studies in coulomb explosion imaging in coulomb explosion imaging ... a walk along the conformation...
TRANSCRIPT
דד
Studies in
Coulomb Explosion Imaging
Thesis presented for the degree of
"Doctor of Phi losophy"
by
D r o r K e l l a
Submit ted to the Scientific C o u n c i l of
the Weizmann Institute of Science
August 1994
׳ ׳ ל ^ 5־ 3 ע / ^ כ־׳ -י /
Studies in
Coulomb Explosion Imaging
Thesis presented for the degree of
"Doctor of Phi losophy"
by
D r o r K e l l a
Submit ted to the Scientific C o u n c i l of
the Weizmann Institute of Science
August 1994
ד !r»,< vjrדי. ׳ ליטי״ד ס מ
This work was done under the supervision of
Professor Zeev Vager Department of Particle Physics
Acknowledgments
I wish to thank all the people who have helped me and made my time pleasant
during the course of my Ph.D. Specifically Prof. Zeev Vager for his guidance. To
Prof. Ron Naaman for his valuable advice. Dr. Daniel Zajfman and Dr. Oded
Heber who worked with me and guided me through most of the experimental work.
To my fellow student Jackob (Yasha) Levin for his cooperation in the development of
the iterative self learning method and general help in all aspects. To my other fellow
students Haim Feldman, David Ben-Hamu, and Oleg Lourie for their help when I
needed it. To Maurice Algranati who built the electronics with Bina Rossenvaser and
Nissan Altstein. To Lome Levinson for his valuable advice and help in the building
of the data acquisition system and to all the rest of the people in the accelerator
laboratory. Last and very much not least I wish to thank my wife Elisabetta boy
Eyal and baby Iris who made it all worthwhile.
3
Contents
1 Introduction 1
1.1 Coulomb Explosion Imaging 1
1.2 The Coulomb Explosion Imaging(CEI) process and interpreting results 5
1.2.1 Introduction 5
1.2.2 C E I Transformation 5
1.2.3 Coordinates 8
2 Coulomb Explosion Imaging system 10
2.1 Introduction 10
2.2 Experimental Set Up I 1
2.2.1 General Description 11
2.2.2 Laser photo-detachment 13
2.2.3 Detection Chamber 14
2.2.4 Detector 15
2.2.5 Detector Electronics 21
2.3 Control , Data Acquisi t ion and Analysis 26
2.3.1 Synchronization 26
2.3.2 Data Acquisi t ion 27
2.3.3 Data Handling 29
2.3.4 Analysis 30
2.3.5 Error Estimate 34
2.4 Conclusion 39
3 Triatomic Clusters 4 0
3.1 Introduction 40
i
3.2 C 3
4 1
3.2.1 Introduction 41
3.2.2 Experimental 43
3.2.3 Analysis 43
3.2.4 Results and Discussion 46
3.3 B 3 49
3.3.1 Introduction 50
3.3.2 Experimental 50
3.3.3 Analysis 51
3.3.4 Results and Discussion 53
3.4 Comparison between B 3 and C 3 CEI and Discussion 54
4 Tetra Atomic Clusters 5"7
4.1 C 4 linear or rhombic? 57
4.1.1 Introduction 5*
4.1.2 Experimental 59
4.1.3 Analysis 59
4.1.4 Results and Discussion 6/
4.1.5 Conclusion 69
4.2 First measurement of the structure of tetra-atomic boron cluster . . . 70
4.2.1 Introduction • 70
4.2.2 Experimental 71
4.2.3 Analysis 71
4.2.4 Results and Discussion 75
4.2.5 Conclusion 76
4.3 Conclusion and comparison of C 4 and B 4 CEI measurements 77
ii
5 A detailed study of conformations in the ground state of C H j 79
5.1 Introduction 79
5.2 The CEI experiment of the Ground State of the Methane ion 80
5.3 Choice of coordinates, density functions and errors 81
5.4 Comparison of the measured and the simulated densities 83
5.5 The "reaction path" - a walk along the conformation density ridges. . 86
5.6 Concluding remarks 93
6 Conclusion 94
A C 3 history 98
111
1 Introduction
1.1 Cou lomb Exp los ion Imaging
When describing a quantum mechanical system as a solution of the Schrodinger
wave equation two properties are defined. One is the energy level of the system and
the other is the wave function related to that level. Molecules are treated as such
quantum mechanical systems and any attempt to understand their structure must
take into consideration both properties. Various methods have been devised in order
to calculate these systems and in all of them some form or other of approximation has
been applied. The results of these calculations need to be compared with experiments
but up to now, most experiments which have been carried out on free molecules deal
only with energy levels, and it seems that part of the story is missing. To fill this gap
a new method called the Coulomb Explosion Imaging (CEI)[1] was devised more than
a decade ago at both Argonne National Laboratory (U.S.A.) and at the Weizmann
Institute of Science (Israel).
Should a molecule be suddenly stripped from its electrons, the bare positive
nuclei would repel each other due to the Coulomb interaction. If we were to measure
the correlated velocities of these outgoing nuclei it would be possible to reconstruct
the initial molecules nuclear configuration before stripping. Based on this rather
simple idea a system for measurement of molecular structure was devised(figure
1). In this method a fast molecular beam is prepared in an accelerator. Molecules
from the beam traverse a very thin foil and through collisions with the atoms of
the foil lose some or all of their electrons. Due to the repelling Coulomb force the
positive fragments fly apart. A few meters away from the foil, the distance between
the fragments has grown from a "micro scale" of several Angstroms to a "macro
1
"micro scale"
Detector
Figure 1: Schematic i l l u s t r a t i o n of C o u l o m b E x p l o s i o n I m a g i n g .
scale" of a few centimeters. At this stage, the velocities of the fragments of the
exploding molecule are measured by a time and position sensitive detector. Using
the knowledge of the Coulomb force which caused the explosion, it is possible to
reconstruct from the measured velocity the initial positions of the fragments in the
molecule before it exploded.
The time which it takes for the molecule to lose its binding electrons is approx-
imately 10 - 1 7 seconds. This time span is much shorter than rotational or vibrational
periods which are usually in the order of 10 - 1 2 and 10~14 seconds respectively. Thus
the measured fragments are, in fact, "snapshots" of the parent molecule. Applying
this measurement to many molecules, one molecule at a time, leads to a large sample
of "snapshots" of the nuclear position coordinates of a certain molecular system. If
the molecular population was prepared in a single quantum mechanical state, the
distribution of nuclear position coordinates of this large sample should converge to
the probability density function of these coordinates. This function may be mathem-
atically described as the nuclear wave-function squared, and since we do not measure
electrons, integrated over the electronic coordinates. Thus a direct measurement may
be taken of a spatial property of the wave-function of the molecular system.
It is at this stage that we must point out that a completely rigorous treatment
of the CEI process should deal with the Coulomb explosion as a propagation of the
molecular wave-function in a Coulomb potential Hamiltonian and the measurement,
or sampling, of this function should be assumed only in the asymptotic region. This
treatment was performed on diatomic and several triatomic examples[2] and it was
shown that for cases where there is no classical ambiguity in interpretation of CEI
results, the classical treatment is a good approximation to the quantum mechanical
one. In cases where there is classical ambiguity, i.e. more than one classical tra-
jectory reaches the same asymptotic result in velocity space, a quantum mechanical
treatment should relieve the ambiguity in the case of a single state measurement. In
our measurements there is no single state selection, therefore the information gained
by a classical treatment is as good as we can get today.
In this dissertation a rather complete picture of the CEI will be presented. In the
second section of this chapter the general concept of the imaging will be described.
The following chapter deals with the experimental method both from the technical
point of view and the analysis tools used in acquiring the CEI results. The measure-
ments and analysis of the carbon and boron trimers and tetramers will be presented
in the third and fourth chapter. The last chapter will be devoted to a measurement
3
that exhibits the full power of the CEI , CH4 .
The two different types of molecules which will be presented display two extreme
regimes for CEI. In the case of clusters the CEI image which arises in velocity
space is very different from that in nuclear position space. Moreover problems such
as ambiguity in interpretation of part of the features and difficulty in acquiring a
large sample for good statistics make this a poor case for CEI. Nevertheless, a new
algorithm for the analysis of such data was devised and important information was
acquired from the cluster measurements. In opposition to the cluster case the C H !
is an ideal case for CEI . As will be explained, the velocity space picture is very
similar to the one in position space, there is very little ambiguity and there is no
real problem in collecting a large sample. To add to the beauty of this measurement
this is a molecular ion which undergoes internal rotation and the CEI resolves this
internal motion. The results, which will be described, are very similar to theoretical
predictions but also contain a very interesting conflict with those calculations.
4
1.2 The Coulomb Explosion Imaging(CEI) process and inter-
preting results
1.2.1 Introduction
A CEI measurement results in data which are sets of correlated 3n velocity coordin-
ates (V-space) for the n fragments of the molecule which exploded. In order to
interpret these results as a probability density function of the position coordinates
of the molecule under study(R-space) we must first understand what are the special
features of the CEI transformation. Furthermore, we must tackle questions such as
how to display these results, or in which coordinate frame to work, in order to ex-
tract useful information. For example, the analysis of diatomic results is best done
by looking at the inter-atomic velocity and from that almost immediately we can in-
fer the inter-atomic distance through the Coulomb potential. When analyzing multi
dimensional cases we must find as good a representation and in what follows we will
deal with this matter.
1.2.2 CEI Transformation
The CEI may be described as a transformation which maps coordinates in R־space
to coordinates in V-space. In what follows we will describe this transformation. A
molecule that undergoes Coulomb explosion reacts to a force which is dependent on
the inverse distance squared between fragments. Therefore this transformation is
nonlinear. For the single dimensional case we can see in figure 2 the transformation
from distances in R-space to velocities in V-space on a Coulomb 1/r potential. There
are two general features which are important in the understanding of CEI. One is
the inversion of coordinates, long bond-lengths are imaged to "short1' velocities and
5
>
1.4
1.2
1
0.8
0.6
0.4
-
>
1.4
1.2
1
0.8
0.6
0.4
>
1.4
1.2
1
0.8
0.6
0.4
>
1.4
1.2
1
0.8
0.6
0.4
>
1.4
1.2
1
0.8
0.6
0.4 . i i . . i . . , ו , . . , , ,
2 4 6 8 10 R
Figure 2: T h e mapping f r o m r to v of a 1/r potential i n a r b i t r a r y units.
vice versa. The other feature may be described as a Jacobian in the transforma-
tion. Notice that two equal size regions in R, 0.5-1. and 2.-2.5, are transformed to
very different sizes of regions in V , 1.-1.4 and 0.63-0.71 respectively. In this simple
example it is evident that the transformed image can be very different from the ori-
ginal. Another fact is that since any measurement device has finite resolution, the
quality of measurement of R coordinates is different for different regions of R due to
the different compression of the image.
The features which are exhibited by a single dimensional CEI are important but
they do not introduce any special difficulty in the interpretation of results. When
dealing with more than two atoms, i.e. multi dimensional events, similar effects
become a true disturbance. The two worst features which the CEI imaging of multi
dimensional cases produces are ambiguity[3] and infinite compression. Figure 3
shows the effect of CEI on the bending angle of a triatomic molecule. The upper
curve belongs to a H X H molecule, where X is a heavy nucleus, below is a triatomic
6
80 ו>
160
140
120
100
80
60
40
20
0 0 20 40 60 80 100 120 140 160 180
Figure 3: The m a p p i n g f r o m b e n d i n g a n g l e i n R t o b e n d i n g a n g l e i n V f o r t r i a t o m i c
m o l e c u l e s a n d c l u s t e r s
homonuclear cluster X 3 . Both figures display infinite compression and ambiguity but
the extent of these effects is dramatically different. The turning point for the X H 2
bending is at approximately 50° while that of the cluster is at 80°. The ambiguity
which is depicted, i.e. two different QR lead to the same 0 y , is not a true ambiguity
since there are two other coordinates of which one is the bond-length and the other
Coulomb energy. In order to recreate the same V-space angle and energy we may
select two different angles, one from either side of the turning point, but then to
recreate the same Coulomb energy we will need to select different bond-lengths. In
this case physical constraints can lift the ambiguity. For example a nearly linear
(180°) V-space configuration can come from a nearly linear configuration in R or
from almost 0° angle in R. But in order to recreate the 0° angle situation we need
nearly infinite bond-lengths in order to keep the Coulomb energy at a given value.
At this stage it is possible to see the large difference between the two cases. A 50°
turning point would give ambiguous results which are, usually, unphysical since the
ambiguous result is with very small angle. On the other hand an 80° turning point is
very bad. The region of 40-60 degrees is ambiguous with 100-120 degrees and both
are perfectly physical.
As for the infinite compression, one can see that for the X H 2 case the region
between 40° to 60° in R is imaged to a very small region around 80° in V. Thus a
result of 80° in V-space may be interpreted as anything in that small region. On
the other hand, in the X 3 case, the region between 60° and 100° is mapped to 60° in
V-space. Thus for clusters which are of triangular shape there is not much more to
extract than the fact that they are in the region of 60°-100°.
Clearly, with many coulomb interacting particles the final velocities and directions
tend to spread evenly in space. The three body coulomb interaction has shown the
typical folding characteristics in the ambiguity. It is fortunate that with one or two
heavy atoms and several light atoms the conversion is single valued in almost all of
phase space. For clusters, the ambiguous phase space becomes quite large.
Finally, using the same considerations as in the diatomic and triatomic cases it
is possible (and most recommended) to, a priori, learn the quality of results which
may be available from any CEI measurement.
1.2.3 Coordinates
A crucial part of analyzing CEI data is the decision on the set of coordinates in which
the results will be expressed. In order to completely define the positions of the nuclei
in a n atomic molecule, 3n coordinates are needed. Since the internal structure is
invariable under displacement of center of mass and under change of orientation 3n-
6 internal coordinates are needed to completely define the internal structure of a
8
molecule. Using the same arguments we need that many coordinates to completely
describe the correlated inter-atomic velocities of the exploding fragments. The six
redundant coordinates, C O M velocity and orientation, are used as important checks
for the quality of the data and data acquisition system.
In general, any set of orthogonal coordinates may be used but it is advisable to
use a set which enhances the features which are in question. Another consideration in
selection of the coordinates is the symmetry of the problem. If there are permutational
equal atoms, such as in cluster measurements, we must use symmetry coordinates of
either the appropriate point group or a set of coordinates generated by permutation
symmetry (see, for example, [4] and references therein). In this case each event
generates as many points in the symmetry coordinate space as there are permutations.
9
2 Coulomb Explosion Imaging system
2.1 Introduction
In the following we present the system used for the CEI measurements and specifically
the system at the Weizmann Institute. In the first part of the chapter the hardware
is described in detail. The main component of the setup is the Weizmann Institute
Pelletron M U D tandem accelerator. Other major features which will be discussed
are the pulsed laser photo-detachment system which allows selective neutralization
of accelerated ions for Coulomb explosion imaging, and a new type of multi-particle
detector used in this system will be presented. It allows the simultaneous measure-
ment of position and time of molecular fragments with a spatial resolution of ~ 0.1
mm and ~ 100 ps time accuracy.
The second part contains a presentation of the data acquisition and the methods
by which the relevant experimental information is extracted from this data. Both
hardware and software used for dealing with this task are described and finally an
error analysis of results is presented.
10
2.2 Exper imenta l Set U p
2.2.1 General Description
Fig. 4 depicts the experimental setup of the CEI at the Weizmann Institute. A
negative molecular ion beam is generated by a cesium sputter source (Hiconex 834).
Preparation of the beam for injection into the 14 UD Pelletron includes extraction
by a positive 10 to 17 kV potential followed by chopping using electrostatic deflectors
(Chopper I) to produce pulses of 1 to 3 /JS duration at a repetition rate of 25 Hz.
The ion pulses are further accelerated by an additional 90 kV and mass selected by
a 90° magnet (Magnet I). The resulting pulses of ions are chopped again (Chopper
II) to a time width r of 100 to 400 ns. The negative ion pulses are then injected into
the HUD Tandem Pelletron accelerator and are accelerated toward the high voltage
region (HV terminal) to an energy of 6 to 12 MeV. The time width r is selected in
such a manner that the corresponding length of an accelerated ion pulse is smaller
than the 3 meters of the field free region at the H V terminal.
When the negative ions reach the H V terminal, a pulse from the laser is fired down
the accelerator tube in order to photo-detach the extra electron. The neutralized
molecules drift through the second part of the tandem accelerator toward Magnet
II, which is used to purge any charged ion left in the neutral beam. A set of slits
mounted upstream of Magnet III collimates the beam to a rate of about one molecule
per pulse on the detector. The neutral molecules are then stripped from electrons by
passing through a thin Formvar foil, and the fragment atomic ions emerging from the
foil repel each other via their Coulomb interaction, a process called the "Coulomb
Explosion". The fragments are then charge and mass separated by Magnet III and
collected on a multi particle position and time sensitive detector. The time and
position pulses from the detector are digitized and analyzed by a dedicated computer.
11
liming Unit
ative ion source Chopper I
ז15 K V 90 K V
Negative Ions
(100-400 nsec pulses),
14UD Pelletron-
Positive Ions
Foil Stripper
\ ] - -*-Laser Power Meter
-Magnet II
Neutral Molecules ־
-Magnet III
To Timing Unit
Detector— "* * D a t a A c q u i s i t i ° n a n d Control ו To Network
Figure 4: C o u l o m b E x p l o s i o n I m a g i n g system a t t h e Weizmann I n s t i t u t e
12
2.2.2 Laser photo-detachment
The conventional methods of electron stripping from projectiles in the H V terminal
of a tandem accelerator via gas or thin foil strippers is not suitable for the CEI
method. In the case of foil stripping, the molecules would simply break up, and
the gas stripping method is also unsuitable since the collisions tend to insert energy
into internal degrees of freedom of the molecule in an uncontrolled way. In order to
control the process of electron stripping, a new method has been developed, based
on laser photo-detachment inside the tandem accelerator[5, 6].
The laser beam from a Nd:YAG or N d : Y A G dye lasers (Quantel Datachrome
5000 system) or Excimer pumped dye laser (Lambda Physics LPX315+LPD3000) is
introduced into the accelerator through a window mounted on top of Magnet I, (see
Fig. 4) and is aimed co-linearly with the accelerated beam. The distance between
the window and the H V terminal stripping region is about 30 m. The laser pulse
(~ 10 to 30 ns long) is timed relative to the ion pulse so that they overlap only in
the H V terminal (see Fig. 4). At the other end of the accelerator, a movable prism
can deflect the laser light to an energy meter (Precision RJP 375) which is used for
aligning the laser beam through the accelerator.
In order for the laser neutralization process to be efficient, the ratio between the
photo-detachment probability (P p ) and the neutralization probability due to residual
gas collisions (P 3 ) in the high voltage terminal has to be large. Assuming a cross
section for collisional neutralization of ~ 1015־16-10־ cm 2 and the H V terminal
vacuum better than ~ 107־ Torr, the probability for a molecule to be stripped along
the 3 meter flight path is P p ~ 103־4-10־. On the other hand, the typical photo-
detachment cross section at a wavelength of 532 nm (Nd-YAG second harmonic) is
cm 10-18-1016־ ~ 2 . A typical laser output is 5 to 100 mJ per pulse. For the lower
13
MAGNET III (CHARGE SELECTION MAGNET)
ROTATEABLE JOINT AND BELLOWS
DETECTORS CHAMBER
Figure 5: D i a g r a m of t h e e x p l o s i o n a n d d e t e c t i o n c h a m b e r
limit of 5 mJ, the probability for photo-detachment is P p ~ 10~ 2-1. It is important
to point out that although in an ideal case P p / P s 1 <§נ, when the power output of the
dye laser is very weak or at photon energies close to thresholds, where the photo-
detachment cross section drops considerably, the two processes may compete, and
measurements with and without laser are needed for better accuracy.
2.2.3 Detection Chamber
The neutralized molecules emerging from the accelerator are collimated by a set
of slits, and impinge on a very thin (approximately 1 / /g/cm 2 ,~ 100 A) Formvar
foil[7, 8, 9]. After the Coulomb explosion is initiated by the foil, the fragments drift
into the detection chamber (Fig. 5) in a cone of approximately 10 to 20 mrad. The
charge separation is made using a 30 cm wide magnet (Magnet 3) which is located
15 cm below the stripping foil (see Fig. 5). Since the distance between the various
exploding fragments in the magnetic field region is of the order of few mm, one can
14
assume that all the fragments are affected by the same field. The magnet deflects
the different fragments according to their charge and mass, and by setting the angle
of deflection of the different charge states to be larger than the Coulomb explosion
cone, the identity of each fragment can be easily deduced from its position on the
detector.
The lower part of the chamber is rotateable in a plane perpendicular to the field
of Magnet III and is made of two parallel 6" tubes with a detector at each end. The
angle of rotation of the detectors and the magnetic field can be changed in such a
way that the desired charge states will fall on the detectors. The distance from the
stripping foil to the detectors is 2195 mm.
2.2.4 Detector
A basic part of the CEI system is the detector which enables three dimensional ima-
ging of multi-particle events. After the dissociation in the thin target, the velocities
of the fragments can be described as follows. The Coulomb explosion velocities are
added vectorially to the beam velocity, typical Coulomb explosion and beam kinetic
energies are ~ 50 eV/amu and ~ 250 KeV/amu respectively. The relative velocity
change from the initial center of mass velocity is thus ~ ^50/250 x 103 = 0.014.
Hence, good time and position resolution are needed for the detector. It can easily
be seen that the larger the distance between the foil and the detector system, the
smaller is the constraint on the absolute time and position accuracy of the detector.
In the past, a system of individual solid state detectors was used to scan the multi-
dimensional coincidence space[10]. Later, multi-wire position and time gas detectors
were developed[ll, 12] and used at the Argonne National Laboratory Coulomb ex-
plosion system. In this system, the flight path between the target and the detectors
15
ז זAluminized Mylar
2 stage MCP
Phosphor screen
and Wires
Window
Figure 6: Schematic d i a g r a m of d e t e c t o r l a y e r s
is 6 m. The fact that, in the present case, the beam has a defined duty cycle (25 Hz)
allowed a different approach to the detection system. It is based on a micro-channel
plate which transforms the impact of each fragment into a scintillation on a phosphor
screen and an electronic signal on wires for timing. The screen is then imaged by
a C C D video camera and read out digitally. The excellent time resolution possible
from this system allows the reduction of the explosion path length to about 2 m.
The simplicity of the two dimensional position analysis is another attractive feature
of this new detector system.
For each molecular fragment, the detector extracts the time of arrival and position
for all the fragments that hit the detector simultaneously. The limit on the number
of fragments measured is set by the condition that each fragment will induce a signal
on a separate set of wires. This limitation sets the maximal number of fragments
for the small detector at 6 while the number for the large detector is much higher
due to both it's size and cross-wire configuration. The detector is made of four
16
different layers (Fig. 6): The first layer is a foil used as an ion-electrons converting
stage, ejecting a few electrons for each particle hit. The second layer is an electron
multiplier micro-channel plate (MCP) in a chevron assembly. The third layer is a
multi wire anode consisting of independent anodes for fast timing, and the last layer
is a phosphor screen emitting visible photons for position imaging. The operations
of the different layers will be described in more detail in the following.
The first stage is made of an aluminized Mylar foil 1.5 jam thick, the aluminum
coating facing the M C P . Without additional special coating, this stage generates
about 5 to 20 electrons for each ion hit (depending on the stopping power). Higher
amplification can be achieved by coating the aluminized side with Csl . A typical Csl
coating of 3000 A results in electron multiplication of 50 to 100 electrons for each
ion hit. The first foil increases the efficiency of single ion detection from ~ 50% for a
bare M C P ion detector to ~ 100% for this foil-MCP assembly. A voltage difference
of 200 V is maintained between the aluminized foil and the M C P in order to extract
the electrons with good focusing properties and optimal kinetic energy for the second
multiplication stage.
The second layer, located 2 mm from the first one, is an electron multiplier
consisting of two micro channel plates in a chevron assembly. Two types of M C P
dimension are used: a 42 mm active diameter and a 77 mm active diameter (Hama-
matsu models F2225 and F2226). A typical potential of about 1700 V is maintained
between the top and the bottom planes of the M C P . An additional electron multi-
plication of about 104 to 105 is achieved at this stage, and more amplification can
be obtained by increasing the M C P voltage up to 2000 V . As a result of these two
first amplification stages, for each fragment hitting the detector, a bunch of ~ 106
fast and well focused electrons is created with a time width of ~ 1 ns.
17
The third layer is located 3 mm after the MCP and consists of an array of
independent thin (50^m diameter) conductive wires used as fast anodes, welded to
a common printed board. The distance between wires is 0.83 mm for the small
detector (48 wires). The configuration of the large detector is slightly different. It
has two layers of wires instead of one, one above the other, at a distance of 50/.m1
between layers. The distance between wires in each plane is 1mm and the directions
of the wires in one plane are orthogonal to those of the other plane.
The use of such an XY configuration enables a much higher efficiency of the
detector. The small, unidirectional wires, detector is limited by the probability of
two fragments hitting the same wire. In this case there is no possibility of extracting
timing information for such an event. For the large detector, should such an event
occur, one can try to extract the timing of each one of these fragments from the other
set of wires. Thus the large detector is limited by the probability that two fragments
hit the same wire and that at least one of them is correlated with a third fragment
that hit the appropriate wire on the second plane.
Since the bottom part of the MCP is about 2000 V below ground potential, each
electron bunch is accelerated towards the wire array, which are coupled through an
impedance of 82 0 to the emitter of grounded base transistors. This electron bunch
induces a fast signal on the wire array, which is used as the timing signal for each
fragment hit. The focusing conditions are adjusted so that each electron bunch can
induce signals on several wires simultaneously. A typical histogram for the number
of wires which produce a signal for a single ion hit on the small detector(i.e. a mono
atomic beam), is shown in Fig. 7. It can be seen that an average of 3 wires produce
signals per ion hit.
By using a weighted mean of all the wire signals in the final timing analysis,
18
500ב»E 3
'3000
2500
2000
1500
1000
500
° 0 2 4 6 8 10 number of wires hit per event
Figure 7: N u m b e r of w i r e s h i t f o r events w i t h a s i n g l e f r a g m e n t o n t h e s m a l l d e t e c t o r
optimal time resolution is obtained. Another timing signal which serves as a common
signal for all the fragments from an event, is taken out through a capacitor from the
last stage of the M C P .
The anode wires are located 0.2 mm from an isolated P-20 phosphor screen.
Each electron bunch hitting the screen generates a light spot of ~ 1 mm diameter.
These are recorded by a C C D camera (Javelin Electronics model JE-7242X) looking
at the screen through a sealed window. The video output of the camera is connected
to the data acquisition system and also produces the master clock for the system
timing (see Fig. 4). The C C D output information is made of 625x215 pixels with
their intensities for each half frame (the camera produces interleaved output). The
position resolution is as good as one pixel and with standard lens magnification, this
corresponds to 100 /um in the detector plane. The number of pixels with intensities
above the C C D noise for each ion hit is typically ~ 60. An example of a distribution
for the number of pixels that are generated by the fragmentation of a 12 MeV beam
19
Number of pixels per event
Figure 8: T h e distribution of number of pixels per event. T h e i n i t i a l beam was 0 2
at 1 2 M e V . T h e distribution shown includes single and coincidence counts on the
detector.
of 02 molecules, is shown in Fig. 8.
Two peaks can be seen: The first one corresponds to single ion events (when the
second ion did not hit the detector due to a different charge state), while the second
peak is from double ion hits. The number of pixels depends mostly on the focusing
condition, the lens position and magnification. Fig. 9 shows a two dimensional picture
of the accumulated fragments from the Coulomb explosion of B 2 , as seen on the
phosphor screen. Two circles can be seen which correspond to different charge
states as separated by Magnet III.
In summary, each multi-particle hit on the detector results in fast timing signals
coming from the wires and position signals from the C C D output. A l l this inform-
ation is transferred to the data acquisition system for accurate 3D imaging of each
event (see sections 2.3.2 and 2.3.3 ).
20
6 55
50
45
40
נ 5
30
25
20
5ז
י 0 2 65־ 60 55 50 45 40 35 30 25 0
Figui'e 9: Two dimensional contour plot for the distribution of accumulated f r a g -
ments f r o m the Coulomb explosion of B2• T h e distribution shown is for the coincid-
ence of two B + g where q=3 ( r i g h t side) and q—4 (left side). T h e beam energy zuas
12 M e V .
2.2.5 Detector Electronics
The most limiting factor for the overall resolution of the three dimensional image is
the accuracy of the time measurement: A resolution of 100 ps in time is equivalent
to an error of ~ 1 mm in position, which represents a few percent for a typical
event where the distances between the fragments on the detector are of the order of
a few centimeters. In order to accurately process timing information coming from
the wires of the detector, a special electronic set-up combined with a Time to Digital
Converter (TDC) system was built(see Fig. 10). An ion hitting the detector creates
two types of signals: The first, coming directly from the last stage of the M C P (see
section 2.2.4) is used, after appropriate delay, as a common stop (STOP) for all
(i.e. all fragments) timing measurements. The second type is the group of signals
induced on the anode wires. These signals are amplified and reshaped for optimum
timing determination. A common threshold is set for all the wire outputs using fast
21
WIRE
F R O M M O P -AMPLIFIER
+CFD
LINE RECEIVER
DISCRIM-
INATOR II DISCRIM-
INATOR = D Q =
A A M .
DISCRIM. L E V E L
D E T E C T O R ELECTRONICS
F.F
F.F
"STOP" LOGIC
C O N S T A N T
C U R R E N T G E N .
C O N S T A N T C U R R E N T G E N .
C H A R G E A/D
C H A R G E A/D
M A I N ELECTRONICS IN " C A M A C " C R A T E
Figure 10: O p e r a t i o n a l d i a g r a m of one c h a n n e l of t h e t i m e t o c h a r g e system.
discriminators; the crossings of this threshold result in the initialization of two (one
for cross on rise and the other for cross on fall) timing signals for each wire. The
threshold is set so that they are above the noise and the signals cross it at the point
of steepest ascent. Each crossing of the threshold enables a constant current source
into a charge A D C . The gates are closed by the STOP signal from the M C P which
is common to all wires. Measurement of both rise and fall of each timing signal
enables pulse height correction by software (see section 2.3.3). The whole system
can be considered as an effective multichannel digital constant fraction discriminator.
A more detailed description of this system follows.
Each detector wire is matched to a grounded base transistor amplifier for min-
imum reflections, which is coupled to a home made preamplifier-shaper (see Fig.
10). This stage is followed by an emitter follower and a second amplifier with a shor-
ted cable differentiation network. The output is an almost symmetrically shaped
pulse, with 3 nsec rise and fall time and 10 ns width with an amplitude of more than
1 V (see Fig. 11).
The timing is provided by a fast pulse discriminator (LeCroy MVL407 400MHz,
22
-120
w 11.2 0.8 0.4 0
-0.4 -0.8 -1.2
Figure 11: ( a ) : Output signal f r o m a w i r e of the detector, and ( b ) : T h e same signal
after reshaping and a m p l i f i c a t i o n .
4 channel voltage comparator). An externally variable stabilized voltage control is
supplied as a common threshold voltage to all discriminators. If the amplified wire
pulse is above the threshold then the corresponding discriminator is set on at a time
t! and set off at a time t 2. The accurate measurement of these two times (t! and
t 2) provides the information needed for the extraction of the ion arrival time and the
height of the pulse.
The fast complementary E C L outputs from the discriminators are sent through
long (3 meters) twisted pair flat cables to the main electronic chassis. Because fast
electronics is needed, all the IC's are of the E C L family. Signals from the twisted
pairs are fed into a line receivers with differential input and complementary output.
An array of flip flops are set at both t! and t 2 of each discriminator pulse. The reset
of all the flip flops is done simultaneously by the STOP signal, which serves as a
reference for all the wire timing.
10 20 30 40 50 60 70 nsec
23
£ 80 z כ
o ° 7 0
60
50
40
30
20
10
0 - 3 0 0 - 2 0 0 - 1 0 0 0 100 200 300 PS
Figure 12: Typical t i m e r e s o l u t i o n f o r a s i n g l e w i r e . The s o l i d l i n e d r a w n t h r o u g h
t h e h i s t o g r a m i s a G a u s s i a n fit w i t h F W H M = 1 4 0 ps.
The stop signal originates from the M C P in the form of a fast positive pulse, as the
result of the first hit. This pulse is amplified and shaped before being introduced into
a constant fraction discriminator (CFD - Tennelec T C 455). After a proper delay,
the signal is fed into the E C L STOP drivers which reset the flip flops. If rather
than a number of fragments, only a single ion hit the detector, then the STOP signal
timing is accurately related to the time of arrival of this ion. Such events are used
for the calibration of the wires parameters which are used later for the extraction of
time difference between the individual hits on different wires, independently of the
individual pulse heights.
The flip flop output signals enable constant current generators, injecting charge
into charge sensitive ADCs. The charges in the A D C channels are proportional to the
time differences T! =t!-t s or T 2 = t ־ 2 t s , where t s is the time of the common STOP.
After appropriate processing of T! and T 2 using calibration parameters (see section
24
3.3), the FWHM resolution per wire is approximately 140 ps (see Fig. 12). Taking
into account that between 2 and 3 wires participate in the measurement of each
fragment, the overall time resolution per fragment is about 80 to 100 ps (FWHM).
25
10 msec
CHOPPER 1
CHOPPER 2
LASER
ADC CLEAR
ADC GATE
CAMERA FIELD
20 msec 40 msec י
On sec
sec גן 0
TRANSFER FROM 144 ׳ msec CCD TOFTS
־ ;1
n
I8nsec
30|1sec
35nsec
.DETECTOR READOUT
Figure 13: The t i m i n g sequence of t h e C E I e x p e r i m e n t . The l o w e r p a r t of t h e figure
i s a n expanded v i e w of t h e t i m e sequence as c o n t r o l by t h e T I M E R (see t e x t ) , r e l a t i v e
t o t h e c a m e r a field s i g n a l .
2.3 Con t ro l , D a t a Acqu i s i t i on and Ana lys i s
2.3.1 Synchronization
A n accurate time sequence, starting at the ion source chopper and ending at the A D C
gates and C C D camera frame initialization, is essential for the proper synchronization
of this special CEI set-up. It is preferable to choose the starting point of such a
sequence to be in phase with the 50 Hz main power frequency. This avoids the
problem of random noise pick-up from the main power lines, and can easily be
achieved by using the time sequence of the C C D camera as the start signal for the
sequence of accurately delayed pulses.
A home built timing unit (TIMER) generates a series of outputs corresponding to
26
the different delays needed between the different stages of the experiment. The unit
is based on a 40MHz (25 ns steps) crystal clock, 20 bit counter and 6 comparator
units each with 20 bits (maximal delay ~26 ms). The timer is remotely controlled
by an RS232 connection driven by the data acquisition computer.
The timer is used for controlling the following devices (see Figs. 4 and 13): Chop-
per I, Chopper III, up to three lasers, the clearing and gating of the ADCs in the
detection system, and automatically matching the C C D frame signal. The delays
between the different components of the system are chosen so that the ion pulses are
synchronized with the two choppers and overlapped with the laser pulses exactly in
the H V terminal. The result is a pulsed neutral molecular beam of 100-400 ns dur-
ation with a well defined kinetic energy which arrives at the detector in coincidence
with the 1 /J,S gating of the ADCs.
2.3.2 Data Acquisition
The functions of controlling the different devices of the experiment, acquiring data
and storing it, creating and displaying the histograms needed to monitor the ex-
periment, analyzing and processing the data are distributed between two computer
systems (see Fig. 14). The "Real Time" Computer (RTC) is a single board V M E
bus computer Motorola MVME147, connected to an ethernet network. This system
controls the various devices of the set up such as the T I M E R (through one of its
RS232 ports), the clear and readout of the ADCs of the detector through the C A -
M A C crate controller and the setting and reading of the frame threshold suppressor
(the video information of the detector, see below) via the V M E bus. The storage of
event data is done (after minimal processing) on a 600 Mbyte disk which is backed
up by an Exabyte tape (2.3 Gbyte per tape). One of the disks of the RTC is a remote
27
ETHERNET
WORKSTATION
VME BUS 2 < 3 > DISK 2
3 2.
1חEXABYTE
m
SYNCH FROM CAMERA TO TIMER
CLEAR AND GATE FROM TIMER
VIDEO SIGNAL TIMING MODULES & ADCS
TIMING SIGNALS I I I I I I ililllllili
CAMAC CRATE
Figure 14: Schematic diagram of the data acquisition system.
NFS mounted disk. On this disk the RTC writes during idle time random copies
of event buffers. Thus any computer on the network can access this NFS disk and
process a sample of the data on-line.
The video signal arriving from the detection system is analyzed by a home built
Frame Threshold Suppressor (FTS). This V M E bus device analyzes the video signal
by digitizing it with a 10 Mhz, 8 bit A D C and comparing it with a preset digital
threshold. When a pixel amplitude passes the threshold, both its amplitude and its
position (row and line) are stored in the internal 8 Kbyte memory of the FTS which
can be read later by the RTC. This filtering leaves approximately 10 to 100 pixels
per hit on the detector (see Fig. 8), reducing the amount of relevant data from 1/8
Mbyte for the full frame to a few hundred bytes. Another function of the FTS is
the synchronization of the RTC with the video cycle. When the FTS has finished
analyzing a frame it sends an interrupt signal to the RTC which starts the data read
out cycle.
28
For on-line monitoring and analysis any computer on the network may mount the
disk where the RTC writes a random sample of the data on-line. The full analysis is
almost the same with the only difference that the RTC writes all of the preprocessed
data on the NFS disk. Processing and display use the C E R N written packages PAW
and H B O O K .
2.3.3 Data Handling
The final velocities V x , V y and VZ of the fragments of a molecule are measured by
the detector as distances between hits in the X Y plane (parallel to the detector plane)
and the time differences between them. The velocities in the X Y plane are extracted
by multiplying the beam velocity by the ratio between these distances and the flight
path from the stripping foil to the detector. In a similar manner the velocity in the
Z direction is found by multiplying the beam velocity by the time difference between
hits and dividing the result by the flight time from the foil to the detector.
In order to extract the final velocities VX V y and VZ for each fragment, two data
sets have to be handled and matched. The first set is composed of the coordinates
(x,y) and amplitude of each camera pixel, after the digitized hardware filtering from
the C C D camera. A computer program recognizes the different fragment positions
(X,Y) by clustering the pixels. The second set of data results from the timing
information. This information consists of two data points, T! and T 2 , for each wire
hit. In order to find the exact time of arrival T of a fragment on the detector, a
calibration process is done before the experiment using single hit events from the
beam. For each wire i , a graph of Wj=T t
1-T t
2 versus T\ is plotted. For a perfectly
symmetric pulse shape and idealized discriminators, this would result in a straight
line with slope=l. In practice, the data set is best fitted with a polynomial Fj ,
29
usually of the second order. The function F; is used for the correction of the time
due to pulse height differences, such that T,-=T1+F,(W,•). In order to get an absolute
calibration of the ADCs for each wire, a similar measurement is done with a known
additional delay of the STOP signal. Finally, the time information (T) and the spatial
information (X,Y) is combined to give the full three dimensional velocity image of
each molecule.
2.3.4 Analysis
The data resulting from CEI measurements are sets of 3n velocity coordinates that
represent, event by event, the correlated velocities of the n fragments of the Coulomb
exploding molecule. Since the force governing the explosion is a Coulomb force, it
is possible, at least for a diatomic molecule, to analytically reconstruct the initial
positions of the nuclei for each molecule before the explosion. For 3 atoms and
more this reconstruction may not be done analytically. Nevertheless, it is possible
to perform a numerical solution of the Coulomb Explosion useing a set of initial
position parameters. Then ,by iterations, correct this set so that the final outcome of
the numerical solution will resemble the experimental result. However, this approach
is at most a good guess due to the random processes which occur during the passage
through the thin foil and the finite detector resolution.
The first set of random processes which is related to the interaction with the foil
are multiple scattering(MS) of ions in the foil and to a lesser extent gradual charge
exchange in the foil. Both processes occur during the passage through the stripping
foil within the first fsec of the explosion and are convoluted into the Coulomb ex-
plosion process. A second effect which is convoluted into the imaging is the finite
resolution of the detector.
30
Much effort was expended to minimize the thickness of the CEI foil and minimize
the temporal error in the detection system(sections 2.2.3 and 2.2.5).
The object of the analysis of CEI is to supply a probability density function
of the positions of the molecular nuclei. The distribution of velocities measured
by CEI may be described as a convolution of the position density function. The
convolving function involves the effects of random foil interactions, the Coulomb ex-
plosion propagation and the resolution of our detecting device. Assuming that the
nature of the random processes involved in the interaction of the exploding molecule
with the foil is known, at least phenomenologically, a forward propagation calcula-
tion may be performed. A Monte Carlo computer code that takes a single initial
configuration of the nuclei in the molecule (a point in R-space) and calculates the
final velocities of the exploding fragments (V-space) has been developed[8]. Since
we are dealing with a classical approximation of the explosion (see section 1.1) the
resulting V-space from the simulation of CEI of a certain point in R-space may be
linearly added to the simulation of another point in R-space. Thus, using this code,
an initial distribution in R-space may be projected into a distribution in V-space
and several of these distributions may be linearly added to fit the data. There are
several ways to apply the CEI simulation code in order to retrieve the initial R-space
distribution and these methods will be briefly described in the following.
The first and simplest method is applicable when a previous knowledge regard-
ing the structure under study can be used. This knowledge can be the result of
theoretical calculations or some other measurement. In this case it is simplest to
perform a sampling of this "known" density function, run the samples through the
CEI simulation, and then compare the results with the data. In fact any one of the
methods must terminate in this fashion, that is once we have found the "correct"
31
R-space distribution we must check this by independently simulating the CEI of the
distribution, and compare it with the data in a statistically meaningful way.
The second method deals with cases that are either single dimensional or mul-
tidimensional cases with very little correlation between dimensions[86]. Examples
are diatomic molecules or the angle of bending of an XH2 molecule. In this case
we may simulate points on a "grid" in R-space, with enough repetition at each grid
point to recreate its CEI response to a 8 function at this point. The result of this
is a response function which can be symbolized as V = M[R] and R ?a Y%=1 c־'<^•
In general, direct inversion of the functional M is impossible due to amplification
of statistical errors. A simple method of retrieving the R-space distribution is by
assuming some parameterization of R-space, such as a polynomial expansion, and
then minimizing the difference between the data and M[R] as a function of these
parameters.
The last, and most difficult case is when the initial R-space is truly multi-
dimensional. In this solution by simulating the response function to 8 func-
tions on a grid that covers phase space is usually impossible since the number of grid
points grows as a power law of the number of dimensions. We have devised a self
learning, unbiased, iterative algorithm to solve this problem[13]. The essence of the
method is as follows. We begin with some sample of R-space which approximately
covers the region of interest and simulate its CEI . A position dependent weight func-
tion is defined in V by dividing the density of the data at a point by the density of
the result of the simulation at the same point. Each point in V which resulted from
the simulation has its associated point in R. The new weighting which was found
for the points in V is applied to the initial points in R and a new density function
in R is created. This function is the beginning of the new sample of R and so on
32
till the process converges and a good fit is achieved between the data and the CEI
simulation result.
A major point that must be raised at this stage is the physical requirement of
the smoothness of R-space. The results of the backward transformation can create
a jagged R-space with high amplitude high frequency spatial components that when
sampled and simulated to Coulomb explode would give a good fit with the data.
Nevertheless, the spatial frequency of R-space is directly related to the kinetic energy
in the system and this entity has an upper bound. Thus, we must impose the concept
of smoothness on the data. As a rule of thumb, the standard procedure is to find the
smoothest-lowest frequency description of R-space that still gives a good fit to the
data.
In the case of single and two dimensional problems there are standard tools to
smooth data. Among these methods are a fit to truncated polynomial expansion or fit
to a small amount of Gaussians. This method of solution is sometimes possible also
in multi-dimensional cases when there is already some existing information regarding
the structure of the molecule, but it is very difficult to apply a-priori. Another method
of smoothing low dimensional problems is by low pass Fourier transform filtering.
When the dimension grows beyond two, the application of the previous methods
of smoothing become difficult. Therefore a different method of smoothing was used
in all the multi-dimensional analysis. The discrete set of points is smoothed by
convolution with a multi-dimensional Gaussian. The simplicity of the method is
attractive. There are,nevertheless, some pitfalls. The major problem of this method
is that a-priori there is no simple way of defining the width of the convolving Gaussian
in R-space. Choosing a width which is too small will result in highly oscillating
results and even regions which are not covered. On the other hand, widths which
33
are too wide will create non-zero density in regions which really should be empty.
Another problem is that the density is defined as a function of all the data points.
This property entails the disadvantage of taking a long computer calculating time.
Another difficulty which arises from this type of smoothing is that the fit has as
many parameters as simulation points. In this case it is very difficult and practically
impossible in terms of computer time to extract the covariance matrix associated
with the fit.
Several new solutions are proposed to solve the previous two problems. The
proposed method to solve the first problem is by ordering the points in some logical
way that will cut down on the number of relevant points accessed for each of these
density calculations. This is not very difficult and initial tests show an improvement
of a factor of 5-20 in computation time, depending on the problem. The solution
to both problems may be attained by clustering the separate Gaussians, resulting
from the convolution, into a smaller number of wider multi-dimensional structures
(for example Gaussians which are not spherically symmetric).
2.3.5 Error Estimate
When the final R-space solution is found we must associate with it some error estim-
ate. A complete error analysis would include the calculation of the error covariance
matrix and from it the x 2 test could be performed. As stated in the previous section
such a test is not possible at this stage of development of the method, but, several
statistically independent calculations where carried out and the main quoted results
in this work are robust.
Even without a complete error analysis it is possible to estimate the intrinsic
errors embedded in the analysis of CEI. There are several sources for these errors.
34
>0.07 >0.07
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 CH 4.5MeV 0 . 6 / i g / c m ' foil ״ B 1 ־ 2 M e V 1 .4 / ig /cm 2 foil R
Figure 15: Output i n t e r - a t o m i c v e l o c i t y vs i n p u t b o n d - l e n g t h . Bond l e n g t h i s i n
A n g s t r o m s a n d v e l o c i t y i n a t o m i c u n i t s . The effect of m u l t i p l e s c a t t e r i n g i s s h o w n
as t h e a v e r a g e a n d w i d t h of t h e r e s u l t s f o r each i n p u t b o n d l e n g t h .
One major contributor is the incomplete knowledge of charge screening effects on the
Coulomb explosion while the exploding molecule is still in the foil which appears as
a systematic error of 10% in the final bond-length definition[8]. The two other main
contributors are the multiple scattering (MS) process in the foil and the temporal
resolution of the detector. Other factors are the charge exchange process in the foil
and finally the statistical significance of the features which are measured. Figures
15 and 16 represent the ambiguity of backward transformation of a single event in
V-space to an event in R-space due to MS. The figures were simulated for conditions
similar to the cluster measurements and the molecular measurements described in
this thesis. As we can see, the error in defining the C H bond for a single event is
approximately 13% compared with the error of 10% for B 2 bond. For the angular
case the error is approximately 15° for both. The "contribution" of the MS to the
ambiguity problem (see section 1.2) is extremely strong for the cluster case. As
35
0 2 0 4 0 6 0 8 0 1 8 0 1 6 0 1 4 0 1 2 0 1 0 0 8 0 6 0 4 0 2 0 0 1 8 0 1 6 י 0 0 ז 0 2 י 0 4 0
CH 2 4.5MeV 0 . 6 / i g / c m 2 foil e' B, 12MeV 1 .4 / ig /cm 2 foil °־
Figure 16: Output angle i n V-space vs input angle i n R-space. T h e effect of m u l t i p l e
scattering is shoiun as the average and width of the results for each input angle.
shown, a V-space result of less than 70° may be transformed to an R-space angle of
anything between 30° and 140°.
The detection error may be divided into temporal and positional resolution. The
temporal resolution is approximately lOOps which is equivalent to approximately 5%
error in inter-atomic velocity. The contribution of this element to the error in bond
length may be substantial but is orientation dependent. If we define x and y as the
velocity extracted from spatial coordinates, and z as the velocity extracted from the
time measurement, then for a diatomic molecule the bond-length is calculated as:
R + A R c x i x ' + y' + iz + S ) 2 ) - 1
Where 5 is the error induced by the temporal measurement. To understand the effect
of the error we may look at two extreme cases. In the first case R is in z direction
and z S נ§> S 2 , then the error is:
A R _ 2z8 _ 2 5 ~ R ~ 1? ~ ~
36
The other case R is in the X Y direction and zS <C 52, then the error is:
A R 82
״ ׳-יי*״<-v_׳
R v2
Where v is y / x 2 + y 2 . While the error in the first case is approximately 10%,
assuming the 5% error in inter-atomic velocity, the error in the second case is only
0.25%. Both phase space and the configuration of the detector, which biases against
the first type of events, make the the overall error much closer to the second case
than to the first one. The positional resolution is better than 0.1mm which is ap-
proximately equivalent to 1% of the inter-atomic velocity and therefore the detection
error is usually dominated by the temporal error.
In view of the above error analysis we can see that analytically the error estimate
of the MS and screening effects are a good limit to the total error estimate and the
other effects are relatively small.
The above estimates of the error may be narrowed by statistics and at this point
it is important to state the conditions preventing or allowing the collection of large
samples. The results, described in this work, all deal with events in which all equal
atoms have equal charge and this charge is the most probable charge state after
stripping. One example for sample collection conditions is as follows: a molecule
of C 4 passing through a t h i c k foil at 12 MeV would have the following charge state
distribution[14]:
Q 2 3 4 5
Part 0.018 0.332 0.559 0.090
Thus the probability that all four atoms will come out at charge state q=+4 is
only 9.8%. If we take into consideration the efficiency of the detector, which is less
37
than 1, this number is even lower. On the other hand measuring C H 4 at 4.5 MeV
is much easier. Here the probability for the H coming out as a proton is nearly 1
and the probability for the C to come out at q=+4 is approximately 58%. The foil
used in CEI is a thin foil and therefore the most likely charge state is lower and
percentages change but these numbers are qualitatively indicative.
38
2.4 Conclusion
The experimental system and analysis methods which were described in this chapter
were used, and in part developed, for the cluster measurements which are described
in the following chapters of this thesis.
At this stage the system is still developing. A new cold molecular source has
been installed for the production of vibrationally cold molecular ions for the CEI. In
order to improve the identification of the state in which the ions are left after photo-
detachment an electron spectrometer was installed under magnet 2 (see figure 4).
At the detection level, the two detectors, small and large one, have been connected
and are now being used in correlation to measure X ״ H m type molecules.
As for the software, the new iterative method is being improved by implementation
of the ideas proposed in this chapter.
39
3 Triatomic Clusters
3.1 Introduction
A cluster may be defined as the aggregate of 3 to a few thousand atoms or molecules.
The family of triatomic clusters is the simplest type of clusters and as such we begin
the description of the CEI of clusters with them.
The structure and properties of triatomic carbon has been under study for more
than a century[15]. It is known to have a linear structure with a very soft, 63 c m - 1 ,
bending mode. In comparison, very little is known about triatomic boron. The
structure of B 3 has been calculated by a few groups but the only measurement from
which structure information may be extracted was done by Electron Spin Resonance
in a frozen matrix. Thus, the structure of gas phase B 3 was never measured before.
Both theoretical predictions and the measurement indicate that the structure of B 3
is an equilateral triangle.
In terms of configuration, these are two extreme examples. We will show what
is the qualitative and quantitative information that can be extracted from such CEI
measurements.
40
3.2 C 3
Abstract
A warm C 3 cluster population was measured using the Coulomb Explosion
Imaging method. A fully correlated distribution of the three internal coordin-
ates of the measured clusters was found. The main features of this distribution
indicate a linear structure with a wide distribution in bending angle. The bond-
length was found to be 1.2A ± 10% in agreement with a b - i n i t i o calculations.
The width of the distribution of the bending mode was found to be more than
3 times wider than what is predicted for the vibrational ground state, which is
consistent with a warm population measurement.
3.2.1 Introduction
The triatomic C'3 cluster has been under observation for more than a century[15].
Appendix A recalls the history of spectroscopic measurements which were performed
on this species. The general consensus today is that C3 is a linear molecule[16] with
bending vibration 1/2 of 63 cm _ 1[17, 18]. The size of the bond-lengths is 1.297A[19]
and the breathing mode and anti-symmetric vibrations, v\ and z/3 are determined at
1224.5 cm20]־ 1]and 2040 c r n " 1 ^ ] respectively.
In comparison with the vast amount of experimental data existing on the prop-
erties of neutral C3 very little was done regarding the ionic forms of this cluster. A
CEI measurement done at Argonne Nat. Lab.[22] measured the structure distribu-
tion of a warm population of C 3 and a preliminary analysis indicated an average
cyclic structure. However, a latter analysis[23] concluded that the experimental
data is also consistent with a linear structure exhibiting large amplitude vibrational
motion similar to that of neutral C 3 and the ambiguity is a result of thermal excit-
ation. Theoretical calculations regarding C 3 claim that it has a cyclic ground state
41
structure[24, 25].
The geometry of the negative ion C j has not been determined experimentally.
Based on the uniform trends in the observed electron affinities of small carbon
clusters, Yang et al.[26] concluded that all the negative ions up to Cg are prob-
ably linear. In addition, photo-electron experiments[27, 28] and photo-depletion
experiments[29] on C 3 have been used to derive the electron affinity of 1.98±0.02
for C 3 . Sunil et a l . [30] have calculated the structure of C 3 to be linear. In a later
calculation Raghavachari[31] predicted the C 3 anion to be linear with a 2 l i e , ground
state which is considerably more rigid than neutral C 3 . A result of his calculation is
that the bond-length of the anion is longer by approximately 0.02A than that of the
neutral, and the three vibrational constants are v \ = 1175 c m - 1 1/2 = 251 c m - 1 , and
1/3 = 1754 c m - 1 . These predictions were found to be consistent with the the analysis
of photo-electron spectra measured by Neumark's group[28].
In view of the vast compilation of measurements and calculations performed on
the C3 cluster it is a good candidate for CEI as a test case to check the reliability of
the measurement method. Already in 1986 our group performed a CEI measurement
of C3[32], using gas stripping and coincidence of different charge state fragments. The
results of this measurement were consistent with a linear structure.
In the following paragraphs we report the CEI measurement of C 3 . The results of
this measurement were analyzed using the unbiased iterative method of section 2.3.4
fully correlated in 3 internal dimensions and the results of this analysis will be
described.
42
3.2.2 Experimental
A vibrationally warm negative carbon beam was produced by the Cs sputter source.
The beam was chopped and mass separated for C 3 mass 36. Further chopping
reduced the pulses to 400 nsec pulses which in turn were injected into the Tandem
H U D Pelletron. The acceleration voltage in this experiment was 11 M V . A laser
pulse from the Nd-YAG second harmonic (A = 532 nm, hv — 2.35eV) was fired
in time to photo-detach the cluster bunch while in the terminal of the accelerator.
Neutralized clusters continued through the accelerator and impinged on a 1.4/xg/cm2
Formvar foil located downstream. After traversing the foil, the stripped molecules
Coulomb exploded and fragments with charge states q=+3 and q=+4 were directed
by the mass selection magnet towards the detector. The most popular charge state
for the outgoing fragments was q=+3. For the present analysis only events that
contained 3 fragments of charge state q=+3 were taken.
3.2.3 Analysis
As was described in section 1.2.3, due to the permutation symmetry of the different
carbon atoms, the analysis of C 3 results must start by choosing symmetry coordinates
to describe the inter-atomic structure in both V-space and R-space. The choice here
is:
S\d) = {xW + x } $ + x$)/y/3
{ 2 X $ - X l $ - X $ ) / y / E
( x j $ - x $ ) / y / 2
Where
d = < r R-space
v V-space V
43
and XJ-J is the absolute distance between atom i and atom j in atomic units for
R-space and X ־ - j the absolute relative velocity divided by the beam velocity(V& e a m)
for V-space. The significance of S!, the symmetric coordinate, is equivalent to a
vibrational breathing mode and the two dimensional representations ( 5 2 , <S3) are
equivalent to a symmetrized version of the symmetric and antisymmetric vibrations.
In this representation, the origin represents an equilateral triangle configuration,
while points on the coordinate S3 — 0 correspond to isosceles triangles.
The data of 1331 events, of charge q=+3 for each atom, were used and since all
six permutations of atomic indexes are included, 7986 points were included in the
analysis.
The search for the nuclear position distribution function of the cluster was per-
formed as described in chapter 2.3.4. The transformations of the discrete points into
a smooth function were done by convoluting the points with a spherically symmetric
3d Gaussian(see section 2.3.4). The width of the Gaussian convoluted with the velo-
city space was taken as 5 X 1 0 - 4 and for R-space was taken as 0.2 in atomic units of
distance. Notice in figure 1 that these values are approximately one tick mark and so
the widths of the convoluting Gaussians are much smaller than any intrinsic width
of the functions.
Figure 1 shows the R-space distribution function which was found from the ana-
lysis and the fit between the data and the CEI simulation of this function. On the
top left and right and on bottom left we can see the quality of the fit, on the bottom
right the R space result is depicted in the ( S 2 " \ S ^ ) coordinates. The velocity space
picture is one of an equilateral triangle with small tails of the distribution which
spread toward the linear configuration. The result of the unbiased analysis which is
depicted for R-space is dramatically different. Here we see 3 permutationally equi-
44
0 . 0 0 6
0 . 0 0 3
- 0 . 0 0 3
- 0 . 0 0 6
0 . 0 0 6
0 . 0 0 3 -
0 -
- 0 . 0 0 3
- 0 . 0 0 6 - 0 . 0 0 6 - 0 . 0 0 3 0 0 . 0 0 3 0 . 0 0 ־ 6 - 0 . 0 0 6 - 0 . 0 0 3 0 0 . 0 0 3 0 . 0 0 6
S2 vs S3 - Data(V) S2 vs S3 - Simulation(V)
1 2 0
1 0 0
8 0
6 0
4 0
2 0
0
ן; |
\ J V
- j .
: I I I - ! | 1 J dtHirc * ! • • ׳ ׳ • "• 1 • 1 • " ^ ! ^ *
1 .5 -
0 -
־ 1 . 5 -
0 . 0 0 7 5 0 . 0 1 5 0 . 0 2 2 5 0 . 0 3
S1 - 3 - 1 . 5 0 1 .5 3
S2 vs S3 - Simulation(R)
Figure 17: F i t between simulation and data of C 3 . Coordinates of velocity space a r e
plotted i n V / V 1 e a m and of c o n f i g u r a t i o n space i n atomic units. Top left: Plot of data
i n 5 ^ vs coordinates. Top right: same plot as top left for the results of the
s i m u l a t i o n . Bottom left: E r r o r bars a r e the data i n 5!^ coordinates, the dashed l i n e
is the result of the s i m u l a t i o n . Bottom right: plot of the R-space distribution i n
vs coordinates. Notice the 3 peaks away f r o m ( 0 , 0 ) and on the 3 symmetry axes.
T h i s shows that the distribution is a r o u n d a n isosceles t r i a n g u l a r structure.
45
valent peaks away from the origin which represents an equilateral configuration. If
we focus our attention on the upper peak it is on the symmetric axis which indicates
that it is centered on an isosceles configuration.The two modes of vibration, the sym-
metric bending and antisymmetric stretching, are almost uncorrelated. It is further
noticeable that there is very little penetration into the equilateral region.
The R-space results were parameterized in terms of a bending angle and the
symmetric and antisymmetric bond stretching. The results are described in the
following section.
3.2.4 Results and Discussion
The average bond-length obtained in this work is 1.2A ±10%, where the error is
the result of our lack of knowledge for the exact screening of electrons affecting the
explosion while the molecule is still in the stripping foil[8]. This result is in agreement
with the predicted value of 1.29A.
The bending angle distribution was fit with a two dimensional Gaussian distri-
bution with a mean of 180° and 0־ of 70° ± 5 ° . In this case the error is the result of
the uncertainty in the reconstruction of initial angle due to multiple scattering and
detector resolution(see sect. 2.3.5). In order to compare harmonic frequency and
measured width we may use the relation [33]:
where \1 is the oscillator reduced mass and u> the frequency, which describes the
relation between the spread of the ground state harmonic oscillator wave-function
and the frequency of the oscillator. Thus the 63 c m - 1 assigned to this mode is
equivalent to a 16° width of the wave-function squared. This value is approximately
one fourth of the width found in this measurement. Thus the measured width is
46
Figure 18: Results of C E I of cold and hot populations of C 3 depicted i n
Notice that the coordinates a r e i n a scaled energy not velocity.
consistent with a thermal excitation of approximately 400° Kelvin. The two main
contributors to this excitation are the Cs sputter source which produces vibrationally
hot negative ions[34, 35] and the mismatch between the 63 c m - 1 bending frequency
of the neutral cluster and the 250 c m - 1 predicted for the negative cluster.
The width of the other vibrations is comparable with the experimental error and
therefore we will not describe their analysis.
A preliminary measurement has been carried out in order to measure the structure
of C 3 resulting from a cold molecular source. The results of this measurement are
depicted in figure 2. The coordinates are similar to those used in the analysis of hot
C 3 except that the X - j are now defined as Vfj multiplied by an arbitrary constant.
Here we can see on the left side the familiar figure of hot C 3 . On the right side, the
cold C 3 shows the beginning of a peak in the region of the linear tail. The analysis and
complete measurement of this data are out of the scope of this thesis and therefore
were not carried out.
47
In conclusion, the measurement of C 3 shows the possibility of measurement of
linear or quasi linear X 3 clusters even for hot populations. The results of such
measurements can conclusively define the geometry of a cluster. Our results, due to
the spread of the bending angles into the high uncertainty region, do not support
more than a Gaussian fit to the width of the angular distribution, but this situation
could be corrected by measuring cold populations.
48
B 3
Abstract
The gas phase triatomic boron cluster was measured, for the first time, us-
ing the Coulomb Explosion Imaging method. A 3-dimensional, fully-correlated
distribution function describing the positions of the nuclei of the clusters meas-
ured was found. The main features of this distribution indicate a triangular
structure and these results are consistent with the equilateral configuration pre-
dieted by theory and measured by Electron Spin Resonance in frozen matrices.
The bond-length was found to be 1.45A ± 1 0 % in agreement with predictions
of theoretical works. An upper limit was set to the electron affinity of B 3 at
2.35 eV.
49
3.3.1 Introduction
Although the ground-state structures, electronic properties, and, in some cases, vi-
brational frequencies of many metal trimers are now known[36], very little is known
experimentally on the structure of B 3 . Several groups have calculated the structure
of the boron trimer [37, 38, 39]. They predict that the ground state of B 3 is of 2 A[
symmetry at its triangular equilibrium geometry with a bond-length of 1.55A to
1.6A, depending on method of calculation. Anderson and co-workers[40] have done
a combined Collision Induced Dissociation and ab-initio study of boron cluster ions
but the only attempt to measure structure was done by Hamrick, van Zee, and
Weltner[41] by means of ESR spectra of B 3 in frozen matrices. They found that the
configuration is an equilateral triangle and a 2 A [ ground state.
In view of this lack of experimental information, we have measured the structure
of B 3 using CEI. Because the boron trimer is predicted to be in a equilateral trian-
gular configuration, it is a typical example where the results of CEI are difficult to
analyze due to ambiguity and insensitivity of the CEI method in that region. In the
following, we will describe the measurements and results acquired for the B3 cluster.
In view of the above problems, although a complete 3 dimensional fit was performed,
only the general shape and sizes will be described.
3.3.2 Experimental
A negative boron beam was produced by the Cs sputter source. The beam was
chopped and mass separated for 1 0 B % mass 30. Further chopping reduced the pulses
to 400 nsec pulses which in turn were injected into the Tandem M U D Pelletron.
The acceleration voltage in this experiment was 11.5 M V . A laser pulse from the
N d - Y A G second harmonic (A — 532 nm, h.v — 2.35eV) was fired in time to photo-
50
detach the cluster bunch while in the terminal of the accelerator. Neutralized clusters
continued through the accelerator and impinged on a 1.4/Ug/cm2 Formvar foil located
downstream. After traversing the foil the stripped molecules coulomb exploded and
fragments with charge states q=+3 and q=+4 were directed by the mass selection
magnet towards the detector. Since the most popular charge state was q=+3, only
those events that contained 3 fragments of charge state q=+3 were used in the
analysis.
3.3.3 Analysis
The coordinates used for the analysis and the parameters for the creation of the
smooth functions are exactly the same as in the analysis of C 3 and the reader is
referred to section 1.2.3 for a complete description. 2513 molecules were measured
with good position and time resolution. Thus, due to the permutation symmetry of
indexes, 15078 points were included in the analysis.
Figure 3 shows the R-space distribution function which was found from the ana-
lysis and the fit between the data and the CEI simulation of this function. On the
top left and right and on bottom left we can see the quality of the fit, on the bottom
right the R space result is depicted in the ( S 2 \ 53^) coordinates. The velocity space
picture is one of a symmetric Gaussian distribution centered at the origin, corres-
ponding to an equilateral triangle geometry. The result of the unbiased analysis
which is depicted for R-space (figure 3, bottom right) is very similar. The irregular
features are an artifact due to the ambiguity of analysis of CEI in this region.
51
0 . 0 0 6
0 . 0 0 3
- 0 . 0 0 3
- 0 . 0 0 6 - C
2 2 5
2 0 0
1 7 5
1 5 0
1 2 5
1 0 0
7 5
5 0
2 5
0
0 . 0 0 6
0 . 0 0 3
- 0 . 0 0 3
- 0 . 0 0 6 - 0 . 0 0 6 - 0 . 0 0 3 0 0 . 0 0 3 0 . 0 0 6 - 0 . 0 0 6 - 0 . 0 0 3 0 0 . 0 0 3 0 . 0 0 6
S2 vs S3 Data S2 vs S3 Simulation
! +t ו ::1::1EI:I:
1 i d i
M•: * ן
1 .5 -
- 1 . 5 -
0 . 0 0 7 5 0 . 0 1 5 0 . 0 2 2 5 0 . 0 3
S1
- 3 - 3 - 1 . 5 0 1 . 5
S2 vs S3 Simulation(R)
Figure 19: F i t between simulation and data of B 3 . Coordinates of velocity space a r e
plotted i n V / V b e a m and of c o n f i g u r a t i o n space i n atomic units. Top left: Plot of data
(velocity) i n ^ v s $3י^ coordinates. Top right: same plot as top left for the results of
the s i m u l a t i o n . Bottom left: E r r o r bars a r e the data i n 5^ coordinates, the dashed
l i n e is the result of the s i m u l a t i o n . Bottom right: plot of the R-space distribution
f u n c t i o n of the simulation i n
4r) vs 4r) coordinates. Notice the broad peak at ( 0 , 0 )
which indicates that the c o n f i g u r a t i o n of the cluster is a r o u n d e q u i l a t e r a l t r i a n g l e
52
3.3.4 Results and Discussion
It is very difficult to determine the exact shape of the distributions due to ambiguity
in the interpretation of the results and the strong nonlinearity of the CEI of triatomic
clusters in the region of interest. As described in sections 1.2 and 2.3.5 there
is a problem of ambiguity in interpretation of results that indicate an equilateral
structure. Therefore there is no simple way to conclude if the vibrations (or spread
in the density function) are real vibrations or if it is simply a result of the ambiguity
of CEI .
We can conclude from the results of the CEI that the configuration is far from a
linear one and the data is consistent with an equilateral structure. The bond-lengths
found for the equilateral structure are 1.45A ± 10% , where the error is the result
of our lack of knowledge for the exact screening of electrons affecting the explosion
while the molecule is still in the stripping foil[8]. Both the approximate shape and
the bond-length are consistent with the predictions of theoretical works.
The results of this measurement do not produce much information. Nevertheless,
the triangular B 3 is a "bad" example of CEI and as such teaches us the limitations of
the method. We have shown that though it is possible to measure the existence and
very approximate shape of such configurations, the CEI is not an adequate source of
information in these cases.
53
3.4 Compar i son between B 3 and C 3 C E I and Discuss ion
The main goal of carrying out the triatomic cluster CEI measurements was achieved.
It was shown that it is possible to measure a multi-particle event with simultaneous
recording of all particles, their charge state, X - Y position and time with an accuracy
which is equivalent to the natural bond-length spread. This opened up the possibility
of measuring structural parameters of small clusters.
We have described our measurements and the results of the analysis for the boron
and carbon triatomic clusters. The results of both measurements seem similar at first
glance(compare figure 1 with figure 3). Notice that the results in ( S ^ ^ S ^ ) both
have a peak at (0,0). This means that both cases the results in velocity space are
mostly triangular in shape and centered around the equilateral configuration. The
major, and significant, difference between the two is in the tails of the distribution.
These tails are the indication that the results of C3 may be coming from a linear
molecule which is the case, as we have shown. Thus due to the nonlinearity of the
CEI , which is very strong in cluster cases, the results in the triatomic cluster case
are mostly defined by "tails" of the distribution.
Figure 4 shows a single dimensional graph of the density in velocity distribution,
and the statistical error of this measurements, along the coordinate with = 0.
This coordinate is a cut in the two dimensional plots of ( S 1 ^ , S ^ ) in figures 1 and 3.
The graph is for both C 3 and B 3 . We can see that though the difference is mainly
in the tail of the distribution it is statistically significant.
Another lesson that may be learned from these measurements deals with the
results that may be expected from such data. In the case of the linear C 3 we were
able to extract not only the general shape but also, under certain constraints, the
angular bending distribution. On the other hand in the case of B 3 , all that could be
54
Figure 20: A plot of a single dimensional cut along the S^' coordinate for both B 3
and C3 with the appropriate error analysis.
said is that the cluster is triangular and the approximate value of the bond-length.
Thus, we may divide triatomic cluster measurements into different regions. In the
region where the cluster is linear and the spread of the bending angle doesn't go
much beyond 30° the CEI sensitivity is very high. In the other limit, where the
molecule is triangular in nature, it is impossible to extract much information due to
ambiguity in interpretation of the data and the compression of the CEI image into a
small region around equilateral configuration in V-space.
As was expected(see section 1.2), for triatomic clusters, not all degrees of free-
dom within the internal coordinates are measured with great accuracy with the CEI
method. On top of that, the Cs sputter source is known to produce vibrationally
hot populations[34, 35] . Nevertheless, there are some pronounced features of CEI
which are worthwhile mentioning as scientifically significant. The confirmation of
the linearity of C 3 shows the power of CEI. In the case of C H j [42], spectroscopy
55
could not distinguish between a bent and a linear molecule. Both C H j and an elec-
tronically excited state of C 3 are of the Renner-Teller type[18] and except for J=0
(total angular momentum) the assignment of a single electronic state is doubtful[43].
Moreover, the bending frequency for C 3 is very low, and therefore the standard vi-
brational rotational and electronic separation used in spectroscopy is not valid[44].
Thus, the CEI finding here which support the linear structure for a "hot" ensemble
of molecules is of importance and unique because of the direct method of extracting
the structure.
56
4 Tetra Atomic Clusters
4.1 C 4 l inear or rhombic?
Abstract
The C 4 cluster was measured using CEI a few years ago and the results
seemed to indicate the existence of a rhombic C4 isomer. This identification
was supported both by analysis of the CEI data and by the fact that the neut-
ral C 4 beam was prepared by photo-detachment of C 4 by 2.35 eV photons,
approximately 1.5 below the electron affinity of linear C 4 and 0.3 eV above the
electron affinity calculated for rhombic C 4 . In another separate measurement,
a threshold was found for the photo-detachment energy of C 4 at ~2.1 eV which
strengthened the previous rhombic conclusion.
A new analysis method which was developed lately shows that the data
must be primarily identified as a linear population of C4. The measurement
and results are described in the following section.
4.1.1 Introduction
Among small carbon clusters, the C4 molecule has received considerable theoretical
and experimental attention[45]. In large part this is because of the unexpectedly
low energy of the elusive cyclic, closed-shell rhombic isomer. In contrast to the
suggestion of early theoretical studies of carbon clusters[46, 47, 48], which indicated
that mono-cyclic isomers would not be competitive with linear isomers until C!o,
later calculations[49, 50, 51] have suggested that the closed-shell rhombic isomer of
C4 is comparable in energy with the linear triplet isomer. Still the higher entropy of
the linear configuration and the possible tunneling between isomers would result in
approximately 10-15% rhombic isomers at room temperature.
57
In the vast majority of experiments in which carbon clusters are formed, it is
the thermodynamically preferred linear isomer which is observed[52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 28, 26, 62]. Optical, electron-spin resonance (ESR), and infrared
spectra have been obtained in matrices and high resolution infrared data has been
obtained in the gas phase[61, 62]. As a result of these measurements, the value of the
asymmetric stretching vibration 1/3 was reported as 1549 c m - 1 and the t ׳ 5 bending
vibration was estimated to be 160±4 c m - 1 . High resolution photo-electron spectra,
of linear C 4 [28] have yielded additional information about the vibrational spectrum
of linear C 4 . These and earlier[26] photo-electron experiments have resulted in es-
timates of the electron affinity of linear C 4 at 3.882±0.010 eV[28]. A series of ESR
and infrared matrix experiments by Graham and co-workers[57, 58, 59] has indicated
that triplet C 4 is c i s bent, though deviating from linearity by no more than 3°. This
may be a matrix effect, since there is no evidence of bending from gas phase high
resolution infrared studies[61, 62] or from theoretical calculations.
The only measurements which claim to have detected rhombus C 4 were done
by our group. These experiments may be separated into two kinds. In the first
type of experiment we have measured a clear photo-detachment threshold for C4,
using photo-depletion and photo production methods, at 2.1 ± 0 . 1 eV[6, 29]. This
result is more than 1.6 eV lower than the threshold measured by photo-electron
spectra(3.7-3.8 eV) and it is similar to the electron affinity calculated for rhombic
C4[63]. Another set of measurements, Coulomb Explosion Imaging, was done for
C 4 resulting from C j photo-detached by 2.35 eV photons. It was claimed that the
results indicate a rhombic structure for C4.
The new unbiased analysis of this data, as described in section 2.3.4 has proven
that the previous conclusion was incorrect. In fact, the results are not only consistent
58
with a linear C4 with geometrical parameters similar to theory, but it is impossible
to explain the data only with a rhombic C 4 . In the following we will describe the
results and analysis for CEI of C 4 .
4.1.2 Experimental
A negative carbon beam was produced by the Cs sputter source. The beam was
chopped and mass separated for C 4 mass 48. Further chopping reduced the pulses
to 400 nsec pulses which in turn were injected into the Tandem H U D Pelletron. The
acceleration voltage in this experiment was 11 M V . A laser pulse from the Nd־YAG
second harmonic (A = 532 nm, h u = 2.35eV) was fired in time to photo-detach
the cluster pulse while in the terminal of the accelerator. Neutralized clusters con-
tinued through the accelerator and impinged on a 1.4jug/cm2 Formvar foil located
downstream. After traversing the foil the stripped molecules Coulomb exploded
and fragments with charge states q=+2 and q=+3 were directed by the mass selec-
tion magnet towards the detector. The most popular charge state for the outgoing
fragments was q=+3, thus, for the present analysis only events that contained 4
fragments of charge state q=+3 were used.
4.1.3 Analysis
Due to the permutation symmetry of the different carbon atoms, the analysis of C 4
results must start by choosing symmetry coordinates to describe the inter-atomic
structure in both V-space and R-space (see section 1.2.3). The choice here is a
59
CM
Figure 21: G r a p h i c a l d e s c r i p t i o n of c o o r d i n a t e s used f o r b u i l d i n g t h e t e t r a h e d r a l
symmetry c o o r d i n a t e s
variation of the tetrahedral symmetry coordinates[64]:
£ f =
E f =
B W =
=
x \ d ) + x ) a > + x r + x l
( x W - x t f ) / V 2
( X $ - X f f ) / ^
( X $ - X $ ) / y / 2
r(d) (d) r(d)
where
d = r R-space
v V-space
and X • ^ is the absolute distance between atom i and C M (figure 5) and X \ v > > is the
absolute velocity relative to C M . 0 ; j is the angle between the direction vectors from
C M to atoms i and j . x \ r J is the absolute distance between atom i and atom j and
X\VJ is the absolute relative velocity for V-space. A l l the velocity coordinate are in
V / V . beam and distances are in atomic units.
60
These coordinates, being tetrahedral coordinates, do not have a simple goemet-
rical interpretation away from the origin. Therefore, in order to understand the
meaning of these coordinates, we will show how each of the major tetramer struc-
tures is represented in these coordinates. Starting with the tetrahedron, this structure
falls at the point where all coordinates except E c are 0. The square and rhombus
structures both fall at E t = 0 and Eb = 7r and their permutationally symmetric posi-
tions(PSP). Any deviation from planarity would be interpreted in these coordinates
as a shift from Eb = 7r towards Eb = 0. The difference between the rhombus and
square geometry manifests itself in the " B " coordinates. In these coordinates, for
( E b , E t ) = (7r,0), a square would fall at (0,0,0) while a rhombus would show up as
\ B y \ > 0 while the two other B's are 0. Finally a linear structure would show as
E t = \/3TT and Eb = 7r and its PSP. In the " B " coordinates the linear structure would
show up as two of the coordinates at 0 and one different from 0.
The analysis of the experimental results produced 324 events of 4 C 3 + hits on
the detector. Due to the permutation symmetry each of data is multiplied by 24 by
permuting the indexes of the fragments. Thus 7776 sets of velocity coordinates were
used for the analysis of C 4 .
Figure 6 shows the results of the CEI measurement in symmetry coordinates.
The smooth functions depicted are a result of the convolution of the data with a
6 dimensional Gaussian (see section 2.3.4). The widths of these Gaussians is 0.3
radian for the angular E 6 and E t coordinates, and 0.001 (in units of V / V ^ a m ) for the
velocity " B " coordinates. The symmetric coordinate E c is integrated over, in this
analysis, by using a very large width for the Gaussian in this direction.
Before displaying the complete results and fits it is useful to explain the results of
velocity space which are displayed in figure 6. Beginning with 6(a) we see a display
61
- 2
- 2 0 2 (a) E״ vs E, at (x,x,0,0,0)
- 0 . 0 0 . 0 ־*-0.005 0 0.005 1(b) E vs EL at (x,0,x,0,0) ־
0.01
0.005
0
•0.005
-0 .01
0.01-0.005־ 0 0.005 0.01 -2 0 2(c) E״ vs B at (x,0,0,x,0) (d) 8 ״ . vs E, at (2.3,x,x,0,0)
0.01
0.005
0
0.005
-0 .01
0.01
0.005
0
0.005
-0 .01
0.01
0.005
0
0.005
-0 .01
- 2 0 2 - 0 . 0 - 0 . 0 0 5 0 0.005 0.01 (e) B, vs E, at (2.3,x,0,x,0) (f) B, vs B. a : (2.3,0,x,x,0)
, I , , . , I , . . . I . . . . I . -0 .01-0 .005 0 0.005 0.01
(g) B_ vs B. at (2.3,0,x,0,x)
Figure 22: 2 - D cuts i n the 6-D smoothed velocity space results of C E I of C4׳. The
values i n subtitles are for coordinates of ( E b , Eu B + , By, B - ) . "E" coordinates are
i n radians and "B" are i n units of V/Vb&am•
62
of Eb versus Et on the plane that cuts B x , B y , and B 2 at the origin. In this plot,
the peak is at (E6,Et)=(2.3,0) and the two other permutation symmetric images of
this peak. E t = 0 means that there are four equal angles, that is either the molecule
geometry is a kite shape (two adjacent sides equal but different than the other two
adjacent sides) or square or rhombus. The deviation of E& from 7r is the result of
bending of this structure. Other peaks appear at the origin and at Eb = 7r, these
peaks are at higher symmetry points and therefore the error associated with the
density at these points is higher. In order to further define the nature of this shape
we look at figures 6(b) and 6(d) in comparison with figures 6(c) and 6(e). These
show the correlation between B y , B + and E&, E t at the peak. B + is constructed as
( B x + B z ) / \ / 2 and thus is the measure of the difference between the sum of two
adjacent bonds and the sum of the opposite adjacent bonds. B_ is constructed as
( B x — B z ) / y / 2 . We can see that both B y and B + peak at 0. This means that the
difference of diagonals and difference of sum of opposite bonds is 0. The same result
exist for B_ and thus the peak of the distribution describes a square bent from the
plane. In the above description we have only considered the peak position. However,
as shown in the C 3 vs B 3 case (section 1.4) the transformation from V-space to
R-space can be very much affected by the distribution around the peak as well as
by the tails of the distributions. In figure 6(a) we can see that the peak extends
towards 7r with another peak there, thus describing a vibration toward the plane.
Comparing the extension of ( B - B + ) at B y = 0 (figure 6(g)) with the extension of
By at B + = B - = 0 (figure 6(e)) we can see that the configuration undergoes strong
vibrations towards a rhombus (the difference of diagonals changes without changing
by much the difference between sides). Finally we turn our attention to figure 6(f).
This figure describes the correlation between B y and B+. This figure holds a major
63
2 bending angles twist angle
F r o n t View Side V i e w
Figure 23: A n g l e s i n use f o r 4 a t o m l i n e a r / r h o m b u s c l u s t e r s i m u l a t i o n
key to the interpretation of the results. We can see that besides the vibration towards
rhombus we have a correlation that may be described by a vibration towards a kite
configuration. It may be described graphically by moving atoms 2 and 4 in figure 5
to the left or right.
The complete configuration of C 4 in velocity space may be described by a square
bent from planarity undergoing vibrations toward the plane and correlated vibration
toward rhombus with extension towards kite shape.
The reconstruction of R-space coordinates for C4 from the V-space described
above was done in a few steps(see section 2.3.4). The first step was the creation
of an approximate grid-like simulation. We assumed that the configuration may be
either linear or rhombic. Thus we simulated the CEI of a large number of config-
urations where the variation between the different configurations was the angle of
each outer bond to the inner bond and the angle of twist between the two outer
bonds(figure 7), the bond-lengths were kept constant at 1.29 A according to the
theory[51]. This approximation was needed because creating a grid on the full 6
dimensional space would have entailed too much computer resources. The results of
this grid simulation were input into the algorithm described in section 2.3.4. When
the algorithm converged to a steady state we convoluted the bond-lengths with a
Gaussian and continued the configuration searching program with all parameters
64
free. The result of the program was checked against the data and was found to be in
good agreement with it. The R-space distribution which was found was very similar
to a linear distribution. The parameters of this distribution were extracted and a
new simulation was created. This simulation samples the distribution of the appro-
priate linear configuration and simulates the Coulomb explosion of these samples.
The results of this simulation are depicted in figure 8, where we can see the quality
of the fit, by comparison with figure 6.
It is possible to see that the general features and distributions are recreated quite
well. Figure 8(a) shows the peak at E t = 0 with Eb shifted from 7r. At the peak we
can see that cuts (d)-(g) are recreated quite well. At this position B y is elongated
while B + x £?_ are narrow. Finally the correlation between B y and B + shows the
same "kiteward" motion as in the data. The positions and widths are similar to those
of the data. A feature which is different between data and simulation is the absence
of peak at the origin (the tetrahedral structure, figure 8(a)-(c)). In order to check
if this discrepancy is significant we must analyze the statistical error of the data(see
sectionl.2).
Figure 9 shows the comparison between the simulation and data including error
analysis. The error bars belong to the data and the dashed line is the simulation.
Notice the large error associated with the peak at 0. It is also possible to see that
there is some discrepancy between the simulation and the data at the region Eb = 1•
This "spillout" is due to to incomplete characterization of the vibration of the linear
cluster and it will be dealt with in the next section.
65
-2 0 2 -0.01-0.005 0 0.005 0.01 (a) E״ vs E, at (x,x,0,0,0) (b) E״ vs B. ot (x,0,x,0.0)
-0.01-0.CC5 0 0.005 0.01 (c) E״ vs B, at (x,0,0,x,0) (d) B. vs E, at (2.3,x.x,0,0)
-2 0 2 -0.01-0.005 0 0.005 0.01 (e) B, vs E, at (2.3,x,0,x,0) (f) 3, vs B, ct (2.3,0,x,x,0)
-0.01-0.005 0 0.005 0.0; (g) B. vs B. at (2.3,0,x,0,x)
Figure 24: 2 - D cuts i n the 6-D smoothed results of simulated C E I of C4- T h e values
i n subtitles a r e for coordinates of ( E b , E t , B + , B y > B _ ) . T h e "E" coordinates a r e i n
radians and the "B" a r e i n units of V / V ) , e a m
66
Figure 25: Comparison between data(with error bars) and simulation (dotted l i n e )
on the Eb coordinate where the rest of the coordinates a r e at o r i g i n
4.1.4 Results and Discussion
Theoretical calculations predict that the bond-lengths of the neutral linear C4 would
be 1.282 A for the inner bond and 1.299 A for the outer bond with 2% variation
depending on method of calcuIation[51]. The negative cluster is predicated to have
the opposite configuration with a longer inner bond, 1.325 A, and shorter outer bonds,
1.273 A . The results of our analysis show an inner bond-length of 1.24 A and an
outer bond-length of 1.35 A with an error of 10% due to our incomplete knowledge
of the electron screening effects at the initiation of the Coulomb explosion in the
foil[8], the multiple scattering in the foil(see section 2.3.4), and the limited temporal
resolution of the detector. Referring to figure 9 we may notice that it is also possible
to create a simulation in which the peak appears in V-space at Eb — זד while still
staying in the statistical error boundary. In this case the internal and external bond-
lengths become equal for the simulation and the "spillout" less pronounced. Due to
the small number of events resulting in poor statistics this feature of the data may
not be completely characterized. Thus the results are in agreement with what is
67
currently known and assumed for linear C 4 . Though the simulation distribution is
completely defined it is impossible to extract relevant vibrational widths due to the
small amount of events which entail a large error in these parameters.
We now turn our attention to a question that is still open. Our measurements
were done using neutral clusters which were prepared by photo-detachment of a C 4
beam by 2.35 eV photons(see experimental section). In a separate experiment we
have measured a threshold in the photo-detachment energy of C 4 at 2.1 ± 0.1eV[29].
The 2.35 eV photo-detachment energy is approximately 1.5 eV lower than the meas-
ured electron affinity(E.A.) of linear C4 [26, 28]. At the same time the calculated
E .A . of rhombic C 4 is approximately the same as the threshold we have found[63].
This puzzling result together with the conclusive analysis of the CEI data needs
to be resolved. Several possible explanations to this inconsistency between photo-
detachment energy and structure have been proposed.
One possibility is that the linear population is highly excited. Normally we do
not think of an excess of 1.5 eV as possible in vibrational excitation. But one
should remember that the linear 4 atomic cluster has 7 separate vibrational modes
(3 stretching and 2 doubly degenerate bending). Thus each vibration has to carry
an excess of approximately 1600cm - 1, which is not out of the question. The main
difficulty in this interpretation is that it does not explain the threshold that we have
found.
Another explanation proposed, again relates to thermal excitation. It was cal-
culated that at high temperatures rhombic C 4 would be able to tunnel through the
potential barrier and come to an equilibrium with the linear species[50]. The equi-
librium would favor only a small percent of rhombic clusters and thus the rhombic
species would be lost. We have tried in our simulations to "force" some of the data
68
to be rhombic and some linear and it was found that the data is consistent with a
of rhombic structures. The problem is that the distribution produced by 5%10%־
CEI of rhombic C4 is completely overlapped by part of the distribution which is
produced by CEI of linear structure. Thus, at this stage, there is no way to prove
or disprove this explanation.
Finally, the last proposed explanation is that the barrier of transition from neutral
rhombic to linear structures was overestimated. In this case the neutral rhombic
isomers would immediately after neutralization traverse the barrier and become a
linear isomer, due to the higher entropy of this isomer.
4.1.5 Conclusion
In the research we have described, we have reaffirmed the existence of the linear C4
isomer with bond-lengths of approximately 1.3 A. The data may not be explained
solely by a rhombic structure but it may contain a small percentage of this struc-
ture. An open puzzle still remains as to whether the population measured was a
warm linear population or a rhombic population that traversed the barrier to linear
structures after neutralization. In future experiments our group intends to try the
same measurements using a source that produces vibrationally cold clusters. At the
time of writing of this thesis, the source has only produced initial results(see section
1.2.4) and therefore no decisive conclusions may be reached yet.
6 9
4.2 F i r s t measurement of the structure of tetra-atomic boron
cluster
Abstract
The structure of the B 4 cluster was measured for the first time using CEI.
The results of this analysis show a planar square with 1.45±0.15A bonds and
small bending vibrations. The full results will be described in the following and
compared with theoretical predictions.
4.2.1 Introduction
With only five electrons per atom, boron clusters are particularly amenable to a b -
i n i t i o theoretical calculations, and boron is the lightest element which forms chemic-
ally bound clusters and is not prohibitively toxic to handle. The only measurement
of boron clusters of size beyond the tri-atomic was reported by Hanley e t . al.[40].
They examined the bonding in boron cluster cation B2~_1 3 by measurement of the ap-
pearance potential and fragmentation pattern in collision-induced dissociation (CID).
They found a characteristic size dependency of different fragmentation channels.
A few groups have calculated the optimized geometry and vibrational frequencies
of B 4[40, 39, 65, 66]. They predicted that the ground state is a singlet square ( 1 A i g )
which undergoes pseudo-Jahn-Teller distortions to a rhombic structure ( 1 A g ) but the
energy difference is so small that the effective structure is a square for any practical
temperature. The bond-length is predicted to be 1.508 A[39] and the bending to the
rhombic structure from a square one is predicted to be only a mere 5° change from
90°.
70
4.2.2 Experimental
A negative boron beam was produced by the Cs sputter source. The beam was
chopped and mass separated for 1 0B± mass 40. Further chopping reduced the pulses
to 400 nsec pulses which in turn were injected into the Tandem H U D Pelletron.
The acceleration voltage in this experiment was 11.5 M V . A laser pulse from the
N d - Y A G second harmonic (A = 532 nm, hv — 2.35eV) was fired in time to photo-
detach the cluster bunch when it is in the terminal of the accelerator. Neutralized
clusters continued through the accelerator and impinged on a 1.4/ug/cm2 Formvar
foil located down stream. After traversing the foil the stripped molecules coulomb
exploded and fragments with charge state q=+3 where directed by the mass selection
magnet towards the detector.
4.2.3 Analysis
The definition of coordinates and the method for retrieving the R-space coordinates
was performed exactly as in the case of C4 (see section 2.1.3) with the exception
that the initial bond-lengths were taken as 1.5 A according to reference [39].
The analysis of the experimental results produced 468 good events of 4 B 3 + hits
on the detector. Due to the permutational symmetry each event is multiplied by 24
by permuting the indexes of the fragments. Thus 11232 sets of velocity coordinates
were used for the analysis of B 4 .
Figure 10 shows the results of the CEI measurement in symmetry coordinates.
The smooth functions depicted are a result of the convolution of the data with a 6
dimensional Gaussian. The width of this Gaussian is 0.3 radian for the angular " E "
coordinates and 2*103־־ in units of V / V f e e a m for the velocity " B " coordinates. The
symmetric coordinate E c was integrated over, in this analysis, by using a very large
71
- ־ 2
. . ו . . . . ו . . . . ו . . . - 2 0 2
(a)E״ vs E, at (x,x,0,0,0) - 0 . 0 - 0 . 0 0 5 0 0.005 0.01 (b)E״ vs B, at (x,0,x,0,0)
l l l l l l l l l l l l l l l . i l
0.01
0.005
0
0.005
-0 .01
-0 .01-0 .005 0 0.005 0.01 (c)E״ vs By at (x,0,0,x,0)
-2 0 2 (d)B״ vs E. at (7r,x,x,0,0)
0.01
0.005
0
0.005
0.01
0.01
0.005
0
0.005
-0 .01
0.01
0.005
0
- 0 . 0 0 5
0.01
- 2 0 2 (e)B, vs E, at (7r,x,0,x,0)
- 0 . 0 V 0 . 0 0 5 0 0.005 0.01 (f)B, vs b, at (n,0,x,x,0)
-0 .01-0 .005 0 0.005 0.01 (g)B־ vs b״ at (n,0,x,0,x)
Figure 26: 2 - D cuts i n t h e 6 - D s m o o t h e d r e s u l t s of C E I of B 4 . The values i n s u b t i t l e s
a r e f o r c o o r d i n a t e s of ( E b , E t , B x , B y , B z ) . W h e r e "E" c o o r d i n a t e s a r e i n r a d i a n s a n d
" B " a r e i n u n i t s of V/Vbeam
72
width for the Gaussian in this direction.
Before displaying the complete results and fits it is helpful to explain the results
of velocity space which are displayed in figure 10. Beginning with 10(a) we see a
display of E& versus E t on the plane that cuts B x , B y , and B z at the origin. We can
see that the peak is at ( E b , E t ) — (7r,0) and the two other symmetric images of this
peak. E t = 0 means that there are 2 pairs of equal angles, that is either kite shape
(two adjacent sides equal but different than the other two adjacent sides) or square
or rhombus. Eb — 7r means that the structure is planar. Further identification of the
structure requires the analysis of the "B" coordinates. Looking at figures 10(d)-(g)
we can see that B x , B y , B z are peaked at the origin for the cut that passes through
the peak at Eb x E t . Thus the highest probability structure in velocity space is a
square. The "vibrations" in velocity space may be extracted from the distributions
and correlations of the smoothed function. In figure 10(a) the peak extends from
Eb = 7r towards the origin. This extension is the bending from planarity motion. On
the figures showing the correlations between B^, B y , and B z no correlation appears.
The width of the peaks is almost equal in all directions with a slightly larger spread in
the By direction which is the difference between the diagonals. Thus the full analysis
of the distribution in velocity space indicates a square figure with small vibrations
and a slightly larger vibrations in the direction which transforms the square into a
rhombus.
The R-space coordinates distribution function which was found by the iterative
simulation program was very similar to a square-like distribution. The parameters
of this distribution were extracted and a CEI simulation was performed by sampling
this distribution function. The results of this simulation are depicted in figure 11 ,
where we can see the quality of the fit by comparing with figure 10.
73
-2 0 2 (0)E״ vs E, at (x.x,0,0,0)
-0.01-0.005 0 0.005 0.01 (b)E״ vs B, at (x,0,x,0,0)
-0.01-0.005 0 0.005 0.01 (c)E״ vs 3 , at (x,O,0,x,O)
-2 0 2 (d)B, vs E, at (7r,x.x,0,0)
-2 0 2 (e)B, vs E, at (7r,x,0,x,0)
-0.01-0.005 0 0.005 0.01 (f)B, vs B, at (7r,0,x,x,0)
0.01 0.005 0 0.005-י0.0-(f)B, vs 3, at (rt,0,x,0,x)
Figure 27: 2 - D cuts i n t h e 6 - D s m o o t h e d r e s u l t s of s i m u l a t e d C E I of B 4 . The values
i n s u b t i t l e s a r e f o r c o o r d i n a t e s of ( E b , E t , B x , B y , B z ) . "E" c o o r d i n a t e s a r e i n r a d i a n s
a n d " B " i n V/Vbeam
74
(I) eu (!)
G - ©
— © a G ־׳ — (j) b 0 b«« 0 (j) ־׳
(j) — © ©
— • Q b2e Q — © — e״ G —
(!) - O
Figure 28: D i a g r a m of t h e D 4 h v i b r a t i o n symmetry c o o r d i n a t e s .
4.2.4 Results and Discussion
In order to compare our results with those of a b - i n i t i o calculations we will use the
point group symmetry coordinates of the D4/! symmetry[67]. These coordinates are
depicted in figure 12.
The results which were found are a D4 . symmetry square configuration with
a bond-length of 1.45A and an error of 10% on the bond-length which is in good
agreement both with the predicted structure of the thermally excited cluster and with
the predicted 1.508 A bond length[39]. Our results also give an estimate of the width
of the R-space distribution in various coordinates. The width of the distribution
may be compared with the ground state width of these coordinates by using the
relation[33]
V flLO
The CEI is sensitive to the out of plane bending B! u . The out of plane bending
vibrational frequency is predicted by the theory to be to = 290cm-1. This amounts
to a width of 4.07° in the probability for bending angle which is defined as the angle
75
between the line connecting the two atoms marked by + in figure 12 and the line
connecting one of the atoms marked by + and the center of mass of the atoms marked
by -. The result of our analysis shows a width in the distribution of bending angle
of 14° ± 2° where the error is limited by the width of the multiple scattering in
this coordinate and the detector resolution(sect. 2.3.5 and 2.2.4). An attempt to
explain this data only with thermal excitation would result in approximatly 5 times
room temperature. This is very high and also very different than the result we have
obtained for C3(sect. 1.2.4). Other explanations could be that the structure of the
negative cluster, which has not been calculated nor measured yet, is much different
than the neutral or that the calculation of the out of plane bending frequency is
erroneous.
4.2.5 Conclusion
The results show that the average structure measured for a distribution of vibrational
states of B4 is a square D4h structure. This result is the first experimental evidence
for the structure of this cluster.
This measurement also shows the power and limitations of CEI . The vibrational
modes related to in plane deviation from the symmetric square configuration are
difficult to resolve for reasons similar to those which prohibit the measurement of
bending angle of triatomic clusters near 60°(see sections 1.2, 2.3.5 and 1.3). In
contrast the bending B ! u mode is accurately measured (though it is far from the
predicated ground state width). In fact, measurement of this mode is very similar
to the measurement of the bending from linearity of a triatomic cluster. This was
shown to be a region where the CEI is extremely sensitive.
76
4.3 Conc lus ion and comparison of C 4 and B 4 C E I measure-
ments
The tetra-atomic clusters of C4 and B 4 where measured using the CEI method. It
was found that the structure of C 4 is consistent with a linear isomer population
and the one of B 4 with a square configuration. The result of boron is in agreement
with the theoretical predictions and serves as the first experimental evidence for the
structure of this cluster. The measurement of C 4 leaves a, yet, unsolved puzzle.
Of the two isomers predicted for C 4 the electron affinity of the linear isomer was
measured by several groups to be in the region of 3.8 eV. The electron affinity of
the rhombic isomer is predicted to be in the region of 2.1 eV and our measurement,
which was done using 2.35 eV photons should have resulted only in the rhombic
isomers. Nevertheless the analysis shows that the data is primarily identified as a
linear isomer population.
The comparison between both measurements holds some important lessons re-
garding CEI measurements. Careful inspection of the CEI results in V space of the
boron and carbon cases shows that the difference is mainly hidden in the tails of the
distributions. This case is similar to what was found in the triatomic case. The main
difference is that in the triatomic case there were only 2 dimensions to explore (ex-
eluding the fully symmetric coordinate). Therefore the analysis is much simpler as
the whole picture may be displayed graphically. In the 4 atomic case, again excluding
the fully symmetric coordinate, there are already 5 dimensions and the analysis is
far from intuitive. The iterative algorithm which was developed solves the problem
of intuition since the computer program can "see" all dimensions at once.
Another feature which is evident is the role of statistics, or the number of events
measured. It is obvious that as the number of atoms measured grows the difficulty in
77
achieving detector coincidence grows which is why the number of events collected for
the 4 atomic clusters is approximately 20% of what was measured for the tri-atomic
cases.
Inspection of the results which were obtained in the two cases poses a question:
the C E I images very well the bending angle from linearity. If this is the case, why
was the out of plane bending of B 4 resolved and the linear bending modes of C 4
were not? The number of events which were measured for both cases is similar but
sti l l the statistical consideration is very different. The out of plane bending of B4 is
a single dimensional one. Therefore it is easy to see that if the structure is planar
most of the events would be in or near the planar configuration. In contrast there
are 2 sets of doubly degenerate modes which bend the linear C 4 out of the linear
configuration. The result of this is that the average configuration is bent and there
are very few events which come from the linear region which is imaged so well by
the C E I .
Once the cold source wi l l function well the group intends to measure a vibration-
ally cold population of C 4 . B y doing that with very good statistics it is hoped that
the puzzle of C 4 w i l l be solved and at the same time the appropriate modes which
are visible through the C E I measurement wi l l be characterized with good accuracy.
78
5 A detailed study of conformations in the ground
state of C H J
Abstract
The results of Coulomb Explosion Imaging for cold CHJ" molecular ions
are converted to molecular conformation probability density. This is the first
complete conversion of such data for a relatively complex molecule. The results
are compared with the corresponding predicted potential energy surfaces which
manifest a Jahn-Teller symmetry breaking. The density of conformations along
a nuclear rearrangement path is deduced and the comparison with theory is
made.
5.1 Introduct ion
The CH4 cation is of interest from several aspects. For example, it is an important
element in the natural course of building large hydro-carbon chains and it is one of
the simplest systems subject to a Jahn-Teller effect in its electronic ground state.
Among the many theoretical and experimental studies related to the methane
cation, two are of main interest to the present contribution. Frey and Davidson
[68] have studied the potential energy surfaces of CH4 and investigated the Jahn-
Teller effect of the three lowest electronic states. A possible pseudo-rotation was
proposed and the Fukui[69] reaction path between equivalent C 2 v potential min-
ima was calculated. Experimentally[70], the CEI method demonstrated for the first
time the capability of observation of reaction paths in molecules. It was shown[70]
that the raw CEI results of CH4 exhibit a reaction path which resembles closely
the theoretical predictions [68]. A qualitative comparison between the experimental
findings and the theoretical expectations strongly support the underlying theoretical
79
approach. Yet, the CEI data contain far more information than this qualitative com-
parison. In principle, quantities like absolute bond lengths and bond angles as well
as density of the different conformations can be extracted. The development of the
iterative algorithm for the unbiased reconstruction of a multi dimensional R-space
from V-space (sect. 2.3.4) has enabled the deduction of such detailed information. In
this chapter, analysis based on this new method is carried out for the data of CEI of
CH4־. The density of conformations along a nuclear rearrangement path is deduced.
As far as conformations along the path, the results are comparable to the theoret-
ical predictions [68], but there seems to be an inconsistency between the measured
density and the predicted potential along the path.
5.2 The C E I experiment of the Ground State of the Methane
ion .
A full account of the CEI experiment, which was done at Argonne Nat. Lab., can
be found in reference [70] and further references given therein. A short description
of the experiment is merited here. Cold C H J ions[71] were accelerated to an energy
of 4.5 MeV. A Coulomb explosion process was initiated by the passage of the ions
through a 0.6/ig/cm 2 Formvar target. Coincidence of 4 protons and a carbon with
charge q=-f 3 were collected using two position-and-time resolving detectors[72, 12].
Fifteen velocity components were recorded for each of the 2500 measured ions. In
the previous publication[70], the topology of a 5-dimensional space spanned by linear
combinations of the H C H angles in the measured velocity space (V-space ) was
described. A physically justified assumption was employed that the events can be
considered as due to random sampling of a smooth probability function in V-space .
This assumption is at par with traditional quantum mechanics measurement theory.
80
This smooth function is defined as the physical density in V-space, where the velocity
measurements were recorded. A "walk" along the ridge of that density enabled
locating the "reaction-path" which is the main result of reference[70]. No attempt
was made to express this V-space density in terms of a new function which is defined
in the space of the conformations of the measured molecule, the R-space , because
of lack of appropriate tools for performing such a transformation. This last task is
undertaken here.
The 5-dimensional space spanned by linear combinations of the HCH angles of
reference [70] was extended with 4 CH relative velocities to a 9-dimensional space.
This space completely defines the internal degrees of freedom in V-space. The re-
maining 3 center of mass coordinates and 3 orientations, though highly important for
the purposes of experimental tests of possible systematic errors, will not be discussed
further.
5.3 Choice of coordinates, density functions and errors.
The choice of internal coordinates in both V-space and the molecular conforma-
tion space (R-space) for the methane cation are similar to those used by Frey and
Davidson[68] and to those used in the analysis of the tetra-atomic clusters(sect. 2.1.3).
Ai = (X1+X2 + X3 + X 4 ) / 4 :
Tx = [ X 1 + X 2 - X 3 - X 4 ) / 2
Ty = (Xl-X2 + X 3 - X 4 ) / 2
Tz = { X l - X 2 - X 3 + X 4 ) / 2
B x = (01 ,2 03 , ־ 4 ) / V ׳ 2
B y = ( 0 1 1 3 - 0 2 , 4 ) / v ׳ 2
81
Bz = (01 ,4 02 ־ 3, ) / V ׳ 2
Eb = ( 2 ( 0 1 1 2 + 03 I2/</((־(01,3 + 01,4 + 02,3 + ©2,4 (4,
Et = (01 , 01 ־ 02,3)/2 3 ,4 + 02 ־ 4,
Where 0־j is the H ־ C H j angle in either V-space or R-space. The X are defined as
proton carbon bond-lengths in R space and as the proton carbon relative velocity in
V-space.
The ensembles in both V-space and R-space are comprised of events which are
defined as points in the 9-dimensional space defined above. A l l the statistical er-
rors in R-space propagate from the measured independent events in V-space. For
each measured event, all the proton permutational events are included and the error
propagation is calculated accordingly. More details on the error propagation in the
analysis is given in section 2.3.5.
As mentioned above, a physically justified assumption is made that the events
can be considered as due to random sampling of a smooth probability function in V
or R spaces. These smooth functions are defined as the physical densities in their
appropriate coordinate space.
In order to define a smooth function based on the 9-dimensional discrete events,
the set of events is convoluted with a 9-D Gaussian as described in section 2.3.4.
Since every point of each density function is defined as a linear combination of
individual contribution of events, then the correlated error of the density can be
easily calculated. As a result, the density at points which are close to each other
have comparable correlated error. A detailed discussion of optimal choice of the a
parameters for a given ensemble is given in section 2.3.4. The choice selected here
is presented in Table 1.
82
Table 1: Gaussian widths used in the smoothing of V and R spaces.
A x T B E
R 0.3 0.1 0.13 0.13
V 0.004 0.004 0.13 0.13
W i d t h s of A i a n d T a r e i n a t o m i c u n i t s f o r R-space a n d i n V / V b e a m f o r V-space,
t h e o t h e r u n i t s a r e i n r a d i a n s .
5.4 Comparison of the measured and the simulated densities.
A statistically distinct (unambiguous) density in R-space was found and it is the
measured R-space probability density function for the conformations of the methane
cation. The quality of the result can be displayed by the comparison of the measured
and the simulated densities in V-space. This is given here in two styles, either by
one parameter graphs of the measured density with (correlated) statistical errors
and, for comparison, the simulated density, or by two dimensional contour levels of
measured and simulated densities. In both styles, only cuts, rather then projections,
are shown. Though this is a meager display of the immense 9-dimensional space, it
is hoped that the most crucial 1- and 2-dimensional cuts of the densities are chosen.
Three conformations of significance will be discussed and it is important to point
out their approximate root in the measured V-space. The first is the C 2 y conform-
ation where the theory predicts the minimum of the potential ([68] and references
therein). The second is the transition state point (TS) - the theoretical prediction
of a point of no return on the reaction path for a permutational switch between two
equivalent C2v potential minima. The third is the conformation with the highest
density ( P E A K ) in R-space found by this experiment. The cuts shown in the figures
were chosen to display the V-space density around the images of these conformations.
83
DATA SIMULATION
Figure 29: 2 - D cuts i n t h e V-space of t h e data(left) a n d s i m u l a t i o n ( r i g h t ) . Top: B z
vs B y a t E b = E t = B x = 0. M i d d l e : E h vs B z a t E t = B x = B y = 0. The l i n e s
m a r k e d a,b a n d c a r e t h e l i n e s of t h e 1 - D cuts. B o t t o m B z vs B y a t E t = B x = 0
a n d E b = 0.46.
84
״ 7 ״ 0 7 0
־ 10
20.4-0 .2 0 0.2 0.4 0.6 0.8 1 Eb(radian)
״ 7 0
0- 1- 1.5צ .5 0 0.5 1 1.5 Bz(radian)
Figure 30: 1 - D cuts i n V space t h r o u g h l i n e s s h o w n i n figure 2 9 m i d d l e r i g h t . The
s o l i d l i n e s b o u n d t h e s t a t i s t i c a l e r r o r r e g i o n of ±0־ i n t h e m e a s u r e d density. The
d o t t e d l i n e s a r e t h e s i m u l a t e d density.
Inspired by the cuts in V-space shown in figure 2(a,b,c) of reference [70], figure 29
displays 2-D cuts in V-space of the measured and the simulated densities. The V-
space C2v peaks appear in figure 29 (top). Two of these C2v peaks in addition
to two of the R-space PEAK images in V-space appear in figure 29(middle). In
figure 29(bottom) the transition state images between four of the R-space PEAK
permutational images are exhibited. Such a comparison between two dimensional
cuts may be misleading since the experimental error is not shown. Therefore, one
dimensional cuts were created through lines which are shown in figure 29 (middle-
right). The one dimensional cuts are shown in figure 30. The graphs representing
the data are shown as a band enclosed by full lines which represent a 10־ confidence
level of the measured density and, for comparison, the simulated density in a broken
line. Notice that the error near symmetry points is larger. Every event near such a
symmetry point contributes several times because of the permutational multiplicity.
But, the error is still due to a single event(see sect 2.3.5).
Though the method of deducing the R-space probability function is supposed
85
to be unbiased and statistically consistent with the measured data, it is reassuring
to notice that indeed the above V-space cuts from the measured and the simulated
results are statistically consistent with each other, at least where the bulk of the
data is concentrated. Further consistency checks were carried out by varying the
smoothing parameters. This will be discussed in the next section.
5.5 The "reaction path" ־ a walk along the conformation
density ridges.
As was discussed earlier, the density in R-space is the primary result of the CEI
experiment. This density can be used for estimates of observables which depend
on the internal coordinates of the studied species. When a molecular state has a
well defined maximum probability, then parameters of interest may be those which
define the equilibrium geometry and the second moments describing the breadth
of the distribution. For a floppy structure, the expansion around a single point
describing the maximum of the density is not satisfactory. The simplest extension is
to consider the elongation near the maximum along the largest width. This can be
expressed mathematically as follows. The density is expanded around the maximum
by a Taylor series. This is easily performed by the use of the smoothing prescription
as described above. The second order term in the expansion is a bilinear form which
can be transformed to a diagonal form. The direction associated with the largest
diagonal term is called here the ridge direction. A step is taken in this direction
and the point of maximum density is found in the space which is perpendicular to
the ridge direction. The newly arrived point is used for the initiation of the next
step. Except for points of bifurcation of the ridge where additional care should be
taken, this process defines a line in space along the ridges of the density. The process
86
is similar to finding a reaction path on a potential surface except in that case one
chooses the smallest diagonal term for the step direction.
Classically, there is a simple relationship between the density of a micro-canonical
ensemble at an energy E and the potential energy V.
( n - 2 ) Density(x) oc ( E - V ( x ) ) 2
where n is the number of degrees of freedom of the coordinates x. Thus for n > 2, if
the level of E is above a barrier between two potential minima, then the path along
the density ridge is identical to the reaction path. Also, the peaks of the density
are at the minima of the potential. The quantum mechanical ground state density of
such a system contains corrections to the above coming from the kinetic energy term,
therefore the two paths do not have to be identical. But, if a potential minimum
is symmetric along a reaction path, such as C2v minimum for the methane cation,
then the ground state density should have a maximum at that point. Therefore, it is
interesting to compare the experimental density ridge path and corresponding density
to the theoretical reaction path for nuclear rearrangement in the methane ion ground
state as well as the potential along the path. These features are shown in figures
31 and 32. The resemblance of the 6 HCH angles along the experimental and the
theoretical paths is astounding in view of the large range span by the conformations.
The relation of the density and the potential along the paths (fig. 32) are surprising
because the C2v conformation is the minimum in the theoretical potential surface and
therefore expected to have a maximal density, yet, it is a local minimum (a saddle
point) along the ridge of the measured density.
The experimental significance of this surprising result was tested in two ways.
Several statistically independent calculations were carried out for arriving at the final
R-space density. The variance in the ratio of density at C2v and at PEAK changed
87
40 1.5 •1 -0.5 0
Theory
0.5 1 1.5 path length ־ 1.5 40 0־ 1 .5 0 0.5 1 1.5
path length Experiment — R space
•1.5 - 1 -0.5 0 0.5 1 1.5 path length
Experiment — V space
Figure 31: The c h a n g e i n t h e H - C - H a n g l e a l o n g t h e r e a c t i o n p a t h f r o m one C2v
s t r u c t u r e t o a n o t h e r p e r m u t a t i o n a l l y e q u i v a l e n t C2v s t r u c t u r e t h r o u g h t h e t r a n s -
i t i o n s t a t e . Left: angles a l o n g t h e c a l c u l a t e d F u k u i r e a c t i o n p a t h f r o m F r e y a n d
D a v i d s o n [ 6 8 ] . R i g h t : angles a l o n g t h e e x p e r i m e n t a l density r i d g e p a t h i n R-space .
B o t t o m : angles a l o n g t h e " r a w " V-space e x p e r i m e n t a l density r i d g e p a t h as p u b l i s h e d
i n reference [70]
88
Energy ( K c a l / m o l )
Density
יz. -
1.8 \ \ 0.9
1.6 7 / \ \ —_ 0.8
1.4 y A 0.7
1.2 / \ 0.6
1 0.5
0.8 0.4
0.6 0.3
0.4 0.2
0.2
0 " i^r , , , 1 1 , , ו I 1 j , 1 , , , . 1 . , , 1
- 0.1
0
0.2
0 1.5 -1 -0.5 0 0.5
Path Length 1 1.5
0.1
0
Figure 32: The s o l i d l i n e i s t h e c h a n g e i n t h e energy a l o n g t h e F u k u i r e a c t i o n p a t h
f r o m one C2v s t r u c t u r e t o a n o t h e r p e r m u t a t i o n of t h e same s t r u c t u r e t h r o u g h t h e
t r a n s i t i o n state as c a l c u l a t e d by F r e y a n d D a v i d s o n [ 6 8 ] . The dashed l i n e i s t h e
e x p e r i m e n t a l p r o b a b i l i t y density a l o n g t h e r i d g e p a t h . The peak of t h e density i s
a r b i t r a r i l y n o r m a l i z e d t o u n i t y .
89
1
8 0.9 c CL
S 0.8
> 0.7
0.6 03־
£ 0.5
0.2
ל 0.1
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 P
Figure 33: Densities of the C2v conformation and transition point conformation
divided by the density at the peak as a f u n c t i o n of smoothing parameter p(see t e x t ) .
A suggestive l i n e a r extrapolation is drawn to "no-smoothing" p=0.
by less than 10%. Another test was carried out and is described below. The R-space
densities at the C2v and at the transition state conformation were divided by the
density at P E A K and plotted as a function of a smoothing parameter p which is
defined as:
a ( p ) = p . a 0
where cr(p) are the standard deviations parameters used to find the densities and
a0־ are given in table 1.
These ratios are expected to approach 1 when p is large, but, the extrapolation of
p towards zero should yield the 'no-smoothing5 ratios. The extrapolations are shown
90
in figure 33. Clearly, such a procedure increases the errors of these 'no-smoothing5
ratios, but, the significance of the statement that the density at the C 2 v configuration
is a deep local minimum is reinforced by the extrapolation lines.
It is worthwhile to add here that the C2v maximum in V-space which can be
clearly observed in figure 29(top) and at E!,=0 in figure 29(middle) is an artifact of the
Coulomb transformation distortion. This property of the Coulomb transformation is
automatically taken into account within the CEI simulation. The C 2 v saddle point
along the ridge in R-space has a corresponding image in V-space which is NOT at
the mentioned artifact peak. Another point of interest is that the ridge path here
departs from the C2v conformation by a pure twist mode. This has been predicted
by the theory [68]. As might be expected this is incorrect for the Coulomb distorted
V-space ridge path found by [70] and depicted in figure 31(bottom). Thus, although
the "raw" V-space reaction path is in some aspects similar to the final R-space path
these last two points emphasize the need for a complete transformation of the CEI
data to R-space if an undistorted density function is required.
For completeness, an additional extrapolation graph is plotted in figure 34. for
the 3 different angles of the 6 HCH angles at the C2v , saddle point in R-space. The
extrapolated values for the HCH angles are almost exactly the predicted findings
in [68]. Table 2 lists the experimental CH bond length for the main conformations
and the corresponding theoretical estimates [68]. The above consistency with theory
stresses the peculiarity of the saddle point in the conformation density at C 2 v •
Finally, according to the CEI experiment, the most probable conformation of the
methane cation vibrational ground state is almost like the predicted C 2 v structure
except for a twist along the line of intersection of the two C2v planes. The twist
breaks the four equal angles into two pairs of small and large angles. At the most
91
Figure 34: T h e 3 different angles of the C2v conformation as a f u n c t i o n of smoothing
parameter p(see t e x t ) . A suggestive l i n e a r extrapolation is drawn to "no-smoothing"
p = 0 .
Table 2: C H bond-lengths of the various conformations in A .
Experiment Theory
R! R2 R3 R4 R! R2 R3 R4
1.13 1.09 1.09 1.13 1.155 1.075 1.075 1.155
PEAK 1.14 1.08 1.08 1.14 - - - -
TS 1.05 1.05 1.12 1.23 1.117 1.117 1.100 1.147
T h e error i n the r a t i o of the bond-lengths is 1 % and the absolute error i n bond-lengths
is about 5%
92
probable configuration the pair of large angles become equal with the largest C 2y
angle. Two kinds of rearrangements occur. The first kind is a twist back through the
C 2 v into a mirror image of the original conformation. The second kind is a splitting
of the small pair into a large and small angles which exchange roles with the smallest
angle and with one of the three large angles.
5.6 Concluding remarks
Configurational densities of a prepared ensemble of molecules are highly informative
and, if available, should be a challenge to contemporary theories. The case of the CEI
of the methane cation is an example of the experimental realizability of this. The close
resemblance of the experimental path and the theoretical reaction path of nuclear
rearrangement in an elaborate Jahn-Teller case such as the case here is reassuring
for both the experimental and the theoretical methods. The curious differences,
especially the density depression at the C 2 v conformation could be of dynamic origin.
For example, the intense connections between permutationally equivalent structures
enforces a tetrahedral symmetry which invokes three equal moments of inertia. Thus,
the choice of a nuclear fixed frame of reference is highly degenerate. Any small
perturbation, like Coriolis terms, might mix the initially highly degenerate possible
low lying molecular states. As a consequence, different total angular momenta may
break the internal symmetry and result in different structures. Here there is an
experimental challenge, namely, to measure the CEI for different angular momentum
windows.
93
6 Conclusion
In the work, which was reported in this thesis, we have measured and analyzed
the fully correlated structures and internal motions of the clusters C 3 , B 3 , C 4 , B 4 ,
and of the CH4 molecular ion, which was measured at Argonne National Laborat-
ory(U.S.A.). The different results were studied from two aspects. From the chemistry
point of view the results were compared with theory and, where possible, with other
types of measurements. These measurements provide an important direct observa-
tion of the geometric structure of these molecules. From the CEI point of view, a
study was carried out on the intrinsic transformations of the CEI from V-space to
R-space and the power and limitations of the method were analyzed.
During the course of this work both the system and computational tools available
for CEI measurements were developed. The new type of M C P position and time
sensitive detector was put into operation and the peripheral hardware and software
were developed. The laser stripping method was perfected and implementation of a
new cold molecular source has begun. In terms of analysis a new method, described
in this thesis, was developed and the fruits of this development have been shown.
We have produced the first experimental observation of the structures of gas
phase B 3 and B 4 . B 3 was found to have an average triangular configuration with
average bond-lengths of 1.45 A ± 0.15 A . The structure of B 4 was found to be
a planar square with 1.45 A ± 0.15 A bond-lengths. Both results were found
to be consistent with theoretical predictions. The bending angle of the square B 4
from planarity was analyzed and the distribution of this bending angle was found
to be approximately 3 times that of the predicted ground state. At this stage it is
impossible to say whether this width is only the result of a vibrationally excited
population. Nevertheless, we have shown that this feature may be resolved from the
94
CEI data and in future experiments the use of vibrationally cold beams will help to
extract the bending width.
We have found that the analysis of CEI data of a vibrationally excited population
of C 3 shows a structure which is consistent with a floppy linear structure with bond-
lengths of 1.2 A ± .1 A and a width in the bending vibrational distribution 3.5
times that of the predicted ground state. In the case of C 4 we have also found a linear
structure with bond lengths of approximately 1.3 A . This result is in agreement with
what is predicted by theory for the linear structure. The puzzle in this measurement
is that the linear population was created by photo-detachment of C j using photons
of 2.35 eV. This energy is about 1.5 eV lower than the photo-detachment threshold
for linear C 4 . Moreover, it is predicted that there will be a threshold for photo-
detachment of the rhombic isomer of C 4 at about 2 eV. Our group, in two separate
measurements, has measured a clear threshold for photo-detachment of C 4 at 2.1
± 0 . 1 eV. This result seemed to indicate the existence of the rhombic C 4 . Thus it
is yet unclear whether there were rhombic isomers that transformed into a linear
configuration or some other explanation.
In the case of C H | we have reanalyzed the data which was acquired at Argonne
Nat. Lab. of a cold population of this molecule. This molecule was predicted
to perform a pseudo-rotational motion with a minima of the potential surface at a
C2v configuration and appreciable tunneling probability between the permutation-
ally symmetric minima. The data was previously analyzed only in terms of the
V-space results. Using the iterative self learning algorithm, developed in the work
described by this thesis, we were able to acquire the complete 9-dimensional R-
space distribution describing the correlated motions and configuration probabilities
of this molecule. A maximal probability reaction path was found on this R space
95
distribution between the permutationally equivalent C2v conformations. This path
is very similar to the one which was predicted by theory. There is a disagreement
between the one dimensional potential along the reaction path which is predicted
and the probability density which was found experimentally. The theory predicts
a minimum in the potential at a C2v conformation. The data shows a large dip in
the probability at the same place where the theory predicts a saddle point in the
potential but there is a secondary dip in the distribution at the C2v conformation as
well. Thus the maximum of the density is shifted from the predicted minimum of the
potential. An attempt to explain this discrepancy is based on the Coriolis coupling
of the highly symmetric average structure of this very floppy molecule(sect. 5.6).
Through the comparison of the two different types of measurements, clusters
versus XH״, an insight may be gained on the quality of measurements which may
be attained from the CEI method. Most measurements which are done today in the
molecular and sub-molecular scales produce results which can be very nonintuitive.
Generally these measurements result in spectra or sets of correlated points which
after certain analysis procedures give a picture of the system which was measured.
The CEI method falls between the cases which are very intuitively analyzed and
those which need certain transformations before they can be understood. As was
demonstrated, the results of the X H n molecule, in terms of the V-space distributions
of angles, are very similar to the final results in R. Therefore part of the analysis was
already possible at the "raw data" stage without much transformation. The clusters
are very different. The results which we have shown exhibit a large difference between
the picture which is measured in V-space and that of the final R-space. This does
not mean that it is impossible to, at least partially, understand the results at the
V-space level. A l l it says is that when we come to understand this cluster data in
96
V-space we must adjust ourselves to the "distorted glasses" through which the image
is shown.
In conclusion, the Coulomb Explosion Imaging method is emerging as an import-
ant tool for understanding molecules. The work, which was described here, included
a critical stage of the preparation of a stable and functioning system at the Weizmann
Institute.
It is my hope that the work which I have invested in this research will be of value
to others that will come after me. I believe that the basic knowledge of the CEI
of complex systems is now much clearer and will allow new and better work to be
performed in the future.
97
A C.3 history
The history of measurements of C 3 goes back more than a century. Huggins [15]
recorded a band spectrum at around 4000 A from a comet. The same band system
has been repeatedly studied in various comets since then[73]. McKellar[74] observed
the band in cool carbon stars. Herzberg[75] reproduced the band system in the
laboratory for the first time. However, the species responsible for the band spectrum
was not identified.
The first conclusive identification of the band system was achieved by Douglas[16]
who observed the 4050 A system in the laboratory at high resolution. The rotational
constants for the upper and lower states were determined, and it was concluded
that the C 3 molecule is linear both in the upper and lower states. Later, Gausset,
Herzberg, Lagerqvist, and Rosen[17, IS] made a more extensive and detailed analysis,
and established the assignment of the transition to be A 1 ^ - X ^ E * . From the
vibrational analysis, they found that the v 2 vibrational frequency is very small(~
63cm - 1 ) in the ground state. In the upper electronic state, a large Renner-Teller effect
was observed upon excitation of the bending vibration. From detailed analysis, the
bending vibrational frequency and the Renner-Teller parameter were determined[18].
Subsequently, Merer[20] extended the analysis of the A J n X - ״ 1 S j " system, and also
found the t ׳ ! frequency in the ground state to be 1224.5 c m - 1 .
More recently Lemire et a l . [76] observed a new band system in 266-302 nm, but its
assignment is uncertain. Rohlfing applied laser induced fluorescence and dispersed-
fluorescence spectroscopy to jet cooled C3[77] in the U V region, and observed a
vibronically induced band system. In addition, two groups of workers[78, 79] have
used the technique of stimulated pumping from the A 1 ^ to characterize higher
vibrational levels of the ground state. These data may provide a very extensive
98
description of the ground state potential surface.
The C3 cluster also was a subject of extensive investigations in the visible and
infrared regions using low temperature matrix isolation techniques. Barger and
Broida[80] deposited carbon vapor from a Knudsen cell at 2500K, and recorded 21
"lines" in the range between 3797 and 4221 A . Weltner and co-workers carried out
extensive investigations on carbon vapor condensed in rare- gas matrices at 4 and
20K[81, 21, 52]. In addition to the visible bands, they observed the infrared spectra
of C3 for the first time, and determined the u3 fundamental frequency to be 2040
c m - 1 [21]. An important contribution from their work was the discovery of a long
lived (20 ms) emission band at around 5900 A . They proposed that the band was
the a3IIu - X 1 ! !* system. Bondbey and English[82] made additional observations of
this phosphorescence. A vacuum UV spectrum of C 3 trapped in argon at 8 K was
observed by Chang and Graham[83].
Matsumura et a l . [84] generated C3 by ArF Excimer laser photolysis of diacetylene
and observed the u3 fundamental band at around 2040 c m - 1 in gas phase using diode
laser spectroscopy. They also observed the 21/2 + ^3 combination band and various hot
bands[85]. The v3 band was simultaneously discovered in the circumstellar envelope
of a giant carbon star IRC + 10216 by Hinkle, Ready, and Bernath[19].
99
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101
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102
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103
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PUBLICATIONS:
Published refereed articles:
• 1989 - with M. Algranati, H. Feldman, E. Malkin, E. Miklazky, R. Naaman,
Z. Vager and J. Zajfman, "The structure of C 4 as studied by the Coulomb
explosion method," J. C h e m . Phys 90 4617
• 1990 - with H. Feldman, E. Malkin, E. Miklazky, Z. Vager, J. Zajfman and R.
Naaman, "The structure of carbon clusters as studied by Coulomb Explosion
Method," J. C h e m . Soc. F A R A D A Y T R A N S . 8 6 2469
• 1991 - with Z. Vager, H. Feldman, E. Malkin, E. Miklazky, J. Zajfman and R.
Naaman, "the structure of small carbon clusters" Z. Phys. D 19 413
• 1991 - with Z. Vager, H. Feldman, E. Malkin, E. Miklazky, J. Zajfman and R.
Naaman, "the structure of small carbon clusters" R a d i a t i o n Effects and Defects
i n Solids 117 33
• 1992 - with D. Zajfman, H. Feldman, 0. Heber, D. Majer, Z. Vager and R. Naa-
man, "Electron Photo-detachment Cross Sections of Small Carbon Clusters:
Evidence for Nonlinear Isomers," Science 2 5 8 1129
• 1993 - with D. Zajfman, 0. Heber, D. Majer, H. Feldman, Z. Vager and R.
Naaman "Observation of laser excitation of rhombic C4 using the Coulomb
Explosion method" Z . Phys. D 2 6 340
• 1993 - with D. Zajfman, 0. Heber, D. Majer, H. Feldman, Z. Vager and R.
Naaman "The isomers of small carbon clusters" Z. Phys. D 2 6 343
108
-with D. Zajfman, H. Feldman, 0. Heber, D. Majer, Z. Vager and R. Naa ־ 1993 •
man, "Carbon cluster imaging at the Weizmann Institute Coulomb Explosion
system" N u c . I n s t . M e t h B 79 227
.with M. Algranati, H. Feldman, 0. Heber, H. Kovner, E. Malkin, E ־ 1993 •
Miklazky, R. Naaman, D. Zajfman, J. Zajfman and Z. Vager, "A system for
Coulomb explosion imaging of small molecules at the Weizmann Institute,"
N u c . I n s t , a n d M e t h . A 329 440
Articles in Preparation:
• with Z. Vager," A detailed study of conformations in the ground state of CHj","
S u b m i t t e d t o J . C h e m . Phys. J u l y 1 9 9 4
• with J. Levin and Z. Vager "Reverse Monte Carlo simulations for Coulomb
Explosion Imaging"
• with H. Feldman, J. Levin, 0. Heber, D. Zajfman, D. Ben-Hamu, O. Lourie
and Z. Vager "First direct observation of the structure of B3 and B 4"
Contributions to conferences
• 1990 - "The structure of carbon clusters as studied by Coulomb Explosion
Imaging," I s r a e l i Phys. Soc. m e e t i n g
• 1992 - "The coulomb explosion imaging of C 3 ," I n t . Symp. o n S m a l l P a r t i c l e s
a n d I n o r g a n i c C l u s t e r s
• 1992 - "A cold molecular source for the Coulomb Explosion Experiment," I n t .
Symp o n a p p l i c a t i o n s of a c c e l e r a t o r s i n r e s e a r c h a n d i n d u s t r y
109
1993 - "A femtosecond after the little bang: CEI of polyatomic molecules,
polyatomic ion impact on solids and related phenomena
1993 - "first observation of B 4," I n t . Conf. on the Physics of Electronic a n
Atomic Collisions
110
י ג מ ו ל ו ץ ק ו צ י ת פ י מ ד ה ר ! ק ח מ
אר חיבור לשם קבלת התור לפילוסופיה״ ״דוקטו
מאת
דרור קלע
צמן למדע י י ן ו ו מוגש למועצה המדעית של מכגוסט 1994 או
י ב מ ו ל ו ץ ק ו צ י ת פ י מ ד ה ר ! ק ח מ
חיבור לשם קבלת התוארר לפילוסופיה״ ״דוקטו
מאת
דרור קלע
צמן למדע י י ן ו מוגש למועצה המדעית של מכוגוסט 1994 או
עבודה זו נעשתה בהנחיתו של פרופסור זאב וגר
ם י ק י ק ל ח המחלקה לפיסיקת ה
ר י צ ק ת
לפני למעלה מעשר שנים החלו לפתח שיטה להדמית מבנה גיאומטרי של מולקולות, במקביל הן
יצp למדע והן במעבדה לאומית Argonne שבארה״ב. שיטה זו מתבססת על הרעיון הבא: י במכון ו
מולקולה שהואצה במאיץ למהירות גבוהה, עוברת בזמן קצר מאד דרך שכבת חומר דקה. במעבר
ת המולקולה דוחים בשכבה אובדים למולקולה אלקטרונים. כתוצאה מכך האטומים המרכיבים א
ת השני ע״י הכוח הקולומבי. במרחק של מספר מטרים מהשכבה הדקה, במורד מסלול הקליע אחד א
המולקולרי, המרחק בין שברי המולקולה גדל מסדר גדל של מספר אנגסטרום למספר מילימטרים.
ת כיוון התנועה והמהירות של כל בשלב זה השברים פוגעים בגלאי מיקום-וזמן המזהה, בקורלציה, א
אחד מהאטומיםןמרחב המהירות). מכיוון שהכוח שגרם להתפוצצות ידוע - הכוח הקולומבי, ניתן
ת המבנה הגיאומטרי של המולקולה לפני ההתפוצצות(מרחב המיקום). בכך הוגדרה לשחזר א
. 1)Coulomb Explosion 1maging(CEההדמיה של גיאומטרית מולקולות הנקראת
, 1 ה א ל מטרת המחקר, המתואר בחיבור זה, היא לחקור לעומק את ההדמיה של CE ולהציג תמונה מ
במידת האפשר, של השיטה. בפרק הראשון מוצג הרעיון הכללי של CEI והיבטים חשובים של
ההדמיה. בפרק שאחריו נדון הפן הניסיוני מהכיוון הטכני וגם מכיוון שיטות האנליזה הדרושות
לניתוח התוצאות. בחלק הטכני מתוארים פיתוחים חדישים בנושאים של נטרול מולקולות בעזרת
ע תאור כללי לייזר בתוך מאיץ, תאור גלאי מיקום־וזמן מסוג חדש המבוסס על MCP ועל וידאו ו
ומלא של כל רכיבי המערכת. בחלק הקשור באנליזה מתוארות השיטות שפותחו לשם הפיכת
ע השיטות, ישנות וחדישות, לשם רכישת התמונה התוצאות מהגלאי לקואורדינטות של מהירות ו
במרחב המיקום מתוך התמונה במרחב המהירות. בפרק השלישי והרביעי מתוארות המדידות
, 4C0 , ,B4 במדידות אלו זוהו מבנים מרחביים קויים פתוחים של צבירי והתוצאות של הצבירים ן
הפחמן, משולשים ומרובעים של צבירי הבורון. תוצאות אלו מושוות עם ניבויים תאורטיים ובמידה
וקיים עם תוצאות נסיוניות אחרות. הפרק האחרון מוקדש לתיאור מדידה המציגה את יישומה של
. 4 C H , מולקולת + C E I ה ט י ש שיטת במקום בו מתבטא מלא כחה של ה
שני הסוגים השונים של המולקולות, צבירים לעומת XHn , המתוארים בחיבור זה מציגים שני
תחומים קיצוניים במדידות CERI. במקרה של מדידת צבירים התמונה המתקבלת במרחב המהירות
שונה מאד מהתמונה המקורית במרחב המיקום. מעבר לכך צצות בעיות כגון תוצאות שאינן חד
משמעיות לניתוח וקושי במדידת מדגם גדול לשם סטטיסטיקה טובה. נראה איך, בעזרת אלגוריתם
חדש שפותח, ניתן בכל זאת לפענח תוצאות חשובות מתוך מדידות אלו. בניגוד למדידות הצבירים
י שיוסבר, התמונה מרחב המהירות דומה מאד לתמונה פ כ 7 . C E C C מהווה דוגמה אידיאלית 4 H +
המקורית במרחב המיקום, התוצאות בדרך כלל חד משמעיות לאנליזה ואין קושי מיוחד באיסוף
0 מבצעת סיבוב פנימי כאשר הפרוטונים הזהים 1 4
+ מדגם גדול. מדידה זו יפה במיוחד מכיוון ש
מבצעים חילוף תוך כדי מינהור. CEm מותאם במיוחד למדידות מסוג זה. התוצאות, המתוארות
בחיבור זה, מתאימות ברוב תכונותיהן לגיבויים תאורטיים לגבי מולקולה זו. למרות זאת קיימת אי
התאמה מעניינת ביותר בין עומק הפוטנציאל התאורטי החד ממדי לארך המסלול לבין צפיפות
p מוצע הסבר לאי התאמה פונקצית ההסתברות הניסיונית לארך מסלול זה. אי ההתאמה מתוארת ו
זו.