WDS'08 Proceedings of Contributed Papers, Part III, 191–197, 2008.
ISBN 978-80-7378-067-8 © MATFYZPRESS
Rotational Excitations of Polar Molecules
M. Šulc
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
R. Čurı́k
J. Heyrovsky Institute of Physical Chemistry of the AS CR, Prague, Czech Republic.
Abstract. In this paper we propose a simple minimal model dependent only on
a few parameters (one or two), applied to description of rotational excitations of
polar molecules by cold electrons (collision energy is of the order of tens meV) in
the framework of the adiabatic approximation. With this model, we are able to
separate the inelastic and elastic components of the total scattering cross section
from the experimental data. Consequently cooling of electrons by inelastic collisions
can be determined. In the first section we give a short overview of the theoretical
description, which is then followed in the next part of this paper by a comparison
with the experiment performed at the ASTRID laboratory at the University of
Århus (Denmark).
Introduction
The main goal of the presented paper is to develop a relatively simple minimal model
suitable for description of scattering of cold electrons by polar molecules in a gas phase. In this
work, we are closely following the approach used in [Čurı́k, R. et al., 2006] where the authors use
similar techniques to quantitatively describe the rotational transitions in H2 O-like molecules,
whereas this paper focuses on symmetric top molecules, especially methane derivatives (monoand di- halogenides). These species exhibit a three fold axis of geometrical symmetry (i.e.
the group C3v ). An ab-initio approach to rotational excitations of polar molecules has been
proposed, e.g. [Faure A., et al., 2004] using the R-matrix method. In contrast to the R-matrix
method where the electron-molecule interaction is calculated in an ab-initio manner, the aim of
the presented work is to describe low-energy rotational dynamics, while the electron-molecule
interaction is parametrized and determined from the experimental data. Moreover, using the
reported method, we are able to separate easily the measured data of total cross sections into
its elastic and inelastic parts as well as predict directly the cross section energy dependence of
an arbitrary state-to-state rotational transition.
The first experiments concerning low energy electron scattering with observation of rotational inelastic scattering phenomena were first performed in [Randall J., et. al., 1993]. These
inelastic processes exhibit large cross sections of the order of thousands Å2 . In order to study
it properly from the experimenter’s point of view, one needs low energy electron beams with
high energy resolution (cca. 1–2 meV) and sufficient current density. This imposes strong requirements of quality on the experimental apparatus. Necessary details will be given later in
this paper. A nice introductory review of inelastic scattering data for several molecular species
accompanied with a short description of the used experimental setup is given in the transparent
work [Field D., et. al., 2001].
Theoretical overview
The interaction of the incident electron with the target molecule (e.g. chloromethane) is
for our purposes approximated by standard ”two-potential” model approach. In this approach
the dominant long-range interaction of the scattered electron and the target molecule can be
approximated by the dipole term proportional to −2D cos θ/r 2 . Here D is the dipole moment
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ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
of the molecule. At shorter distances (say for r < r0 ), the pure dipole potential is distorted
by an unknown, presumably well-behaved, short-range interaction. We further assume, that
this interaction can be described as a sum of radially symmetric terms. This assumption then
reflects itself in the simple form of the generalised S-matrix (see below).
It is exactly the dipole term which makes the traditional partial waves approach of a very
limited use. Strength of the dipole potential at large distances is comparable to the centrifugal
barrier and consequently it couples large number of partial waves. Therefore one needs to
adopt a different approach, for instance different basis set for expressing the angular part of the
scattering wave function.
The suitable functions come naturally as the eigenfunctions (dipole harmonics) of the angular dipole operator L2 − 2D cos θ (e.g. [Mittleman, M. H., R. E. von Holdt, 1965], [Fabrikant,
I. I., 1976]). The presence of the short-range potential part then causes certain phase shifts for
radial functions while the angular part is described by the dipole harmonics. These phase shifts
are finally used as the fitting parameters of our model. Because we are considering only collision
energies in order of tens of meV, we expect, that only one or two of these ”generalised” phase
shifts (also called short-range phase shifts) will be relevant, instead of tens ordinary (physical)
phase shifts, which would be necessary to satiate the traditional approach.
The foundation stone of our method is the adiabatic approximation adopted for scattering
problems as given in the pioneering work [Chase, D.M., 1956]. It assumes that the characteristic
rotational period of the molecule under consideration is much greater than the typical interaction
time. During the collision, the molecule can be thus regarded as ”frozen” in space. We therefore
first calculate the scattering amplitude in the molecular frame of reference and consequently
transform it to the laboratory frame using the well-known transformation properties of the
j
spherical harmonics via Wigner Dm,k
functions. This laboratory frame scattering amplitude
depends on the variables specifying the spatial orientation of the molecule. In the next step using
the adiabatic approximation, an approximate inelastic scattering amplitude can be calculated
by computing a weighted mean over all possible molecule orientations - eq. (11) in [Chase D.M.,
1956].
Equipped with the knowledge of the laboratory-frame scattering amplitude we can compute
the cross sections for arbitrary state-to-state transition (specified by two quantum numbers J, K
- see e.g. [Hougen J., 1962]). The total cross section including elastic and inelastic channels can
be obtained as the sum over all possible final states and thermal average over the initial states.
The experimental apparatus permits to measure directly the total scattering cross section
and also the total cross section of scattering into the rear hemisphere (i.e. for π > θ > π/2). In
this way two sets of experimental data are available to be compared to the theoretical results.
As already mentioned, the experimental data are measured in the energy region from cca. 10
meV to 0.7 eV. In the present model we assume that only two short-range phase shifts (σ, π)
are relevant. Our strategy is then to fit results of our two-parameter modeling to these two
curves. As the fit can be carried out for every collision energy we obtain the energy dependence
of the short-range phase shifts.
Formal theory
In accordance with the adiabatic approximation [Chase D.M., 1956], we need to evaluate
the elastic scattering amplitude in the molecular frame of reference. This can be done as follows.
We seek the solution Ψ(~r ) of the Schrödinger equation for the incident electron with momentum
~k . The angular part of Ψ(~r ) can be expanded into the eigenfunctions of the dipole operator
L2 − 2D cos θ, i.e.
Ψ(~r ) =
1X
ψλ,m (r)Zλ,m (r̂) , where ψλ,m (0) = 0,
r
λ,m
2
L − 2D cos θ Zλ,m = λ(λ + 1)Zλ,m .
192
(1)
ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
Substituting this expression into the Schrödinger equation gives a set of N coupled differential
equations
X d2
′ ′ λ(λ + 1)
2
′
′
−
+ k δλ ,λ δm ,m − λ , m 2V λ, m ψλ,m (r) = 0,
(2)
dr 2
r2
λ,m
where N denotes the number of terms in the expansion (1) and V stands for the short-range
potential term. With regard to the boundary condition in (1), we have N solutions in general
(each of which has N components forming thus together a square matrix Φα,γ of dimension
N × N ). As in the usual partial wave decomposition, the asymptotic behaviour of Φ can be
required to fulfil
π
π
(3)
Φα,γ (r) → δα,γ e−i(kr−λα 2 ) − α,γ ei(kr−λα 2 ) ,
where α,γ denotes the generalised (a.k.a. short-range) S-matrix. By definition, in a pure dipole
potential this matrix would be a unit matrix. With a general short-range modification to the
dipole, will lose this property. However, for a spherically symmetric short-range interaction, it
is possible to block-diagonalize with respect to m and moreover each m-block will be diagonal
itself. We can thus write in the form
 2iσ

e


e2iπ


0
= diag 0 , 1 , . . . , where e.g.
=
(4)
,
1


..
.
where σ, π are the generalised phase-shifts, i.e. in analogy with ordinary phase shifts (for sand p-wave) they reflect phase changes in the corresponding dipole partial waves caused by the
short-range perturbation.
We seek a combination of these solutions (i.e. a linear combination of the columns of Φ),
which gives the correct asymptotic behaviour, namely a pure plane wave plus spherical outgoing wave modified by the scattering amplitude f (~k , ~r ). By means of rather simple algebraic
manipulations we obtain for the amplitude
1 m
2πi X X
l′
m∗
∗
m
~
′
δλ,λ − λ+λ′ λ,λ′ (−1)
(5)
f (k , ~r ) =
l′ ,λ′ l,λ Yl′ ,m (k̂)Yl,m (r̂),
k
i
m,l,l′ λ,λ′
{z
}
|
ρm
l,l′
where we have introduced the unitary transition matrices m
l,λ between the spherical and dipole
harmonics, i.e.
X
X
m
m∗
|λ mi =
|l mi =
(6)
l,λ |l mi
l,λ |λ mi .
l
λ
Our next step is to transform f (~k , ~r ) into the laboratory frame of reference, where the scattering
amplitude depends on the molecular orientation as
r
2πi X X m 2l′ + 1 l′
l∗
~
D0,m (ω)Dk,m
(ω)Yl,k (r̂).
f (k , ~r , ω) = ρl,l′
(7)
~k 4π
′
m,l,l
k
j
The transformation properties of spherical harmonics are given by Wigner Dm,k
functions being
also the eigenfunctions of the symmetric top Hamiltonian in the representation of Euler angles
ω ≡ (α, β, γ). The state-to-state amplitude is within the adiabatic approximation expressed by
(see eq. (11) in [Chase, D., 1956])
J ′ K ′M ′
(8)
fJKM
(r̂) = J ′ K ′ M ′ f (~k , ~r , ω)JKM .
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ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
Figure 1. Experimental apparatus.
Some algebraic effort yields the final expression
r
X
X
√
′
2π (2J + 1)(2J ′ + 1)
J ′K ′ M ′
(2L + 1)Yl,M̄ (r̂)
(−1)K−M
fJKM (r̂) =
(−1)m 2l′ + 1
ik
4π
L
m,l,l′
(9)
J
J′
L
J
J′
L
L
l
l′ L
l
l′
×
ρm ′ ,
M −M ′ −M̄ K −K ′ −K̄ M̄ −M̄ 0 K̄ −m m l,l
where M̄ = M −M ′ , K̄ = K −K ′ and the direction of ~k in the laboratory frame of reference was
put into coincidence with the third axis. It is then easy to evaluate the differential scattering
cross section via standard formula
dσΓ′ ,Γ =
kΓ′
|fΓ′ ,Γ (r̂)|2 dΩ,
kΓ
(10)
where Γ and Γ′ were introduced as a shorthands for JKM and J ′ K ′ M ′ , respectively.
Experimental setup
The schematic diagram of the used experimental apparatus is depicted in Fig. 1. Technical
details together with more thorough discussion can be found in [Hoffmann, S.V. et. al., 2002]
or [Field, D. et. al., 2001], so we give here only a short overview.
The electrons which are used in the scattering experiment are generated by threshold
photoionisation of Ar at 15.75 eV using synchrotron radiation supplied by the ASTRID storage
ring at the University of Århus. For low energy experiments, the energy resolution of the
electrons (determined mainly by the monochromaticity of the photon beam) is crucial. The
experimental conditions allowed to achieve approximately 1.6 meV FWHM1 .
The obtained electrons are consequently driven out of the source by an electric field of
0.2-0.4 V/cm forming a focused beam thanks to the four element zoom lens. The electric field
must be rather weak in order to not degrade the electron beam energy resolution. The electrons
then enter the scattering chamber filled with the studied gas via the slit denoted in Fig. 1 as
A2. Unscattered electrons are allowed to leave the chamber through the slit A3. The total
1
full width at half maximum
194
ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
0.40
short-range phase shift π [mod π]
short-range phase shift σ [mod π]
3.30
3.25
3.20
3.15
3.10
3.05
3.00
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
20
100
200
300
Collision energy [meV]
20
100
200
300
Collision energy [meV]
Figure 2. Fitted energy dependence of the short-range phase shifts.
scattering cross section σ (including elastic and inelastic channels) is measured via examining
the electron beam attenuation in the chamber.
Moreover, whole apparatus can be immersed in an axial magnetic field of typical strength
∼ 20 G (2 mT). The strength of the magnetic field is chosen in such a way that, for incident
energies up to 0.7 eV, the electrons scattered into the forward hemisphere (i.e. those with
π/2 > θ > 0) exhibit a spiral trajectory with radius smaller than the radius of the exit slit
A3 (1.5 mm) and leave thus the chamber without being detected. Consequently only electrons
scattered into the rear hemisphere contribute to the measured cross section in this case.
In the present study we have focused our attention on chloromethane CH3 Cl molecule. This
molecule can be in a good approximation described as a symmetric top with rotational constants
and dipole moment (according to [Herzberg, G., 1991] and [Lide, D.R., 2004]) summarised in
the following table.
Table 1. Rotational constants of CH3 Cl.
C [a.u.]
A [a.u.]
0.4806 · 10
−4
0.4055 · 10
Dipole moment [a.u.]
−5
0.74606 ± 0.00008
All the constants refer to the ground state. It’s worth to note that the dipole moment of CH3 Cl
is slightly higher than the critical value which is ∼ 0.6393 [Čurı́k, R. et al., 2006].
Results
The basic ingredients of the presented theory are short-range phase shifts σ and π. They
were obtained by fitting the two sets of experimental data shown in Fig. 3. In order to model the
experimental conditions, cross sections (10) were summed over all the final states and averaged
over the initial rotational states. We used the room temperature of 300K to define the thermal
distribution of the initial states. The fitting procedure was carried out independently for each
collision energy point. Resulting energy dependence of the phase shifts σ and π is displayed in
Fig. 2. A comparison of the fit with the actual experimental data can be seen in Fig. 3.
From the left panel of Fig. 2, it can be seen, that the σ phase shift reaches π for E → 0.
This fact suggests that the Levison’s theorem also holds for the short-range phase shifts (as
shown in [Fabrikant, I. I., 1979]). We find this feature important because the application of
the present theoretical model on the experimental data leads to a physically properly behaved
phase shift. This is an independent test of validity of the model. Similar conclusions can be
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ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
3500
domain of the model
σtot experimental values
σback experimental values
3000
σ [Å2 ]
2500
2000
1500
1000
σtot
500
σback
0
20
50
100
150
200
250
300
Collision energy [meV]
Figure 3. Fit of the experimental data based on the phase shifts shown in Fig. 2 displayed
together with the experimental data. Two broken curves represent the upper and lower bound
for the total cross section.
drawn from the right panel of Fig. 2 where π shifts becomes negligible for E → 0.
As the short-range phase shifts are both bounded in the interval [0, 2π] one can explore
what sizes of total cross-sections can be described by the present model. In other words, for a
given dipole moment and rotational constants of a symmetric-top molecule, what is the possible
impact of the short-range interaction on the total cross-section? This question is partially
answered by Fig. 3 where we also show the lower and upper limits on the total scattering cross
section σtot (broken curves). As the measurements and ab-initio calculations of the electronmolecule collisions in the energy domain of 20-300 meV are very difficult to carry on, authors
are not aware of any published data to compare their results with.
Conclusions
We have presented a simple model capable of predictions of elastic as well as inelastic cross
sections for scattering of slow electrons (with incident energy below 0.3 eV) by polar symmetrictop molecules. The model allows to extract relevant physical parameters that describe shortrange part of the interaction between the scattered electron and the molecule (short-range phase
shifts). We have shown, on a practical application of rotational excitations of CH3 Cl, that these
phase shifts exhibit physically correct low-energy behaviour providing some confidence to the
validity of the model.
In the future, we would like to extend our model so as to be able to describe properly more
general molecular species as well as incorporate the influence of vibrational transitions on the
predicted scattering quantities. Moreover, it remains to show whether this model can explain
anomalous behaviour of the total scattering cross section observed e.g. for CClF3 (Freon-13),
nitrobenzene or chlorine dioxine ClO2 . These molecules do exhibit a significant suppression
of forward scattering cross section making the backward scattering dominant, being caused
probably by some kind of an interference phenomenon. The work [Field D. et. al., 2001]
suggests that formation of a metastable negative ion of the target molecule may explain this
effect. Details about this case are to be published in a forthcoming paper.
Acknowledgments.
The authors would like to thank to the team of prof. T. A. Field for
the accurate experimental scattering data gathered at the Institute for Storage Ring Facilities at the
University of Århus (Denmark). The present work was also supported by the Charles University Grant
Agency (grant 257718) and the Grant Agency of the Czech Republic (grant 202/08/0631).
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ŠULC AND ČURÍK: ROTATIONAL EXCITATIONS OF POLAR MOLECULES
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