoretical predictions for collision cross-sections exist for
these benchmark systems, and our experiments are fully consistent
with the most accurate calculations presently possible. The success
attained here implies that for less well-known systems, for which
experimental validation of the PESs and of the approximations
used to calculate them is still necessary, our approach will challenge
theory with an unprecedented level of detail.
In combination with the state-of-the-art imaging techniques
presently available, measurements of DCSs with an angular resolution better than 0.18 and a velocity resolution at the per mille
level appear realistic in the near future. This will open up new possibilities for the investigation of inelastic, reactive and gas–surface
scattering phenomena. For instance, our resolution will allow for
measurements of DCSs at low collision energies, where the scattering image diameter becomes very small. This will result in unique
probes of quantum phenomena such as scattering resonances34
and how they affect the angular distribution of the scattered molecules. The measurement of DCSs for collisions between two
state-selected molecular beams is another exciting avenue. By the
conservation of energy and momentum, collision-induced
rotational energy transfer that occurs in both molecules from individual encounters will lead to multiple concentric rings in the scattering image. Resolving these rings will yield a kinematically
complete picture of bimolecular scattering processes. Ultimately,
using monochromatic beams as employed here, the dominant
factor limiting image resolution will become the resolution of the
imaging system itself.
Methods
The beam of NO radicals was produced by expanding 5% NO seeded in xenon with a
backing pressure of 0.5 bar through a home-built pulsed valve32. The Stark
decelerator was operated using a phase angle of w0 ¼ 08 and selected a packet of NO
radicals with a mean velocity of 370 m s21 from the molecular beam. The velocity
and angular distributions of the packet were derived from trajectory simulations
of the deceleration process and had widths of 2.4 m s21 (yielding a speed ratio of
S ¼ 125 for the beam) and 0.18 (the transverse velocity spread amounts to
0.7 m s21), respectively. The packet of NO radicals intersected with the central axis
of a beam of He, Ne or Ar at a distance of 69 mm from the exit of the decelerator.
The beams crossed at a 908 angle of incidence. The Rg atom beams were produced by
expanding neat He, Ne or Ar with a backing pressure of typically 3 bar into vacuum
using a commercially available pulsed valve (Jordan). The beam source was
located 110 mm from the interaction region and the Rg atom beam was passed
through a 2-mm-diameter skimmer and a 3-mm-diameter collimator that were
located 87 mm and 52 mm from the interaction region, respectively. The arrival time
distribution of the NO packet in the interaction region had a width of 11.8 ms,
whereas the Rg atom beams had a much longer temporal duration, in the
60–80 ms range.
Two pulsed dye laser systems were used. The first dye laser (226 nm, bandwidth
0.08 cm21, 5 ns pulse duration, 3 mm diameter) was used to excite NO to the
electronically excited A2Sþ state by inducing the (0–0) band of the A2Sþ X2P
transition. The bandwidth of this laser was sufficiently large to cover the Doppler
shifts in the spectral profiles due to the recoil velocities of the scattered NO radicals.
The second dye laser (328 nm, bandwidth 0.06 cm21, 5 ns pulse duration, 4 mm
diameter) was pumped by a second Nd:YAG laser, and was used to subsequently
ionize the NO radicals just above the energetic threshold. Both lasers were tightly
focused into the scattering volume to offer a small ionization volume, and attenuated
to 3 mJ and 6 mJ for the first and second colours, respectively. This was done to
prevent Coulomb repulsion effects from excessive signal levels and to prevent direct
(1 þ 1) REMPI by the first dye laser only. It was verified that all ionization signal
disappeared when blocking either of the two laser beams.
The VMI detector consisted of a repeller plate, an extractor plate and a grounded
time-of-flight tube (length, 550 mm). Voltages of 1,000 V and 758 V were applied to
the repeller and extractor plates, respectively, to ensure velocity mapping conditions.
No time-slicing was used. Images were recorded using a charge-coupled device
camera (PCO Pixelfly 270XS, 1,391 × 1,023 pixels) and transferred to a PC for
220
DOI: 10.1038/NCHEM.1860
subsequent averaging and data analysis. Event counting and centroiding were
employed in the data acquisition software (DaVis, LaVision). The electric field in
the ionization region was insufficient to noticeably mix the NO (X2P) L-doublet
levels or to orient the NO radicals. Mostly final states of e parity were probed,
as rotational energy transfer in NO þ Rg collisions is subject to so-called parity
pairs33, that is, the DCS for scattering into a final state with rotational angular
momentum j and parity e is identical to the DCS for scattering into a final state
with rotational angular momentum j þ 1 and parity f, apart from a scaling factor
close to unity.
The experiment ran at a repetition rate of 10 Hz. Images for a given final
product-state were first recorded by overlapping both the atomic and molecular
beams in time. The Rg atom beam was then delayed with respect to the NO packet in
an alternating fashion, such that only background signals were recorded. The
scattering image was inferred from the signal intensity difference of both images.
We observed that the image can move over the detector surface due to slow
charging-up effects of the ion optics assembly. In addition, the Jordan valve used
to generate the atomic beam tended to heat up during experimental runs,
causing a gradient in collision energy and COM location. Typically, between 1 × 105
and 1 × 106 shots were summed and it was found to be a good compromise between
statistics and blurring of the image due to drifting effects. Each image was recorded
within a single measurement day to further reduce the risk of reduction in image
resolution. No collision-induced polarization effects were recorded for transitions
that induced a small change in rotational angular momentum j. A possible collisioninduced polarization is expected to be largely destroyed by nuclear
hyperfine depolarization.
Received 25 September 2013; accepted 20 December 2013;
published online 9 February 2014
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Acknowledgements
This work is part of the research programme of the Foundation for Fundamental Research
on Matter (FOM), which is supported financially by the Netherlands Organization for
Scientific Research (NWO). S.Y.T.v.d.M. acknowledges support from the NWO via a VIDI
grant, and from the European Research Council via a Starting Grant. The authors thank D.
Parker for discussions and general support and the Fritz-Haber-Institute in Berlin for loan
of equipment. The authors thank K. Liu for discussions about the analysis of scattering
images and J. Kłos, H. Cybulski, B. Fernández, Y. Endo and M. Alexander for providing
their PESs. The authors thank B. Nichols and S. Chefdeville for assistance during part of the
measurements, R. Rammeloo for developing software to simulate ion images, and A. van
Roij, L. Gerritsen, C. Berkhout and P. Claus for expert technical support.
Author contributions
The research project was conceived and supervised by S.Y.T.v.d.M. The experiments were
carried out by A.v.Z. and J.O. Data analysis and interpretation was performed by A.v.Z.,
J.O., S.N.V. and S.Y.T.v.d.M. Theoretical calculations were performed by J.O., A.v.d.A. and
G.C.G. The paper was written by S.Y.T.v.d.M. with contributions from all authors. All
authors contributed to discussions about the content of the paper.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints. Correspondence and
requests for materials should be addressed to S.Y.T.v.d.M.
Competing financial interests
The authors declare no competing financial interests.
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