8448 Phys. Chem. Chem. Phys., 2011, 13, 8448–8456 This journal is
c
the Owner Societies 2011
Cite this:
Phys. Chem. Chem. Phys
., 2011, 13, 8448–8456
Crossed beam scattering experiments with optimized energy resolution
Ludwig Scharfenberg, Sebastiaan Y. T. van de Meerakker* and Gerard Meijer
DOI: 10.1039/c0cp02405h
Crossed molecular beam scattering experiments in which the energy of the collision is varied can
reveal valuable insight into the collision dynamics. The energy resolution that can be obtained
depends mainly on the velocity and angular spreads of the molecular beams; often, these are too
broad to resolve narrow features in the cross sections like scattering resonances. The collision
energy resolution can be greatly improved by making appropriate choices for the beam velocities
and the beam intersection angle. This method works particularly well for situations in which one
of the beams has a narrow velocity spread, and we here discuss the implications of this method
for crossed beam scattering experiments with Stark-decelerated beams.
I. Introduction
The crossed molecular beam technique is one of the most
widely used experimental approaches to study collisions
between individual atoms and molecules, and has been seminal
to our present understanding of molecular dynamics at
a microscopic level.
1
Since its introduction in the 1950’s,
the technique has witnessed a remarkable and continuous
development. Its present level of advancement allows for
accurate control over the collision partners prior to the
collision event, and for sophisticated detection of the collision
products.
2–4
One of the most important parameters in a collision
experiment is the collision energy of the colliding particles.
The collision energy can be tuned by controlling the velocity of
the particles prior to the collision, or by changing the angle
between the intersecting beams. For the latter approach,
ingenious crossed molecular beam machines have been
engineered to continuously vary the collision energy.
5
These methods have been used to measure, for instance, the
threshold behavior of rotational energy transfer,
6,7
or to tune
the collision energy over the reaction barrier for reactive
scattering.
8,9
Recently, new molecular beam techniques have become
available that allow for detailed control over the velocity of
molecules in a beam. This control is obtained by exploiting
the interaction of molecules with electric or magnetic fields in a
so-called Stark decelerator or Zeeman decelerator, respectively.
10
The tunability of the velocity allows for scanning of the
collision energy in a fixed experimental geometry. State-to-state
inelastic scattering between Stark-decelerated OH radicals and
conventional beams of Xe, Ar, and He atoms, as well as D
2
molecules,
11–13
has been studied. These beam deceleration
methods hold great promise for future scattering experiments
and offer the possibility to extend the available collision energy
range to energies below one wavenumber.
14
Essential in these experiments is the resolution with which
the collision energy can be varied. High energy resolutions are
particularly important at those collision energies where a
detailed structure in the energy dependence of the cross
sections is expected. At low collision energies, shape or orbiting
resonances can occur that are caused by rotational states of
the collision complex that are trapped behind the centrifugal
or reaction barrier.
15,16
At collision energies near the energies
of excited states of one of the reagents, also Feshbach
resonances can occur.
17
The experimental mapping of these
resonances would probe the potential energy surfaces that
govern the interactions with unprecedented detail.
18
The energy resolution that can be obtained experimentally
depends on the velocity and angular spreads of the molecular
beam pulses. Typical molecular beam spreads are too large to
reveal narrow features like scattering resonances that often
require energy resolutions of about one wavenumber. So far,
only in exceptional cases have resonances been observed,
mostly for kinematically favorable systems in which a collision
partner with low mass has been used.
19–22
Recently, crossed
beam experiments employing a tunable beam crossing angle
have been reported in which the resolution was sufficient to
resolve the contribution of individual partial waves to the
scattering.
23,24
Compared to conventional molecular beams, Stark-decelerated
molecular beams offer superior velocity spreads that typically
range between 1 and 20 m s
1
.
25
This narrow velocity spread
can be exploited in crossed beam scattering experiments to
yield a high energy resolution. Indeed, energy resolutions of
Z 13 cm
1
have already been achieved for the OH–Xe and
OH–Ar systems, which is particularly good in view of the
relatively large reduced mass of these systems. This energy
resolution was sufficient to accurately measure the threshold
behavior of the rotational inelastic cross sections,
11
and to
resolve broader features in the collision energy dependence of
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,
14195 Berlin, Germany. E-mail: basvdm@fhi-berlin.mpg.de
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the cross sections.
12
The sharp resonances that are predicted
by ab initio calculations remained elusive, however.
To further improve the energy resolution in these experiments,
the velocity spread of the collision partner needs to be reduced.
This can be achieved by using a second Stark decelerator to
obtain control over a molecular collision partner, or by using
mechanical velocity selectors to reduce the velocity spread of
the atomic collision partner. However, both approaches would
greatly reduce the number density in the colliding beam.
Here we describe a simple yet effective method to improve
the collision energy resolution that does not rely on velocity
selection of the target beam. We show that for beam crossing
angles smaller than 901, kinematically favorable situations can
occur in which the velocity spread of the target beam does not
contribute to the collision energy resolution. This enables high
collision energy resolutions without sacrificing the number
density of the target beam that is available to the scattering.
This method has been exploited before to improve the
resolution in scattering experiments. To the best of our knowl-
edge, it was described for the first time in a book chapter by
Pauly and Toennies
26
in 1968 and it was part of the disserta-
tion
27
of R. Feltgen (a student of Pauly) in 1970. The method
was used in an experiment by Scoles and coworkers, in which
orbiting resonances were observed in the integral elastic
scattering cross sections for the scattering of velocity selected
H atoms by Hg atoms.
19,20
A beam intersection angle of 731
was used in order to improve the velocity resolution. A similar
investigation was performed by Toennies and coworkers, who
used a beam intersection angle of 461 to resolve orbiting
resonances in the scattering of H atoms by various rare gas
atoms.
21
It is noted that a smart use of the beam kinematics
has also been exploited to optimize the post-collision velocity
spread of the scattered molecules.
28,29
The method is particularly advantageous if one of the
beams has a narrow velocity spread. For collisions between
Stark-decelerated beams and conventional beams of rare gas
atoms, for instance, a very high energy resolution can be
obtained that may well be sufficient to experimentally resolve
scattering resonances, even for systems with a relatively large
reduced mass.
This paper is organized as follows. In Section II the method
is explained in more detail, and the beam properties that
are used throughout this paper are introduced. In Section III
we describe different experimental approaches that can be
followed to vary the collision energy, and their implications
for the collision energy resolution are analyzed. The description
will be held rather generally, although we will emphasize on
the experimental arrangement of one Stark-decelerated beam
colliding with a conventional molecular beam. In Section IV
we illustrate the potential of the method using a recent crossed
beam experiment as an example. In this experiment, a Stark
decelerated beam of OH radicals was scattered with a beam
of He atoms at a 901 crossing angle, and we show that the
future implementation of the method may well lead to the
experimental observation of scattering resonances for this
system. In Section V we will draw conclusions, again with an
emphasis on the advantages this method can have for crossed
beam collision experiments in which Stark-decelerated beams
are employed.
II. Collision kinematics
Consider two colliding particles with mass m
1
and m
2
and with
laboratory velocity vectors v
1
and v
2
, respectively. This situation
is schematically represented in Fig. 1. The collision energy E of
the system, calculated in a frame of reference that is moving
with the velocity of the center-of-mass of the two particles, is
given by:
E ¼
m
2
jv
1
v
2
j
2
¼
m
2
ðv
2
1
þ v
2
2
2v
1
v
2
cos fÞ; ð1Þ
where v
1
and v
2
are the magnitudes of the laboratory velocity
vectors, f is the enclosed angle between both velocity vectors,
and m = m
1
m
2
/(m
1
+ m
2
) is the reduced mass of the system.
This energy E is the total energy that is available for inelastic
processes. Small changes in v
1
, v
2
or f will cause an approxi-
mate change of E that is given by its differential:
dE = m([v
1
v
2
cos f]dv
1
+[v
2
v
1
cos f]dv
2
+ v
1
v
2
sin(f)df).
(2)
The geometric meaning of the partial derivatives is brought
out more clearly if expressed directly by the velocity vectors:
dE = m([v
1
v
ˆ
1
v
2
]dv
1
+[v
2
v
ˆ
2
v
1
]dv
2
+|v
1
v
2
|df) (3)
with the vectors of unit length v
ˆ
1
and v
ˆ
2
.
Two important special cases can occur. If the beams are
parallel on average (f =01 or 1801), the influence of the
angular spread of the beams becomes negligible. If the relative
velocity vector g is, on average, perpendicular to v
1
or v
2
, E is
almost unaffected by small changes in v
1
or v
2
, respectively. In
this case the influence of the velocity spread of one of the
beams becomes negligible. The collision energy resolution thus
strongly depends on the geometry of the Newton diagram that
describes the scattering process. For a suitable choice of the
geometry, this can be exploited to optimize the collision energy
resolution in the experiment. This is the main idea behind the
method.
To make the discussion quantitative, an estimate of the
collision energy distribution is required. This distribution is
determined by the distributions of the vectors v
1
and v
2
and
hence by six independent variables. This number can be
reduced by changing to a more suitable coordinate system
Fig. 1 Laboratory velocity vectors v
1
and v
2
of two colliding
particles. The relative velocity vector of the two particles is given by
the vector g = v
1
v
2
.
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and by making appropriate approximations. If the vectors
v
1
and v
2
are written as functions of spherical coordinates the
collision energy becomes:
E(v
1
,v
2
)=E(v
1
(v
1
,j
1
,y
1
),v
2
(v
2
,j
2
,y
2
)) (4)
where y
1(2)
denotes the polar angle, i.e. the angle subtended by
v
1(2)
and the z-axis and j
1(2)
denotes the azimuthal angle, i.e.
the angle subtended by the orthogonal projection of v
1(2)
onto
the xy-plane and the x-axis. If the averaged velocity vectors lie
exactly within the xy-plane, the first order change of E with
y
1(2)
vanishes
35
so that we need to consider the projection of
the velocity vectors onto the xy-plane only. For the experiment,
this means that it is sufficient to collimate the beams by slits
(rather than pinholes) that are oriented perpendicular to the
xy-plane. We can now identify f in eqn (1) with f = j
1
j
2
,
and we only have to optimize the collision energy resolution
with respect to the three scalar variables v
1
, v
2
, and f.
In the experiment, v
1
, v
2
and f are distributed around their
mean values; let the variance of these variables be denoted by
s.
36
Because j
1
and j
2
are independent, the variance s
f
of the
distribution for f is given by s
2
f
¼ s
2
j
1
þ s
2
j
2
. Hence the
differential (2) can be used to estimate the width of the energy
distribution, and the variance of the collision energy, s(E), is
given to first order by:
s
2
ðEÞ¼m
2
ð½v
1
v
2
cos f
2
s
2
v
1
þ½v
2
v
1
cos f
2
s
2
v
2
þ½v
1
v
2
sin f
2
s
2
f
Þ
ð5Þ
in which v
1
, v
2
and f now stand for the respective mean values.
Because E as well as s(E) is linear in m, it suffices to consider
s(E/m). For convenience, the value of m is listed in Table 1 for a
few selected collision systems. The molecules that are listed in
the top row are typical molecules that are suitable candidates
for Stark deceleration.
III. Overview and applications
If one intends to conduct an experiment at a given mean
energy E with the highest possible resolution, one has to
optimize five parameters: Dv
1
, Dv
2
, Df and the mean values
of two of the three variables v
1
, v
2
, f —the third is always
determined through eqn (1). In the following sections, we will
analyze how the resolution depends on the experimental
parameters, using three different experimental approaches.
In Section IIIA we discuss the situation in which the beam
speeds are held constant, and the collision energy is tuned by
variation of the beam intersection angle f alone. In Section
IIIB, we describe the situation for a fixed beam intersection
angle and target beam speed; the collision energy is tuned by
variation of v
1
. Finally, in Section IIIC we discuss the most
general case in which v
1
, v
2
, and f are allowed to vary to
optimize the energy resolution.
The paramet er s tha t are used in the ex amples are cho sen
to repres ent the collision energy res olution as realistic as
possible and that may be expected in an experiment. The
molecular beam velocity spreads are assumed to be 10% of
the mean speed of the beam. In those cases where the
velocity of the p rimary beam (v
1
) is varied, we assume that
the beam is produced with a Stark decelerator. For a Stark-
decelerated beam, the absolute velocity spread i n the for-
ward direction is (almost) constant and does not depend on
the mean velocity; we will assume here a constant velocity
spread of 10 m s
1
for all cases. The angular spr ead of a
Stark-decelerated beam is generally smaller (typically 11,or
about 20 mrad) than the angular spread of a conventional
molecular beam. To simpli fy the analysis, we assume a
constant angular spread in our examples, but one should
keep in mind that it actually depends on the forward velocity
if a decelerator is used. Angular spreads are assumed to be 0,
20,40or80mrad.
In our analysis, we assume Gaussian distributions for all
variables. In this case the distribution for E, as approximated
by the differential, becomes a well defined Gaussian with s (E)
given by eqn (5). If we denote the full width at half maximum
of the distribution of quantity x by D(x) D
x
, we have D(x)=
2.35s(x) and
DðE=mÞ¼ð½v
1
v
2
cos f
2
D
2
v
1
þ½v
2
v
1
cos f
2
D
2
v
2
þ½v
1
v
2
sin f
2
D
2
f
Þ
1=2
:
ð6Þ
This expression is used for all calculations that are
presented below.
A v
1
and v
2
constant, / variable
In this case, both beam speeds are assumed to be constant, and
the beam intersection angle alone is used to change the energy.
For the kinematic parameters we use v
1
= v
2
= 500 m s
1
, and
Dv
1
= Dv
2
=50ms
1
. The resulting curves for the energy
resolution D(E/m) as a function of E/m are shown in Fig. 2.
Two curves are shown that correspond to an angular spread of
Df = 0 (red dashed curve) and Df = 40 mrad (red solid
curve). The angle f that is needed to obtain a specific E/m is
given by the black curve with respect to the right axis.
If small crossing angles can be realized, fairly low collision
energies are accessible for systems with a small reduced mass.
For example, the system OH/
4
He has m = 3.2 u, so that at 301
a collision energy of B9cm
1
is obtained with a resolution of
B1.9 cm
1
.
The energy resolution D(E/m) depends approximately linearly
on the energy E/m, resulting in a constant relative energy
resolution DE/E. This linear behavior is a consequence of
Table 1 Reduced mass m = m
1
m
2
/(m
1
+ m
2
) (in atomic units) for a
selection of collision systems
m
2
/m
1
7
LiH OH/NH
3
CO
817 28
1 H 0.88 0.94 0.97
2 D/H
2
1.60 1.79 1.87
3
3
He/HD 2.18 2.55 2.71
4
4
He/D
2
2.67 3.24 3.50
8
7
LiH 4.00 5.44 6.22
17 OH/NH
3
5.44 8.50 10.58
20 Ne/ND
3
5.71 9.19 11.67
28 CO 6.22 10.57 14.00
40 Ar 6.67 11.93 16.47
83.8 Kr 7.30 14.13 20.99
131.3 Xe 7.54 15.05 23.08
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the choice of equal velocities v
1
= v
2
= v. With the help of
eqn (6) and (1), the relative energy resolution DE/E for this
special case is given by:
DE
E
¼
D
2
v
1
þ D
2
v
2
v
2
þ
1 þ cos f
1 cos f
D
2
f
!
1=2
; ð7Þ
which is nearly independent of f for small values of D f.
It is noted that the low collision energies and high energy
resolutions that can be obtained for small beam intersection
angles and systems with low reduced mass have been exploited
recently in an experiment by Costes and coworkers, who have
thereby been able to observe oscillations in the integral cross
sections for the reactive scattering of S (
1
D
2
) atoms with H
2
molecules.
24
B v
2
and / constant, v
1
variable
In this case, the experimental geometry and the target beam
velocity are fixed and the collision energy is tuned by varying
the velocity v
1
. This situation pertains, for instance, to a
collision experiment in which a Stark-decelerated beam is
collided with a conventional molecular beam at a fixed beam
intersection angle. Hence we assume in our analysis for beam 2
the parameters v
2
= 500 m s
1
and D
v
2
=50ms
1
; for beam 1
we assume a velocity spread of D
v
1
=10ms
1
for all
velocities. Further, we assume an angular spread D
f
=40
mrad (2.31).
In Fig. 3 the resulting values for D(E/m) are shown for two
different beam intersection angles. The red solid and red dashed
curves (with respect to the axis on the left) show the expected
collision energy resolution as a function of the collision energy
for f =451 and f =901, respectively. The corresponding
primary beam velocities v
1
that are required to obtain this
collision energy are shown as green curves with respect to the
axis on the right.
From Fig. 3 it is evident that beam crossing angles of f =
451 result in lower collision energies, and, perhaps more
important, better energy resolutions. At low collision energies,
there are actually two values for v
1
that result in the same
collision energy. The energy resolution, however, is much
different for both situations. The energy resolution shows a
minimum that occurs for the chosen beam parameters at
E/m = 7.6 cm
1
u
1
and v
1
= 600 m s
1
.
From the analysis given in Section II, one would expect that
the best collision energy resolution is obtained when the
relative velocity vector g is perpendicular to v
2
; this condition
is fulfilled for E/m = 10.4 cm
1
u
1
and v
1
= 707 m s
1
. The
position of the minimum that is found in Fig. 3 deviates
slightly from these values due to the nonzero angular spread
D
f
and the nonzero velocity spread of beam 1. This is
illustrated by the red dotted curve in Fig. 3, labeled
DE
0
(451), that shows the energy resolution that would be
obtained for D
v
1
= D
f
= 0. In this hypothetical situation,
the best collision energy resolution that can be obtained is
indeed found for g > v
2
, and becomes independent of the
velocity spread of beam 2. To first order, the collision energy
spread vanishes in this case.
C Variation of v
1
, v
2
, and / for a fixed energy
In this case the mean collision energy is specified while v
1
, v
2
,
and f are allowed to vary. For a given choice of E, v
1
and f,
there are in general two possible values for v
2
which yield this
energy E. In calculations it is therefore advantageous to vary
v
1
and v
2
and to let f be uniquely determined by eqn (1). To
search for a minimum in DE then has the following geometrical
significance: the vectors v
1
, v
2
and the relative velocity g define
Fig. 2 The dependence of the full width at half maximum D(E/m)on
E/m pertaining to the situation in which both beam velocities are
constant and the collision energy is tuned by variation of f (see
Section IIIA). Beam parameters: v
1
= v
2
= 500 m s
1
, D
v
1
= D
v
2
=
50 m s
1
and D
f
= 40 mrad (2.31) (solid red curve), D
f
= 0 mrad
(dashed red curve). The corresponding beam intersection angle is
shown as the black curve with respect to the axis on the right side.
Fig. 3 The dependence of D(E/m)onE/m for the situation in which
the beam intersection angle and the target beam velocity v
2
are
constant, while the collision energy is tuned by variation of v
1
(see
Section IIIB). Beam intersection angles of f =451 (red solid curve)
or f =901 (red dashed curve) are assumed. Beam parameters: v
2
=
500 m s
1
, D
v
2
=50ms
1
, D
v
1
=10ms
1
, and D
f
= 40 mrad (2.31).
The corresponding primary beam velocities v
1
are shown as the green
curves with respect to the axis on the right. Note that at low collision
energies for 451 there are two possible values for v
1
at a given energy
with differing values for the resolution. The red dotted curve labeled
DE
0
(451) pertains to the hypothetical case in which Dv
1
=0ms
1
,
Df = 0 mrad.
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a triangle, g is held fixed and the vertex opposite to g is allowed
to move over all points within the plane (excluding some areas
which may not be accessible in the experiment).
Again, we calculate the expected energy resolution for an
experiment in which a Stark-decelerated beam collides with a
conventional molecular beam; i.e., we take the beam parameters
D
v
1
=10ms
1
, D
v
2
= 0.10 v
2
and D
f
= 40 mrad. The
collision energy resolution D(E/m) is calculated on a sufficiently
fine grid of values for v
1
and v
2
, where v
1
= 100–1000 m s
1
and v
2
= 400–1000 m s
1
. The subsidiary condition of
constant energy is introduced by letting f be determined by
eqn (1). The surface D(E/m)(v
1
,v
2
) for the fixed collision energy
E/m =10cm
1
u
1
is shown in Fig. 4.
The optimal resolution with D(E/m) = 0.73 cm
1
u
1
is
obtained for v
1
= 627 m s
1
, v
2
= 400 m s
1
and f =511;
note that there is no local minimum, only a global one.
D Applications
In a crossed beam collision experiment, one would like to tune
the collision energy with the highest possible resolution for
each value of the collision energy. As described in Section
IIIC, one would have to optimize the values for v
1
, v
2
, and f to
accomplish this. This is possible in theory, it is however not
practical in an experiment. In this section we discuss to which
extent satisfactory results can also be obtained by a variation
of two parameters only.
First, let us assume that the apparatus allows for a continuous
variation of the crossing angle and the speed of beam one,
while the speed of beam two is fixed. As before, we assume
v
2
= 500 m s
1
and Dv
2
=50ms
1
. We calculate the values
for v
1
and f that result in an optimal energy resolution for the
cases D
f
= 20, 40, 80 mrad (hereafter referred to as case 1, 2
and 3, respectively). In all cases and for all values for v
1
we
assume Dv
1
=10ms
1
. The minimal value for D(E/m) has
been determined by numerically evaluating eqn (6) on a
sufficiently fine grid, subject to the condition of constant
collision energy. In Fig. 5 the optimal values for D(E/m) are
shown (red curves) as a function of the collision energy for all
three cases. The values for f (black curve) and v
1
(green curve)
for a given E/m are plotted with respect to the axis on the right.
To stay on the optimal curve, f and v
1
have to be changed
continuously. It is of practical interest to consider what
happens if we move away from the optimal curve by either
changing only v
1
or only f. In Fig. 6 such deviations are
considered for case 1. The solid lines correspond to a change of
v
1
from 0 to 1000 m s
1
at fixed intersection angles (indicated
on each curve). The two dashed lines correspond to fixed
values for v
1
with v
1
= 575 or 773 m s
1
and variable f with
f =01–901.
All curves touch the optimal curve of case 1, as it should be.
The energy range that can be scanned with a close to optimal
resolution appears limited, both in the case where only v
1
is
varied and in the case where only f is varied. Note that by
changing v
1
alone, the energy range with a satisfactory energy
resolution becomes more and more narrow as f decreases,
finally vanishing at f =01.
Let us now consider an apparatus in which the beam
intersection angle is fixed, but both beam velocities are variable.
We assume f =451, as this beam intersection angle appears
experimentally most feasible. Again, we assume the beam
parameters pertaining to case 1, i.e., D
v
1
=10ms
1
for all
values of v
1
, D
v
2
= 0.10 v
2
, and D
f
= 20 mrad. In Fig. 7 the
optimal values for D(E/m) are shown (red curve, labeled (1
0
)),
together with curve (1) that was shown in the preceding
figures. On the left side of this figure, the corresponding values
for v
1
and v
2
that are required to obtain the optimal value for
the energy resolution are shown in green.
It is observed that by a proper variation of v
1
and v
2
at a
fixed value of f =451 (curve (1
0
)), energy resolutions are
Fig. 4 Contour plot of D(E/m)forafixedE/m of 10 cm
1
u
1
pertaining
to the situation where v
1
, v
2
,andf are varied (see Section IIIC). Beam
parameters: D
v
1
=10ms
1
, D
v
2
= 0.10 v
2
, D
f
= 40 mrad. The
contour lines for the beam intersection angles f are shown as black
curves.
Fig. 5 The minimized values for D(E/m) (red curves with respect to the
left axis) pertaining to the situation in which the target beam velocity v
2
is kept constant, and both v
1
and f are allowed to vary to tune the
collision energy. Beam parameters: Dv
1
=10ms
1
, v
2
=500ms
1
and
D
v
2
=50ms
1
. The assumed angular spreads are Df = 20, 40,
80 mrad corresponding to curves 1, 2 and 3 respectively. The values for
v
1
(green curve) and for f (black curve) that result in the optimal
energy resolution are plotted with respect to the axis on the right.
Downloaded by Fritz Haber Institut der Max Planck Gesellschaft on 29 June 2011
Published on 11 March 2011 on http://pubs.rsc.org | doi:10.1039/C0CP02405H
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