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An analytic model of rotationally inelastic collisions of polar
molecules in electric fields
Mikhail Lemeshko and Bretislav Friedrich
Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, D-14195 Berlin, Germany
(Dated: July 9, 2008)
Abstract
We present an analytic model of thermal state-to-state rotationally inelastic collisions of polar
molecules in electric fields. The model is based on the Fraunhofer scattering of matter waves and
requires Legendre moments characterizing the “shape” of the target in the body-fixed frame as its
input. The electric field orients the target in the space-fixed frame and thereby effects a striking
alteration of the dynamical observables: both the phase and amplitude of the oscillations in the
partial differential cross sections undergo characteristic field-dependent changes that transgress into
the partial integral cross sections. As the cross sections can be evaluated for a field applied parallel
or perpendicular to the relative velocity, the model also offers predictions about steric asymmetry.
We exemplify the field-dependent quantum collision dynamics with the behavior of the Ne-OCS(1Σ)
and Ar-NO(2Π) systems. A comparison with the close-coupling calculations available for the latter
system [Chem. Phys. Lett. 313, 491 (1999)] demonstrates the model’s ability to qualitatively
explain the field dependence of all the scattering features observed.
PACS numbers: 34.10.+x, 34.50.-s, 34.50.Ez
Keywords: Rotationally inelastic scattering, polar molecules, alignment and orientation, Stark effect, models
of molecular collisions.
1

I. INTRODUCTION
Collisions of molecules in electric, magnetic, or radiative fields are nearly ubiquitous in
nature as well as in the laboratory. Molecules colliding in the Earth’s atmosphere or in
interstellar space are commonly subjected to magnetic and radiative fields; in the labora-
tory, collisions in fields appear with particular prominence in stereodynamics [1], coherent
control [2], and molecular trapping and cooling [3]. Molecular collisions in fields have been
the subject of a number of theoretical studies, recently reviewed, e.g., in Ref. [4]. However,
analytic models of such collisions are scarce, and limited to the collision regime near the
Wigner limit, see, e.g., ref. [5]. Here we present an analytic model of state-to-state rotation-
ally inelastic collisions of atoms with polar molecules in electric fields. The model is based
on the Fraunhofer scattering of matter waves and is applicable to collisions at thermal and
hyperthermal energies. We develop the model for the collisions of closed-shell atoms with 1Σ
(linear) or 2Π (symmetric-top equivalent) molecules, and compare it with the close-coupling
calculations of van Leuken et al. [6], available for the latter system.
The field-free Fraunhofer model was developed by Drozdov [7] and generalized by Blair [8]
in the late 1950s to treat inelastic nuclear scattering. The model provided a much-sought
explanation of the experimentally observed phase shifts between oscillations in the elastic
and inelastic differential cross sections for the scattering of protons or α particles by medium-
sized nuclei, later referred to as the “Blair phase rule.” In 1984, the field-free Fraunhofer
model was adapted by Faubel [9] to account for rotationally inelastic thermal collisions
between helium atoms and N2 and CH4 molecules.
In this paper, we extend the model to include the effects of an electrostatic field on ro-
tationally inelastic scattering of polar molecules by atoms. Within the model, these effects
arise due to the orientation of the polar molecules in the space-fixed frame and the concomi-
tant relaxation of the parity selection rule. Although the model – in both its field-free and
field-dependent incarnation – is only semiquantitative, it readily explains all the features
found in the state-to-state differential and integral cross sections and in their dependence
on the strength and direction of the electrostatic field. These features include the phases of
the angular oscillations in the differential cross section and their characteristic variation as
a function of the electric field.
In Section II, we prepare the soil by introducing the field-free Fraunhofer model of matter-
2

wave scattering. In Sections III and IV, we extend the Fraunhofer model to account for
scattering of polar molecules in electric fields. In Section III, we work out closed-form
expressions for the partial and total differential and integral cross sections and the steric
asymmetry for collisions between closed-shell atoms and polar 1Σ molecules, and apply
them to the Ne-OCS(1Σ, J = 0 → J ′) collision system. Section IV develops the theory for
collisions between closed-shell atoms and polar 2Π molecules in electric fields and exemplifies
the results by treating the Ar-NO (2Π, J = 12 → J ′) collision system. The main conclusions
of this work are summarized in Section V.
II. THE FRAUNHOFER MODEL OF FIELD-FREE SCATTERING
We first describe the Fraunhofer model of field-free scattering and discuss its validity.
The model is based on two approximations:
(i) The energy sudden approximation, which represents the amplitude
fi→f(ϑ) = 〈f|fel(ϑ)|i〉 (1)
for scattering into an angle ϑ from an initial, |i〉, to a final, |f〉, state in terms of the elastic
scattering amplitude, fel(ϑ), at fixed values of the internal coordinates. The energy sudden
approximation is well justified when the collision energy exceeds the spacing of the internal
states, Ecoll ≫ ∆Eint [8], [9].
(ii) The elastic scattering amplitude fel(ϑ) in Eq. (1) is replaced by the amplitude for
Fraunhofer diffraction by an impenetrable, sharp-edged obstacle as observed at a point of
radiusvector r from the obstacle, see Fig. 1. In its simplest form, the Fraunhofer diffraction
amplitude is given by the integral
f(ϑ) ≈
∫
e−ikRϑ cosϕdR (2)
where ϕ is the polar angle of the radius vector R which traces the shape of the obstacle,
R ≡ |R|, and k ≡ |k| with k the initial wave vector. Relevant is the shape of the obstacle
in the space-fixed XY plane, perpendicular to k, itself directed along the space-fixed Z-
axis, cf. Fig. 1. The major approximation made in deriving Eq. (2) consists in neglecting
terms non-linear in r. We note that the notion of a sharp-edged obstacle comes close to
the rigid shell approximation. The latter has been widely used in classical [10], [11], [12],
3

quantum [13], and quasi-quantum [14] treatments of field-free molecular collisions, where
the collision energy by far exceeds the depth of any potential energy well.
In optics, Fraunhofer (i.e., far-field) diffraction [15] occurs when the Fresnel number is
small,
F ≡ a
2
rλ ≪ 1 (3)
Here a is the dimension of the obstacle, r ≡ |r| is the distance from the obstacle to the
observer, and λ is the wavelength, cf. Fig. 1. Condition (3) is well satisfied for nuclear
scattering at MeV collision energies as well as for molecular collisions at thermal and hyper-
thermal energies. In the latter case, inequality (3) is fulfilled due to the compensation of the
larger molecular size a by a larger de Broglie wavelength λ pertaining to thermal molecular
velocities.
We note that the Fraunhofer scattering amplitude, Eq. (2), is quite similar to the ampli-
tude for Born scattering [16]. Either amplitude is a Fourier transform of the target’s spatial
characteristic – either its shape or its potential. Both the Fraunhofer and Born amplitudes
comprise averages of the phase factor, exp(ikR), over the target’s surface or volume [17].
For nearly-circular targets, with a boundary R(ϕ) = R0 + δ(ϕ) in the XY plane, the
Fraunhofer integral of Eq. (2) can be evaluated and expanded in a power series in the
deformation δ(ϕ),
f(ϑ) = f0(ϑ) + f1(ϑ, δ) + f2(ϑ, δ2) + · · · (4)
with f0(ϑ) the amplitude for scattering by a disk of radius R0
f0(ϑ) = i(kR20)
J1(kR0ϑ)
(kR0ϑ)
(5)
and f1 the lowest-order anisotropic amplitude,
f1(ϑ) =
ik
2π
∫ 2π
0
δ(ϕ)e−i(kR0ϑ) cosϕdϕ (6)
where J1 is a Bessel function of the first kind. Both Eqs. (5) and (6) are applicable at small
values of ϑ . 30◦, i.e., within the validity of the approximation sin ϑ ≈ ϑ.
A key step required to maintain the analyticity of the Fraunhofer scattering amplitude,
Eq. (6), is to present the shape of the atom-linear molecule potential in terms of a series in
spherical harmonics,
R♭(θ♭, φ♭) =
∑
κν
ΞκνYκν(θ♭, ϕ♭) (7)
4

with Ξκν the Legendre moments. The polar and azimuthal angles θ♭ and ϕ♭ pertain to the
body-fixed frame, defined, e.g., by the target’s principal axes of inertia. However, what
matters is the target’s shape in the space fixed frame, see Fig. 1, which is given by
R(α, β, γ; θ, ϕ) =
∑
κνρ
ΞκνDκρν(αβγ)Yκρ(θ, ϕ) (8)
where (α, β, γ) are the Euler angles through which the body-fixed frame is rotated relative
to the space-fixed frame, (θ, ϕ) are the polar and azimuthal angles in the space-fixed frame,
and Dκρν(αβγ) are the Wigner rotation matrices. Clearly, the term with κ = 0 corresponds
to a disk of radius R0,
R0 ≈
Ξ00√
4π
(9)
Since of relevance is the shape of the target in the XY plane, we set θ = π2 in Eq. (8). As a
result,
δ(ϕ) = R(α, β, γ; π2 , ϕ)− R0 = R(ϕ)− R0 =
∑
κνρ
κ 6=0
ΞκνDκρν(αβγ)Yκρ(π2 , ϕ) (10)
By substituting from Eq. (10) into Eq. (6) and evaluating the integral, we obtain the following
expression for the first-order scattering amplitude,
f1(α, β, γ;ϑ) =
ikR0
2π
∑
κνρ
κ 6=0
ΞκνDκρν(αβγ)FκρJ|ρ|(kR0ϑ) (11)
with Fκρ defined by
Fκρ =









(−1)ρ2π
(
2κ+1
4π
)
1
2 (−i)κ
√
(κ+ρ)!(κ−ρ)!
(κ+ρ)!!(κ−ρ)!! for κ+ ρ even and κ ≥ ρ
0 elsewhere
(12)
For negative values of ρ, the factor (−i)κ is to be replaced by iκ. Finally, by making use of
Eq. (1), we obtain the inelastic scattering amplitude as
fi→f(ϑ) ≈ 〈f|f0 + f1|i〉 = 〈f|f1|i〉 =
ikR0
2π
∑
κνρ
κ 6=0
κ+ρ even
Ξκν〈f|Dκρν |i〉FκρJ|ρ|(kR0ϑ) (13)
5

III. THE FRAUNHOFER MODEL OF ROTATIONALLY INELASTIC SCATTER-
ING OF POLAR 1Σ MOLECULES BY CLOSED-SHELL ATOMS IN AN ELECTRO-
STATIC FIELD
A. The field-dependent scattering amplitude
When a polar 1Σ molecule enters an electrostatic field, its rotational states undergo
hybridization (coherent linear superposition), induced by the interaction of the molecule’s
body-fixed electric dipole moment, µ, with the electric field, ε [18], [19]. Because of the
cylindrical symmetry about the electric field vector, the permanent-dipole interaction couples
free-rotor basis states, |J,M〉, with a fixed value of the good quantum number, M , but a
range of J ’s. Thus the hybrid wavefunctions take the general form
|J˜ ,M, ω〉 =
∑
J
aJ˜JM(ω)|J,M〉 (14)
where the expansion coefficients aJ˜JM depend solely on a dimensionless interaction parameter,
ω ≡ µε/B (15)
which measures the maximum potential energy, µε, of the dipole in terms of the molecule’s
rotational constant, B. The symbol J˜ denotes the nominal value of J that pertains to the
field-free rotational state which adiabatically correlates with the hybrid state,
|J˜ ,M, ω → 0〉 → |J,M〉 (16)
and µ ≡ |µ|, ε ≡ |ε|.
In order to account for an arbitrary direction of the electric field with respect to the
initial wave vector k, we introduce a field-fixed coordinate system X♯Y ♯Z♯, whose Z♯-axis
is defined by the direction of the electric field vector ε. The free-rotor states are thus given
by spherical harmonics whose arguments are the angles θ♯ and ϕ♯ in the field-fixed frame,
|J,M〉 = YJM(θ♯, ϕ♯) (17)
Apart from possessing a sui generis energy level pattern, the |J˜ ,M, ω〉 eigenstates have
an indefinite (mixed) parity and are directional, exhibiting a varying degree of orientation,
which depends on the values of J˜ , M , and ω. In the oriented states, the body-fixed dipole
6

(and thus the internuclear axis) librates about the field direction like a pendulum, and so the
hybrid states are referred to as pendular. It is the directionality of the pendular states along
with their mixed parity that enters the field-dependent Fraunhofer model and distinguishes
it from the field-free model, which assumes an isotropic distribution of the molecular axes
and a definite parity. The directional properties of pendular states are exemplified in Figure
Fig. 2, which shows polar diagrams of the field-free and pendular wave functions at ω = 5.
Hence the scattering process in the field comprises the following steps: A molecule in a
free-rotor state |J,M〉 enters adiabatically the field where it is transformed into a pendular
state |J˜ ,M, ω〉. This pendular state may be changed by the collision in the field into another
pendular state, |J˜ ′,M ′, ω〉. As the molecule leaves the field, the latter pendular state is
adiabatically transformed into a free-rotor state |J ′,M ′〉. Thus the net result is, in general,
a rotationally inelastic collision, |J,M〉 → |J ′,M ′〉.
In order to be able to apply Eq. (13) to collisions in the electrostatic field, we have to
transform Eq. (14) to the space-fixed frame XY Z. If the electric field vector is specified by
the Euler angles (ϕε, θε, 0) in the XY Z frame, the initial and final pendular states take the
form
|i〉 ≡ |J˜ ,M, ω〉 =
∑
J
aJ˜JM(ω)
∑
ξ
DJξM(ϕε, θε, 0)YJξ(θ, ϕ) (18)
〈f| ≡ 〈J˜ ′,M ′, ω| =
∑
J ′
bJ˜ ′∗J ′M ′(ω)
∑
ξ′
DJ ′∗ξ′M ′(ϕε, θε, 0)Y ∗J ′ξ′(θ, ϕ) (19)
which is seen to depend solely on the angles θ and ϕ (and not the angles θ♯ and ϕ♯ pertaining
to the field-fixed frame).
On substituting from Eqs. (18) and (19) into Eq. (13) and its integration, we obtain a
general expression for the Fraunhofer scattering amplitude in the field,
fωi→f(ϑ) =
ikR0
2π
∑
κρ
κ 6=0
κ+ρ even
Dκ∗−ρ,∆M(ϕε, θε, 0)Ξκ0FκρJ|ρ|(kR0ϑ)
×
∑
JJ ′
aJ˜JM(ω)bJ˜
′∗
J ′M ′(ω)
√
2J + 1
2J ′ + 1C(JκJ
′; 000)C(JκJ ′;M∆MM ′), (20)
where ∆M ≡ M ′−M and C(J1, J2, J3;M1,M2,M3) are Clebsch-Gordan coeffients [20]. Since
the atom-linear molecule potential is axially symmetric, only the Ξκ0 coefficients contribute
to the scattering amplitude.
7

Eq. (20) can be simplified by limiting our considerations to special cases. A first such
simplification arises when we let the initial free-rotor state to be the ground state, |J,M〉 ≡
|0, 0〉. A second simplification is achieved by restricting the orientation of the electric field
in the space-fixed frame to a particular geometry.
(i) For an electric field parallel to the initial wave vector, ε ⇈ k, we have θε → 0, ϕε → 0.
As as result, the Wigner matrix becomes Dκ∗−ρ,∆M(0, 0, 0), which equals a Kronecker delta,
δ−ρ∆M . Hence only the ρ = −∆M ′ term yields a nonvanishing contribution to the scattering
amplitude of Eq. (20),
fω,‖
0,0→J˜ ′,M ′(ϑ) = J|M ′|(kR0ϑ)
ikR0
2π
∑
κ 6=0
κ+M ′ even
Ξκ0FκM ′
×
∑
JJ ′
a0J0(ω)bJ˜
′∗
J ′M ′(ω)
√
2J + 1
2J ′ + 1C(JκJ
′; 000)C(JκJ ′; 0M ′M ′) (21)
We see that the angular dependence of the scattering amplitude for the parallel case is
simple, given by a single Bessel function, J|M ′|.
(ii) For an electric field perpendicular to the initial wave vector, ε ⊥ k, we have θε →
π
2 , ϕε → 0. Hence
fω,⊥
0,0→J˜ ′,M ′(ϑ) =
ikR0
2π
∑
κρ
κ 6=0
κ+ρ even
dκ−ρ,M ′
(π
2
)
Ξκ0FκρJ|ρ|(kR0ϑ)
×
∑
JJ ′
a0J0(ω)bJ˜
′∗
J ′M ′(ω)
√
2J + 1
2J ′ + 1C(JκJ
′; 000)C(JκJ ′; 0M ′M ′) (22)
where dκ−ρ,M ′ are the real Wigner rotation matrices. Since the summation mixes different
Bessel functions (for a range of ρ’s), the angular dependence of the scattering amplitude in
the perpendicular case is more involved than in the parallel case.
We note that, unfortunately, the Fraunhofer model does not distinguish between the
parallel and antiparallel orientations of the field with respect to the initial wave vector, as
can be seen by substituting Dκ∗−ρ,∆M(0, π, 0) = δρ∆M(−1)κ−ρ into Eq. (20). This defect is
inherent to the Fraunhofer model, since the diffraction occurs on a two-dimensional obstacle
in the XY plane, which looks the same from either side of the plane, no matter whether
ε ⇈ k or ε ↑↓ k.
8

B. Results for Ne-OCS(1Σ, J = 0 → J ′) scattering in an electrostatic field
We now proceed with the presentation of the collisional model with a concrete collision
system in mind, namely He + OCS(1Σ, J = 0 → J ′). The OCS molecule has been widely
used in experiments with helium nanodroplets [21]. The electric dipole moment µ = 0.709 D,
rotational constant B = 0.2039 cm−1, and spectroscopic amicability make the OCS molecule
a suitable candidate for an experiment to test the field-dependent Fraunhofer model for a
1Σ molecule.
According to Ref. [22], the ground-state Ne-OCS potential energy surface has a global
minimum of a depth of -81.26 cm−1. In order to diminish the effect of this attractive well
in the collision, we choose a collision energy Ecoll = 500 cm−1, which corresponds to a wave
number k = 21.09 A˚−1. The “hard shell” of the potential energy surface at this collision
energy, shown in Fig. 3, we found by a fit to Eq. (7) for κ ≤ 6. The coefficients Ξκ0 obtained
from the fit are listed in the Table I. According to Eq. (9), the Ξ00 coefficient determines
the hard-sphere radius R0, which is responsible for elastic scattering.
1. Differential cross sections
We start by analyzing the field-free state-to-state differential cross section, which is given
by
I f-f0,0→J ′,M ′(ϑ) = |f0,0→J ′,M ′(ϑ)|2 = ΦJ ′|M ′|Ξ2J ′0J2|M ′|(kR0ϑ) (23)
with
ΦJ ′|M ′| =







(kR0)2
4π
√
(J ′+|M ′|)!(J ′−|M ′|)!
(J ′+|M ′|)!!(J ′−|M ′|)!! for J ′ + |M ′| even
0 otherwise
(24)
see Eq. (13) and Ref. [9]. We see that the state-to-state differential cross section is pro-
portional to the square of the ΞJ ′0 coefficient, which means that the shape of the repulsive
potential provides a direct information about the relative probabilities of the field-free tran-
sitions and vice versa. For the Ne-OCS system, the Ξ2,0 coefficient dominates the anisotropic
part of the potential, see Table I. As a result, the corresponding J = 0 → J ′ = 2 transition
is expected to dominate the inelastic cross section.
Recalling the properties of the Bessel functions [24], we see that for kR0ϑ & πJ
′
2 (which
9

corresponds to ϑ & J ′ degrees for the system under investigation), the differential cross-
section has the following angular dependence:
I f-f0,0→J ′,M ′(ϑ) ∼









cos2
(
kR0ϑ− π4
)
for M ′ even
sin2
(
kR0ϑ− π4
)
for M ′ odd
(25)
By averaging over M ′ and taking into account that ΦJ ′|M ′| vanishes for J ′ + |M ′| odd, we
obtain the angular dependence of the differential cross-section for a 0 → J ′ transition:
I f-f0→J ′(ϑ) ∼









cos2
(
kR0ϑ− π4
)
for J ′ even
sin2
(
kR0ϑ− π4
)
for J ′ odd
(26)
The “phase shift” of π2 predicted by Eq. (26) for the oscillations in the differential cross
sections corresponding to even and odd field-free transitions is shown in Figs. 4 and 5. The
elastic scattering amplitude, given by Eq. (5), has a sin2
(
kR0ϑ− π4
)
asymptote, and so is
out of phase with even-J ′-transitions. This latter effect, which is known as the “Blair phase
rule,” can be also seen in Figs. 4 and 5.
The state-to-state differential cross sections for scattering in a field parallel (‖) and
perpendicular (⊥) to k are given by
Iω,(‖,⊥)0→J ′ (ϑ) =
∑
M ′
Iω,(‖,⊥)0,0→J ′,M ′(ϑ) (27)
with
Iω,(‖,⊥)0,0→J ′,M ′(ϑ) =
∣
∣
∣
fω,(‖,⊥)
0,0→J˜ ′,M ′(ϑ)
∣
∣
∣
2
(28)
and are shown for the Ne+OCS collisions at ε = 50 and 100 kV/cm in Figs. 4 and 5. The
figures reveal that an electrostatic field on the order of 10 kV/cm dramatically alters the
cross-sections. In this subsection we only analyze the field-induced “phase shifts” of the
oscillations, and defer the discussion of the amplitudes to subsection IIIB 2 on the integral
cross sections, to which the amplitudes are closely related.
(i) For a parallel field, ε ‖ k, the differential cross section, Eq. (28), has the same explicit
angular dependence as for the field-free case, Eq. (25). However, the field suppresses the
selection rule (24) and so the summation in Eq. (27) comprises all M ′-states. Therefore,
10

the resulting cross section is a field-dependent mixture of the sine- and cosine-contributions
given by Eq. (25). The angular dependence of the differential cross sections in Fig. 4 can be
gleaned from Eq. (21). The first sum in Eq. (21) extends over even κ for even M ′, and over
odd κ for odd M ′. Therefore, the Ξκ0 coefficients, Table I, determine not only the relative
contributions of different J ′ states, but also of different M ′ states in Eq. (27). Since the Ξ20
coefficient eclipses the others, transitions to even M ′ states dominate whenever the field is
high enough, and the field-free cross-section (26) has a cos2 ϑ asymptote. This can be clearly
seen in Fig. 4: for odd J ′, there is a field-induced phase shift of the differential cross section,
which is absent for transitions to even J ′.
(ii) For a perpendicular field, ε ⊥ k, several Bessel functions contribute to the scattering
amplitude. However, since the summation in Eq. (22) requires that κ + ρ be even, it is
the even Bessel functions which, like in the case of a parallel field, can be expected to
dominate the cross section. Indeed, the cross sections shown in Figs. 4 and 5 for parallel and
perpendicular fields are, for J ′ = 1, 2, 3, similar to one another. However, the J = 0 → J ′ = 4
differential cross section in the perpendicular field exhibits an additional phase shift. This
cross section represents a special case as it is not dominated by the Ξ20 coefficient. The Ξ20
coefficient fails to dominate the J = 0 → J ′ = 4 cross section because of the selection rule,
J ′ = J ; J ± 2, that the Clebsch-Gordan coefficients C(J2J ′, 000) impose on the κ = 2 term.
However, the products of the hybridization coefficients, a0J0(ω)b4˜∗J0(ω) and a0J0(ω)b4˜∗J±2,0(ω)
that occur in the term are very small, due to a tiny overlap of the a0J0(ω) and b4˜∗J ′0(ω)
hybridization coefficents. Therefore, a superposition of both even and odd Bessel functions
contributes to the J = 0 → J ′ = 4 differential cross section. A more detailed discussion of
the overlaps of the hybridization coefficients follows in the next subsection.
2. Integral cross sections
The angular range, ϑ . 30◦, where the Fraunhofer approximation applies the best, com-
prises the largest impact-parameter collisions that contribute to the scattering the most, see
Figs. 4 and 5. Therefore, the integral cross section can be obtained to a good approxima-
tion by integrating the Fraunhofer differential cross section, Eq. (27), over the solid angle
sinϑdϑdϕ,
σω,(‖,⊥)0→J ′ =
∫ 2π
0
dϕ
∫ π
0
Iω,(‖,⊥)0→J ′ (ϑ) sinϑdϑ (29)
11

The integral cross-sections thus obtained for the field parallel and perpendicular to the initial
wave vector are presented in Fig. 6. One can see that the state-to-state cross section for
the J = 0 → J ′ = 2 collisions steadily decreases with the interaction parameter ω, whereas
the other state-to-state cross sections show a non-monotonous dependence. These features
reflect the dependence of the integral cross sections on the final M ′ states accessed by the
inelastic collisions, which is shown in Fig. 7 for the ε ‖ k case. The relative weights of
different M ′ states contributing to the cross sections are determined by a combination of
the Ξκ0 coefficients of Eq. (13) and the Fκ,M ′ factor of Eq. (12). Since the Ξ20 coefficient
looms over the rest and the Fκ,M ′ factor is only non-vanishing for κ+M ′ even and κ ≥ M ′,
collisions leading to M ′ = 0, 2 dominate for all J ′ values, see Fig. 7.
But how to explain the field dependence of the cross sections for the prevalent J =
0,M = 0 → J ′,M ′ = 0, 2 collisions? The answer comes from the realization that the
dependence of the integral cross sections on ω is entirely determined by the coefficients
aJ˜JM(ω) and bJ˜
′
J ′M ′ in the scattering amplitude – both for ε ‖ k, Eq. (21), and ε ⊥ k, Eq.
(22). What the field dependence of the coefficients a0J0(ω), b2˜J ′0(ω), and b3˜J ′0(ω) looks like is
shown in Fig. 8. As noted in subsection IIIB 1, for J ′ = 1, 2, 3 the scattering amplitude of
Eq. (21) is dominated by the Ξ20 coefficient, which entails the selection rule J ′ = J ; J ± 2 in
the summation over J and J ′. In the field-free case, this selection rule is only satisfied for
the J = 0 → J ′ = 2 scattering, which indeed governs the field-free collisions. Once the field
is applied, the “distributions” of the a0J0(ω) and bJ˜
′
J0(ω) coefficients undergo a broadening,
see Fig. 8 (b)-(e). For the J ′ = 2 channel, such a broadening reduces the overlap of the
corresponding hybridization coefficients and thus diminishes the J = 0 → J ′ = 2 cross
section. In contradistinction, the overlap of the hybridization coefficients for J ′ = 1, 3
increases with ω, resulting in an increase of J = 0 → J ′ = 1, 3 cross sections, see Fig. 6.
At even higher ω, the spread of the coefficients is so large that the products a0J0(ω)bJ˜
′∗
J ′0(ω),
corresponding to the selection rule J ′ = J ; J ± 2, become very small, cf. Fig. 8(e). As a
result, the cross-sections for the J ′ = 1, 3 channels decrease. The field dependence of the
J = 0 → J ′ = 4 channel is less straightforward, since, as outlined above, its cross section is
governed by Ξκ0 coefficients with κ 6= 2.
A prominent feature in Fig. 6 is the significant influence of the orientation of ε with
respect to k on the cross section for the J = 0 → J ′ = 1 channel. As mentioned above, for
both ε ‖ k and ε ⊥ k, the partial J = 0;M = 0 → J ′ = 1;M ′ = 0 cross section provides the
12

main contribution to the J = 0 → J ′ = 1 transition. However, an inspection of Eqs. (21)
and (22) reveals that the integral cross sections for the J = 0;M = 0 → J ′ = 1;M ′ = 0
transitions is always greater for ε ⊥ k than for ε ‖ k, due to the coefficients dκ−ρ,0(π2 ).
3. Frontal versus lateral steric asymmetry
We define the steric asymmetry as
Si→f =
σ‖ − σ⊥
σ‖ + σ⊥
, (30)
where the cross sections σ‖,⊥ correspond, respectively, to ε ‖ k and ε ⊥ k. The dependence of
the steric asymmetry on the permanent dipole interaction parameter ω is presented in Fig. 9.
One can see that a particularly pronounced asymmetry obtains for the J = 0 → J ′ = 1, 4
collisions. This can be traced to the field dependence of the corresponding integral cross
sections, Fig. 6. Within the Fraunhofer model, elastic collisions do not exhibit any steric
asymmetry. This follows from the isotropy of the elastic scattering amplitude, Eq. (22),
which depends on the radius R0 only: a sphere looks the same from any direction.
IV. THE FRAUNHOFER MODEL OF ROTATIONALLY INELASTIC SCATTER-
ING OF SYMMETRIC-TOP-EQUIVALENT LINEAR POLAR MOLECULES BY
CLOSED-SHELL ATOMS IN AN ELECTROSTATIC FIELD
A. The field-dependent scattering amplitude
Here we consider a symmetric top-equivalent linear polar molecule, such as OH(2Π 1
2
),
NO(2Π 1
2
), colliding with a closed-shell atom. We treat the molecule as a pure Hund’s case (a)
species, characterized by a non-zero projection, Ω, of the electronic angular momentum on
the molecular axis, whose definite-parity rotational wavefunction is given by
|J,M, |Ω|, ǫ〉 = 1√
2
[
|J,M, |Ω|〉+ ǫ|J,M,−|Ω|〉
]
(31)
where the symmetry index ǫ distinguishes between the members of a given Ω doublet. The
symmetry index takes the value of +1 for e levels and of −1 for f levels. The parity of the
wave function is equal to ǫ(−1)J− 12 [25].
13

The rotational states of a Hund’s case (a) molecule with J > 0 and M > 0 can be oriented
by coupling the opposite-parity members of an Ω doublet via the electric-dipole interaction.
Such a coupling creates precessing states, in which the body-fixed electric dipole moment
µ precesses about the field vector. As a result, molecular rotation does not average out
the dipole moment in first order. A precessing state is a hybrid of the two opposite-parity
members of an Ω-doublet, and can be written as
|J,M, |Ω|, w〉 = α(w)|J,M, |Ω|, ǫ = −1〉+ β(w)|J,M, |Ω|, ǫ = 1〉, (32)
with w ≡ µε/∆ an interaction parameter which measures the maximum potential energy of
the electric dipole in terms of the Ω-doublet splitting, ∆, for J = |Ω|. For a precessing state
with w ≫ 1, the coefficients |α(w)| = |β(w)| = 2− 12 , and the mixing of the states within
an Ω doublet is perfect. A less perfect mixing, |α(w)| 6= |β(w)|, obtains when w ≤ 1. The
wavefunction, Eq. (32), reduces for a precessing state with a perfect mixing to |J,M, |Ω|〉.
It is the inherent orientation of the precessing states along with their mixed parity that
enters the Fraunhofer model for the scattering of Hund’s case (a) molecules in an electric
field. The directionality of the precessing states is illustrated in Figure 10. We assume the
hybridization of J-states for a symmetric-top state to be negligible.
The symmetric top wavefunction is given by a Wigner rotation matrix
|J,M,Ω〉 =
√
2J + 1
4π D
J∗
MΩ(ϕ♯, θ♯, γ♯ = 0) (33)
In analogy with Eqs. (18) and (19), we transform the wave function to the space-fixed frame,
DJ∗MΩ(ϕ♯, θ♯, 0) =
∑
ξ
DJξM(ϕε, θε, 0)DJ∗ξΩ(ϕ, θ, 0) (34)
For transitions with |Ω| = |Ω′|, the initial and final precessing states can, therefore, be
written as
|i〉 =
√
2J + 1
4π
∑
ξ
DJξM(ϕε, θε, 0)
1√
2
{
[α(w) + β(w)]DJ∗ξ|Ω|(ϕ, θ, 0)
+ [−α(w) + β(w)]DJ∗ξ−|Ω|(ϕ, θ, 0)
}
(35)
14

〈f| =
√
2J ′ + 1
4π
∑
ξ′
DJ ′∗ξ′M ′(ϕε, θε, 0)
1√
2
{
[α′(w) + β ′(w)]DJ ′ξ′|Ω|(ϕ, θ, 0)
+ [−α′(w) + β ′(w)]DJ ′ξ′−|Ω|(ϕ, θ, 0)
}
(36)
By substituting from Eqs. (35) and (36) into Eq. (13), we finally obtain the scattering
amplitude for inelastic collisions of symmetric-top molecules in precessing states
fwi→f(ϑ) =
ikR0
4π
√
2J + 1
2J ′ + 1
∑
κρ
κ 6=0
κ+ρ even
Dκ∗−ρ,∆M(ϕε, θε, 0)Ξκ0FκρJ|ρ|(kR0ϑ)
× C(JκJ ′; |Ω|0|Ω|)C(JκJ ′;M∆MM ′)
{
[α(w)α′(w) + β(w)β ′(w)]
[
(−1)κ + (−1)∆J
]
+ [α(w)β ′(w) + α′(w)β(w)]
[
(−1)κ − (−1)∆J
]
}
(37)
We note that if both the initial and final precessing states are perfectly mixed, the term in
the curly brackets of Eq. (37) reduces to 2(−1)κ. The scattering amplitudes for different
orientations of the electrostatic field ε with respect to the initial wave vector k are obtained
from Eq. (37) by substituting the appropriate values of the angles: θε = 0;ϕε = 0 for ε ‖ k,
and θε = π2 ;ϕε = 0 for ε ⊥ k. Eq. (37) implies that the integral cross-sections, cf. Eqs. (27)
- (29), for J → J ′ transitions are the same in the parallel and perpendicular fields. However,
the partial integral cross sections for J,M → J ′,M ′ transitions do depend on whether the
field is parallel or perpendicular to k.
B. Results for Ar-NO(2Π 1
2
) scattering in an electrostatic field
We now consider the excitation of NO(J = |Ω| = 12 , f → J ′, |Ω|, e/f) by collisions with
Ar, under conditions similar to those defined in Refs. [6] and [26]: a hexapole state selector
selects the ǫ = −1(f) state, Eq. (31), which adiabatically evolves into a partially oriented
state, Eq. (32), when the collision system enters the electric field. The electric field of 16
kV/cm, directed parallel to the initial wave vector, ε ‖ k, creates a precessing state, Eq. (32),
whose hybridization coefficients are α = 0.832 and β = 0.555. Next, a collision with an Ar
atom excites the NO molecule to a final, field-free state, |J ′,M ′, |Ω|, ǫ′〉. The final, excited
15

state is considered to be exempt from any effects of the electric field, as its Ω-doubling is
large and hence w < 1. As a result, β ′(w) = 1 or α′(w) = 1 for a final state of e or f parity,
respectively. For a collision so defined, the scattering amplitude, Eq. (37) for ε ‖ k, takes
the form
fw,‖i→f(ϑ) =
ikR0
4π
√
2
2J ′ + 1J|∆M |(kR0ϑ)
∑
κ 6=0
κ+∆M even
Ξκ0Fκ,∆MC(JκJ ′;M∆MM ′)C(JκJ ′; |Ω|0|Ω|)
×












β(w)
[
(−1)κ + (−1)∆J
]
+ α(w)
[
(−1)κ − (−1)∆J
]
α(w)
[
(−1)κ + (−1)∆J
]
+ β(w)
[
(−1)κ − (−1)∆J
]












(38)
where the first or second row of the expression in the curly brackets corresponds to a final
state of, respectively, e or f parity. The coefficients Ξκ0 of the Ar-NO interaction potential,
extracted from the data of Sumiyoshi et al. [27], are listed in Table I. According to ref. [27],
the Ar-NO potential surface exhibits a global minimum of −115.4 cm−1 and, thus, the
Fraunhofer model should be valid at collision energies Ecoll > 400 cm−1. In ref. [14], the
rigid shell QQT model was also used at these energies.
The state-to-state integral cross sections for spin-conserving collisions (|Ω′| = |Ω| = 12) at
a collision energy of 442 cm−1 are shown in Fig. 11, along with the close coupling calculations
of Refs. [6] and [26]. The analytic Fraunhofer model provides a simple interpretation of the
features exhibited by the cross sections.
First, let us consider the field-free case for an initial f state, i.e., for α(w = 0) = 1 and
β(w = 0) = 0:
fw=0i→f (ϑ) ∼
J|∆M |(kR0ϑ)
∑
κ 6=0
κ+∆M even
Ξκ0Fκ,∆MC(JκJ ′;M∆MM ′)C(JκJ ′; |Ω|0|Ω|)











[
(−1)κ − (−1)∆J
]
[
(−1)κ + (−1)∆J
]











(39)
Eq. (39) immediately reveals that if the potential energy surface is governed by terms with κ
even, parity-conserving transitions, f → f , will dominate for ∆J even, while parity-changing
16

transitions, f → e, will dominate for ∆J odd. This propensity can be seen in Fig. 11. It was
explained previously in Ref. [28] by a rather involved analysis of the close-coupling matrix
elements.
The qualitative features of the scattering in an electric field can also be readily explained
by the Fraunhofer model. If the field is present and the target molecule oriented, both even
and odd ∆J ’s in the curly brackets of Eq. (38) contribute to the scattering for any value of κ.
For a potential energy surface governed by even-κ terms, the electric field will enhance the
parity-conserving transitions for ∆J odd, and suppress them for ∆J even; parity-breaking
collisions will prevail for ∆J even and subside for ∆J odd.
From Fig. 11 one could see that for ∆J > 2 the Fraunhofer model yields an integral cross
section which is significantly smaller than the one obtained from a close-coupling calculation.
This is supported by the work of Aoiz et al. [23], who found that the diffractive contribution
to the differential cross sections of Ar-NO collisions is much greater for ∆J = 2 than for
∆J = 3.
As noted in subsection IVA, the integral cross-sections for J → J ′ scattering are the
same for ε ‖ k and ε ⊥ k. Therefore, the Fraunhofer model does not distinguish between
the two collisional configurations and yields a zero steric asymmetry as defined by Eq. (30)
for symmetric-top-equivalent molecules.
Eqs. (38) and (39) also reveal the angular dependence of the scattering. In particular, by
making use of the asymptotic forms of the Bessel functions [24], we see that for a potential
energy surface governed either by even- or odd-κ terms, the differential cross sections for
parity-conserving and parity-breaking transitions will be out of phase. This is illustrated
for scattering from |J = 12 , |Ω| = 12 , f〉 to |J ′ = 52 , |Ω| = 12 , e/f〉 states in Fig. 12 (full
curves). We also note that the parity-breaking cross section is much smaller than the
parity-conserving one, since the Ar-NO potential is dominated by even κ-terms, cf. Table I.
When the field is on, the initial parity is no longer defined. Moreover, both even and odd
Bessel functions J|ρ|(kR0ϑ) contribute to the cross section. As a result, the differential cross
sections corresponding to the final e and f states (dashed curves) become similar to one
another.
We see that the field-free differential cross sections, presented in Fig. 12, are qualitatively
similar to the results of close-coupling calculations presented in Fig. 4 of ref. [29], which
also show a phase shift between parity-changing and parity conserving cross sections. When
17

the field is turned on, the close coupling calculations also reveal that the cross sections
corresponding to final e-states exhibit a phase shift and become similar to the cross sections
for the final f -states, see Fig. 8 of ref. [29].
V. CONCLUSIONS
We made use of the Fraunhofer model of matter wave scattering to treat rotationally
inelastic collisions of polar molecules in electric fields. So far, we have worked out the model
for polar molecules in 1Σ and 2Π states interacting with closed-shell atoms. In accordance
with the energy sudden approximation, inherent to the Fraunhofer model, the interaction
must be dominated by repulsion. This limits the applicability of the model to the thermal
and hyperthermal collision energy range. However, the model is also inherently quantum
and, therefore, capable of accounting for interference and other non-classical effects. The
effect of the electric field enters the model via the directional properties of the molecular
states and their mixed parity induced by the field. Even a small orientation of the molecule
is found to cause a large alteration of the scattering observables, such as differential and
integral cross sections. The strength of the analytic model lies in its ability to separate
dynamical and geometrical effects and to qualitatively explain the resulting scattering fea-
tures. These include the angular oscillations in the state-to-state differential cross sections
or the rotational-state dependent oscillations in the integral cross sections as a function of
the electric field. In the face of the absence of any other analytic model of collisions in fields,
the Fraunhofer model is apt at providing a touchstone for understanding such collisions.
Acknowledgments
We are indebted to Gerard Meijer, Bas van de Meerakker, Steven Hoekstra, Joop
Giliamse, and Ludwig Scharfenberg for discussions and encouragement that helped us to
get along with the present work. We thank Steven Stolte for his most helpful comments.
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[2] Special issue on coherent control, J. Phys. B 41, 074001 (2008)
18

[3] Special issue on cold molecules, Eur. Phys. J. D 31, 149 (2004)
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A. R. P. Rau, J. Phys. B. 33, R93 (2000).
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[7] S. I. Drozdov, Soviet. Phys. JETP. 1, 591, 588 (1955)
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Colorado, Boulder, 1966), Vol. VII C, 343-444.
[9] M. Faubel, J. Chem. Phys. 81, 5559 (1984).
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[11] A. Ichimura, M. Nakamura, Phys. Rev. A 69, 022716 (2004);
M. Nakamura, A. Ichimura, Phys. Rev. A 71, 062701 (2005).
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[13] S. Bosanac, Phys. Rev. A 26, 282 (1982).
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C. A. Taatjes, A. Gijsbertsen, M. J. L. de Lange, and S. Stolte, J. Chem. Phys 111, 7631
(2007).
[15] M. Born and E. Wolf, Principles of optics, 7th ed., Cambridge University Press (1997).
[16] L. D. Landau, E. M. Lifshitz Quantum Mechanics, Non-Relativistic Theory, Oxford: Perga-
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[17] J. S. Blair, Phys. Rev. 115, 928 (1959).
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[19] B. Friedrich and D. Herschbach, Z. Phys. D 18, 153 (1991).
[20] R. N. Zare Angular momentum: Understanding spatial aspects in chemistry and physics, Wiley,
New York (1988).
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[22] H. Zhu, Y. Zhou, and D. Xie, J. Chem. Phys 122, 234312 (2005).
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der, J. Chem. Phys.119, 5860 (2003).
19

[24] G. N. Watson, Theory of Bessel Functions, Cambridge University Press (1922).
[25] J. M. Brow., J. T. Hougen, K. P. Huber, J. W. C. Johns, I. Kopp, H. LeFebvre-Brion,
A. J. Merer, D. A. Ramsay, J. Rostas, R. N. Zare, J. Mol. Spec., 55, 500 (1975).
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F. H. W. van Amerom, J. Bulthuis, J. G. Snijders, S. Stolte, J. Phys. Chem. 99,15573 (1995).
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[29] M. H. Alexander, Faraday Discuss. 113, 437 (1999).
20

Figures
FIG. 1: Schematic of Fraunhofer diffraction by an impenetrable, sharp-edged obstacle as observed
at a point of radius vector r(X,Z) from the obstacle. Relevant is the shape of the obstacle in
the XY plane, perpendicular to the initial wave vector, k, itself directed along the Z-axis of the
space-fixed system XY Z. The angle ϕ is the polar angle of the radius vector R which traces the
shape of the obstacle in the X,Y plane and ϑ is the scattering angle. See text.
J=0
(a)
J=1 J=2 J=3
J=0~
(b)
J=1~ J=2~ J=3~
!
FIG. 2: A comparison of the moduli of the free rotor wavefunctions
∣
∣|J,M = 0〉
∣
∣, panel (a), with the
moduli of the pendular wavefunctions
∣
∣|J˜ ,M = 0, ω = 5〉
∣
∣, panel (b). Also shown is the direction
of the electric field vector, ε.
21

02.5
5
0
o
30
o
60
o
90
o
120
o
210
o
240
o
270
o
300
o
330
o
R♭
(Å
)
!
♭
FIG. 3: Equipotential line R♭(θ♭) for the Ne-OCS potential energy surface at a collision energy of
500 cm−1. We note that the equipotential line for the Ar-NO collision system looks similar and is
not shown. The Legendre moments for either potential energy surface are listed in Table I.
22

0.1
1
10
100
1000
10000 (a) J'=0
0.1
1
10
(b) J'=1
0.1
1
10
100 (c) J'=2
0.1
1
(d) J'=3
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.01
0.1
0o 5o 10o 15o 20o
!
(e) J'=4
FIG. 4: Differential cross sections for the Ne-OCS (J = 0 → J ′) collisions in an electrostatic field
ε=50 kV/cm (red dashed line) and 100 kV/cm (blue dotted line), parallel to the relative velocity
vector. The field-free cross sections are shown by the green solid line. The dashed vertical line
serves to guide the eye in discerning the angular shifts of the partial cross sections.
23

0.1
1
10
100
1000
10000 (a) J'=0
0.1
1
10
(b) J'=1
0.1
1
10
100 (c) J'=2
0.1
1
(d) J'=3
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.01
0.1
0o 5o 10o 15o 20o
!
(e) J'=4
FIG. 5: Differential cross sections for the Ne-OCS (J = 0 → J ′) collisions in an electrostatic field
ε=50 kV/cm (red dashed line) and 100 kV/cm (blue dotted line), perpendicular to the relative
velocity vector. The field-free cross sections are shown by the green solid line. The dashed vertical
line serves to guide the eye in discerning the angular shifts of the partial cross sections.
24

0.01
0.1
1
10
0 1 2 3 4 5 6 7 8
0 25 50 75 100 125 150
!
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
" (kV/cm)
(a)
J'=2
J'=1
J'=3
J'=4
0.01
0.1
1
10
0 1 2 3 4 5 6 7 8
0 25 50 75 100 125 150
!
" (kV/cm)
(b)
J'=1
J'=2
J'=3
J'=4
FIG. 6: Partial integral cross sections for Ne-OCS (J = 0 → J ′) collisions in an electric field
parallel, panel (a), and perpendicular, panel (b), to the initial wave vector.
25

0.1
1
0 25 50 75 100 125 150
! (kV/cm)
(a)
J'=1
M'=0
M'=0
0.01
0.1
1
10
(b)
J'=2
M'=2M'=0
M'=1
0.01
0.1
1
(c)
J'=3 M'=0
M'=2
M'=1
M'=3
0.001
0.01
0.1
0 1 2 3 4 5 6 7 8
"
(d)
J'=4
M'=0
M'=2
M'=1
M'=3
M'=4
Pa
rt
ia
l i
n
te
gr
al

cr
o
ss

se
ct
io
n
s
(Å
2 )
FIG. 7: Partial integral cross sections for Ne-OCS (J = 0,M = 0 → J ′,M ′) collisions in an electric
field parallel to the initial wave vector. The red solid lines show the M ′-averaged cross sections for
the J = 0 → J ′ collisions.
26

0.25
0.5
0.75
1
(a) != 0
0.25
0.5
0.75
1
(b) != 0.29
0.25
0.5
0.75
1
(c) != 1.46
0.25
0.5
0.75
1
(d) != 2.92
0
0.25
0.5
0.75
1
0 1 2 3 4 5 6 7
J, J'
(e) != 11.68
FIG. 8: Absolute values of the hybridization coefficients a0J0(ω) (black dashed line), b2˜J ′0 (red solid
line) and b3˜J ′0 (blue solid line) for different values of the interaction parameter ω. The dashed
vertical lines serves as a guide to the eye. See text.
27

-1
-0.5
0
0.5
1
0 1 2 3 4 5 6 7 8
0 25 50 75 100 125 150
!
St
er
ic
a
sy
m
m
et
ry
" (kV/cm)
J'=1
J'=2
J'=3
J'=4
FIG. 9: Steric asymmetry, as defined by Eq. (30), for Ne-OCS (J = 0 → J ′) collisions.
(a) (b) (c)
!
FIG. 10: The moduli of the symmetric-top wavefunction for J = M = Ω = 12 . Panel (a) shows
the field-free wavefunctions, Eq. (31), for ǫ = 1 (blue line) and ǫ = −1 (red line). Panels (b)
and (c) show the wavefunctions of the precessing states in the field, Eq. (32), for, respectively, an
incomplete (α = 0.832, β = 0.555) and perfect (α = β = 1√
2
) mixing of the Ω doublet states. See
text.
28

10



0.001
0.1
10
1 2 3 4 5 6
Pa
rti
al
In
te
gr
al
c
ro
ss
se
ct
io
n
(Å
2 )
!J
Close-coupling
Fraunhofer
(a)
10-5
0.001
0.1
10
1 2 3 4 5 6
!J
(b)
FIG. 11: Integral cross sections for the excitation of NO(J = |Ω| = 12 , f) in collisions with Ar
to higher rotational levels of the |Ω| = 12 manifold. Panels (a) and (b) pertain, respectively, to
parity-conserving and parity-breaking Ar-NO collisions. The results obtained from the Fraunhofer
model are shown by blue curves, those obtained from the close-coupling calculations of Ref. [26]
by red curves. Dashed lines pertain to field-free scattering, solid lines to scattering in an electric
field ε = 16 kV/cm.
0.01
0.1
1
10 (a)
10-5
0.001
0.1
10
0o 5o 10o 15o 20o
(b)
!
FIG. 12: Differential cross sections for the excitation of NO(J = |Ω| = 12 , f) in collisions with
Ar to the J ′ = 52 ,Ω =
1
2 state. Panels (a) and (b) pertain, respectively, to parity-conserving and
parity-breaking Ar-NO collisions. Green lines pertain to field-free scattering, red lines to scattering
in an electric field ε = 16 kV/cm.
29

Tables
TABLE I: Hard-shell Legendre moments Ξκ0 for the Ne-OCS potential at a collision energy of 500
cm−1 and for the Ar-NO potential at a collision energy of 442 cm−1.
Ξκ0 (A˚)
κ Ne-OCS Ar-NO
0 14.7043 11.0407
1 -0.0968 0.1744
2 0.9455 0.5757
3 0.0540 0.0040
4 -0.0384 -0.0713
5 -0.0131 -0.0013
6 0.0012 0.0106
30