Collisions of paramagnetic molecules in magnetic fields: an analytic model based on Fraunhofer diffraction of matter waves
Physical Review A (2008)
- DOI: 10.1103/PhysRevA.79.012718
- arXiv: 0809.3331
Available from
Mikhail Lemeshko's profile on Mendeley.
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Abstract
We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions of open-shell molecules ( 2Sigma, 3Sigma, and 2Pi) with closed-shell atoms. Our treatment makes use of the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric M. Lemeshko and B. Friedrich, J. Chem. Phys. textbf129, 024301 (2008) and radiative fields M. Lemeshko and B. Friedrich, Int. J. Mass. Spec. in press (2008). A magnetic field aligns the molecule in the space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly affects the differential and integral scattering cross sections. We exemplify our treatment by evaluating the magnetic-field-dependent scattering characteristics of the He - CaH (X 2Sigma +), He - O2 (X 3Sigma -) and He - OH (X 2PiOmega) systems at thermal collision energies. Since the cross sections can be obtained for different orientations of the magnetic field with respect to the relative velocity vector, the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions. The steric asymmetry is found to be almost negligible for the He - OH system, weak for the He - CaH collisions, and strong for the He - O2. While odd Delta M transitions dominate the He - OH (J=3/2,f to J', e/f) integral cross sections in a magnetic field parallel to the relative velocity vector, even Delta M transitions prevail in the case of the He - CaH (X 2Sigma +) and He - O2 (X 3Sigma -) collision systems. For the latter system, the magnetic field opens inelastic channels that are closed in the absence of the field. These involve the transitions N=1, J=0 to N', J' with J'=N'.
Author-supplied keywords
Available from
Mikhail Lemeshko's profile on Mendeley.
Page 1
Collisions of paramagnetic molecules in magnetic fields: an analytic model based on Fraunhofer diffraction of matter waves
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Collisions of paramagnetic molecules in magnetic fields: an analytic
model based on Fraunhofer diffraction of matter waves
Mikhail Lemeshko and Bretislav Friedrich
Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, D-14195 Berlin, Germany
(Dated: September 19, 2008)
We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions
of open-shell molecules (2Σ, 3Σ, and 2Π) with closed-shell atoms. Our treatment makes use of the
Fraunhofer model of matter wave scattering and its recent extension to collisions in electric [M.
Lemeshko and B. Friedrich, J. Chem. Phys. 129, 024301 (2008)] and radiative fields [M. Lemeshko
and B. Friedrich, Int. J. Mass. Spec. in press (2008)]. A magnetic field aligns the molecule in the
space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly
affects the differential and integral scattering cross sections. We exemplify our treatment by eval-
uating the magnetic-field-dependent scattering characteristics of the He – CaH (X2Σ+), He – O2
(X3Σ−) and He – OH (X2ΠΩ) systems at thermal collision energies. Since the cross sections can be
obtained for different orientations of the magnetic field with respect to the relative velocity vector,
the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions.
The steric asymmetry is found to be almost negligible for the He – OH system, weak for the He
– CaH collisions, and strong for the He – O2. While odd ∆M transitions dominate the He – OH
(J = 3/2, f → J ′, e/f) integral cross sections in a magnetic field parallel to the relative velocity
vector, even ∆M transitions prevail in the case of the He – CaH (X2Σ+) and He – O2 (X3Σ−)
collision systems. For the latter system, the magnetic field opens inelastic channels that are closed
in the absence of the field. These involve the transitions N = 1, J = 0 → N ′, J ′ with J ′ = N ′.
PACS numbers: 34.10.+x, 34.50.-s, 34.50.Ez
Keywords: Rotationally inelastic scattering, paramagnetic molecules, alignment and orientation, Zeeman
effect, models of molecular collisions.
I. INTRODUCTION
All terrestrial processes, including collisions, take place in magnetic fields. And yet, quan-
titative studies of the effects that magnetic fields may exert on collision dynamics are mostly of
a recent date, having been prompted by the newfashioned techniques to magnetically manipulate,
control and confine paramagnetic atoms and molecules. Theoretical accounts of molecular collisions
in magnetic fields are usually based on rigorous close-coupling treatments [4]. Analytic models of
such collisions are scarce, and limited to the Wigner regime, see, e.g., ref. [5]. Here we present an
analytic model of state-to-state rotationally inelastic collisions of closed-shell atoms with open-shell
molecules in magnetic fields. The model, applicable to collisions at thermal and hyperthermal colli-
sion energies, is based on the Fraunhofer scattering of matter waves [6]–[8] and its recent extension to
include collisions in electrostatic [9] and radiative [10] fields. The magnetic field affects the collision
dynamics by aligning the molecular axis with respect to the relative velocity vector, thereby changing
the effective shape of the diffraction target. We consider open-shell molecules in the 2Σ, 3Σ, and 2Π
electronic states, whose body-fixed magnetic dipole moments are on the order of a Bohr magneton
[13]–[15]. These states coincide with the most frequently occurring ground states of linear radi-
cals, which are exemplified in our study by the CaH(X2Σ+), O2(X3Σ−), and OH(X2ΠΩ) species.
We take, as the closed-shell collision partner, a He atom. Helium is a favorite buffer gas, used to
X
iv
:0
80
9.
33
31
v1
[
ph
ys
ics
.at
om
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9 S
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08
Collisions of paramagnetic molecules in magnetic fields: an analytic
model based on Fraunhofer diffraction of matter waves
Mikhail Lemeshko and Bretislav Friedrich
Fritz-Haber-Institut der Max-Planck-Gesellschaft,
Faradayweg 4-6, D-14195 Berlin, Germany
(Dated: September 19, 2008)
We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions
of open-shell molecules (2Σ, 3Σ, and 2Π) with closed-shell atoms. Our treatment makes use of the
Fraunhofer model of matter wave scattering and its recent extension to collisions in electric [M.
Lemeshko and B. Friedrich, J. Chem. Phys. 129, 024301 (2008)] and radiative fields [M. Lemeshko
and B. Friedrich, Int. J. Mass. Spec. in press (2008)]. A magnetic field aligns the molecule in the
space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly
affects the differential and integral scattering cross sections. We exemplify our treatment by eval-
uating the magnetic-field-dependent scattering characteristics of the He – CaH (X2Σ+), He – O2
(X3Σ−) and He – OH (X2ΠΩ) systems at thermal collision energies. Since the cross sections can be
obtained for different orientations of the magnetic field with respect to the relative velocity vector,
the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions.
The steric asymmetry is found to be almost negligible for the He – OH system, weak for the He
– CaH collisions, and strong for the He – O2. While odd ∆M transitions dominate the He – OH
(J = 3/2, f → J ′, e/f) integral cross sections in a magnetic field parallel to the relative velocity
vector, even ∆M transitions prevail in the case of the He – CaH (X2Σ+) and He – O2 (X3Σ−)
collision systems. For the latter system, the magnetic field opens inelastic channels that are closed
in the absence of the field. These involve the transitions N = 1, J = 0 → N ′, J ′ with J ′ = N ′.
PACS numbers: 34.10.+x, 34.50.-s, 34.50.Ez
Keywords: Rotationally inelastic scattering, paramagnetic molecules, alignment and orientation, Zeeman
effect, models of molecular collisions.
I. INTRODUCTION
All terrestrial processes, including collisions, take place in magnetic fields. And yet, quan-
titative studies of the effects that magnetic fields may exert on collision dynamics are mostly of
a recent date, having been prompted by the newfashioned techniques to magnetically manipulate,
control and confine paramagnetic atoms and molecules. Theoretical accounts of molecular collisions
in magnetic fields are usually based on rigorous close-coupling treatments [4]. Analytic models of
such collisions are scarce, and limited to the Wigner regime, see, e.g., ref. [5]. Here we present an
analytic model of state-to-state rotationally inelastic collisions of closed-shell atoms with open-shell
molecules in magnetic fields. The model, applicable to collisions at thermal and hyperthermal colli-
sion energies, is based on the Fraunhofer scattering of matter waves [6]–[8] and its recent extension to
include collisions in electrostatic [9] and radiative [10] fields. The magnetic field affects the collision
dynamics by aligning the molecular axis with respect to the relative velocity vector, thereby changing
the effective shape of the diffraction target. We consider open-shell molecules in the 2Σ, 3Σ, and 2Π
electronic states, whose body-fixed magnetic dipole moments are on the order of a Bohr magneton
[13]–[15]. These states coincide with the most frequently occurring ground states of linear radi-
cals, which are exemplified in our study by the CaH(X2Σ+), O2(X3Σ−), and OH(X2ΠΩ) species.
We take, as the closed-shell collision partner, a He atom. Helium is a favorite buffer gas, used to
Page 2
2thermalize molecules and radicals produced by laser ablation and other entrainment techniques [17].
The paper is organized as follows: in Section II, we briefly describe the field-free Fraunhofer
model of matter-wave scattering. In Sections III, IV, and V, we extend the Fraunhofer model
to account for scattering of open-shell molecules with closed-shell atoms in magnetic fields: in
Section III, we work out closed-form expressions for the partial and total differential and integral
cross sections and the steric asymmetry of collisions between closed-shell atoms and paramagnetic
2Σ molecules, and apply them to the He–CaH(X2Σ, J = 1/2 → J ′) collision system; in Sec. IV we
present the analytic theory for 3Σ molecules and apply it to the He–O2(X3Σ, N = 0, J = 1 → N ′, J ′)
scattering; in Section V we develop the theory for collisions of 2Π molecules and exemplify the results
by treating the He–OH(X2Π, J = 3/2, f → J ′, e/f) inelastic scattering. Finally, in Section VI, we
compare the results obtained for the collisions of the different molecules with helium and draw
conclusions from our study.
II. THE FRAUNHOFER MODEL OF FIELD-FREE SCATTERING
The Fraunhofer model of matter-wave scattering was recently described in Refs. [9] and [10].
Here we briefly summarize its main features.
The model is based on two approximations. The first one replaces the amplitude
fi→f(ϑ) = 〈f|f(ϑ)|i〉 (1)
for scattering into an angle ϑ from an initial, |i〉, to a final, |f〉, state by the elastic scattering
amplitude, f(ϑ). This is tantamount to the energy sudden approximation, which is valid when the
collision time is much smaller than the rotational period, as dictated by the inequality ξ ≪ 1, where
ξ = ∆ErotkR0
2Ecoll
≈ BkR0Ecoll
, (2)
is the Massey parameter, see e.g. Refs. [11],[12]. Here ∆Erot is the rotational level spacing, B the
rotational constant, Ecoll the collision energy, k ≡ (2mEcoll)1/2/~ the wavenumber, m the reduced
mass of the collision system, and R0 the radius of the scatterer.
The second approximation replaces the elastic scattering amplitude f(ϑ) in Eq. (1) by the
amplitude for Fraunhofer diffraction by a sharp-edged, impenetrable obstacle as observed at a point
of radiusvector r from the scatterer, see Fig. 1. This amplitude is given by the integral
f(ϑ) ≈
∫
e−ikRϑ cosϕdR (3)
Here ϕ is the asimuthal angle of the radius vector R which traces the shape of the scatterer, 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.
We note that the notion of a sharp-edged scatterer comes close to the rigid-shell approxima-
tion, widely used in classical [18]–[20], quantum [21], and quasi-quantum [22] 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 [23] occurs when the Fresnel number is small,
F ≡ a
2
rλ ≪ 1 (4)
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 (4) is well satisfied for nuclear scattering at MeV
The paper is organized as follows: in Section II, we briefly describe the field-free Fraunhofer
model of matter-wave scattering. In Sections III, IV, and V, we extend the Fraunhofer model
to account for scattering of open-shell molecules with closed-shell atoms in magnetic fields: in
Section III, we work out closed-form expressions for the partial and total differential and integral
cross sections and the steric asymmetry of collisions between closed-shell atoms and paramagnetic
2Σ molecules, and apply them to the He–CaH(X2Σ, J = 1/2 → J ′) collision system; in Sec. IV we
present the analytic theory for 3Σ molecules and apply it to the He–O2(X3Σ, N = 0, J = 1 → N ′, J ′)
scattering; in Section V we develop the theory for collisions of 2Π molecules and exemplify the results
by treating the He–OH(X2Π, J = 3/2, f → J ′, e/f) inelastic scattering. Finally, in Section VI, we
compare the results obtained for the collisions of the different molecules with helium and draw
conclusions from our study.
II. THE FRAUNHOFER MODEL OF FIELD-FREE SCATTERING
The Fraunhofer model of matter-wave scattering was recently described in Refs. [9] and [10].
Here we briefly summarize its main features.
The model is based on two approximations. The first one replaces the amplitude
fi→f(ϑ) = 〈f|f(ϑ)|i〉 (1)
for scattering into an angle ϑ from an initial, |i〉, to a final, |f〉, state by the elastic scattering
amplitude, f(ϑ). This is tantamount to the energy sudden approximation, which is valid when the
collision time is much smaller than the rotational period, as dictated by the inequality ξ ≪ 1, where
ξ = ∆ErotkR0
2Ecoll
≈ BkR0Ecoll
, (2)
is the Massey parameter, see e.g. Refs. [11],[12]. Here ∆Erot is the rotational level spacing, B the
rotational constant, Ecoll the collision energy, k ≡ (2mEcoll)1/2/~ the wavenumber, m the reduced
mass of the collision system, and R0 the radius of the scatterer.
The second approximation replaces the elastic scattering amplitude f(ϑ) in Eq. (1) by the
amplitude for Fraunhofer diffraction by a sharp-edged, impenetrable obstacle as observed at a point
of radiusvector r from the scatterer, see Fig. 1. This amplitude is given by the integral
f(ϑ) ≈
∫
e−ikRϑ cosϕdR (3)
Here ϕ is the asimuthal angle of the radius vector R which traces the shape of the scatterer, 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.
We note that the notion of a sharp-edged scatterer comes close to the rigid-shell approxima-
tion, widely used in classical [18]–[20], quantum [21], and quasi-quantum [22] 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 [23] occurs when the Fresnel number is small,
F ≡ a
2
rλ ≪ 1 (4)
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 (4) is well satisfied for nuclear scattering at MeV
Page 3
3collision energies as well as for molecular collisions at thermal and hyperthermal energies. In the
latter case, inequality (4) is fulfilled due to the compensation of the larger molecular size a by a
larger de Broglie wavelength λ pertaining to thermal molecular velocities.
For a nearly-circular scatterer, with a boundary R(ϕ) = R0 + δ(ϕ) in the XY plane, the
Fraunhofer integral of Eq. (3) can be evaluated and expanded in a power series in the deformation
δ(ϕ),
f(ϑ) = f0(ϑ) + f1(ϑ, δ) + f2(ϑ, δ2) + · · · (5)
with f0(ϑ) the amplitude for scattering by a disk of radius R0
f0(ϑ) = i(kR20)
J1(kR0ϑ)
(kR0ϑ)
(6)
and f1 the lowest-order anisotropic amplitude,
f1(ϑ) =
ik
2π
∫ 2π
0
δ(ϕ)e−i(kR0ϑ) cosϕdϕ (7)
where J1 is a Bessel function of the first kind. Both Eqs. (6) and (7) are applicable at small values
of ϑ . 30◦, i.e., within the validity of the approximation sinϑ ≈ ϑ.
The scatterer’s shape in the space fixed frame, see Fig. 1, 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, Dκρν(αβγ) are
the Wigner rotation matrices, and Ξκν are the Legendre moments describing the scatterer’s shape
in the body-fixed frame. 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 combining Eqs. (1), (7), and (10) we finally obtain
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) (11)
where
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.
latter case, inequality (4) is fulfilled due to the compensation of the larger molecular size a by a
larger de Broglie wavelength λ pertaining to thermal molecular velocities.
For a nearly-circular scatterer, with a boundary R(ϕ) = R0 + δ(ϕ) in the XY plane, the
Fraunhofer integral of Eq. (3) can be evaluated and expanded in a power series in the deformation
δ(ϕ),
f(ϑ) = f0(ϑ) + f1(ϑ, δ) + f2(ϑ, δ2) + · · · (5)
with f0(ϑ) the amplitude for scattering by a disk of radius R0
f0(ϑ) = i(kR20)
J1(kR0ϑ)
(kR0ϑ)
(6)
and f1 the lowest-order anisotropic amplitude,
f1(ϑ) =
ik
2π
∫ 2π
0
δ(ϕ)e−i(kR0ϑ) cosϕdϕ (7)
where J1 is a Bessel function of the first kind. Both Eqs. (6) and (7) are applicable at small values
of ϑ . 30◦, i.e., within the validity of the approximation sinϑ ≈ ϑ.
The scatterer’s shape in the space fixed frame, see Fig. 1, 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, Dκρν(αβγ) are
the Wigner rotation matrices, and Ξκν are the Legendre moments describing the scatterer’s shape
in the body-fixed frame. 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 combining Eqs. (1), (7), and (10) we finally obtain
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) (11)
where
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.
Page 4
4III. SCATTERING OF 2Σ MOLECULES BY CLOSED-SHELL ATOMS IN A
MAGNETIC FIELD
A. A 2Σ molecule in a magnetic field
The field-free Hamiltonian of a rigid 2Σ molecule
H0 = BN2 + γN · S (13)
is represented by a 2 × 2 matrix, diagonal in the Hund’s case (b) basis, |N, J,M〉. Here N and S
are the rotational and (electronic) spin angular momenta, B is the rotational constant and γ the
spin-rotation constant. Its eigenfunctions
Ψ±(J,M) =
1√
2
[
∣
∣S, 12
〉∣
∣J,Ω,M
〉
±
∣
∣S,− 12
〉∣
∣J,−Ω,M
〉
]
, (14)
are combinations of (electronic) spin functions |S,MS〉 with Hund’s case (a) (i.e., symmetric top)
functions |J,Ω,M〉 pertaining to the total angular momentum J = N+S, whose projections on the
space- and body-fixed axes are M and Ω = ± 12 , respectively. The Hund’s case (a) wavefunctions
are given by:
|J,M,Ω〉 =
√
2J + 1
4π D
J∗
MΩ(ϕ, θ, γ = 0) (15)
The Ψ+ and Ψ− states are conventionally designated as F1 and F2 states, for which the
rotational quantum number N = J − 12 and N = J + 12 , respectively. Equation (14) can be recast
in terms of N instead of J :
|Ψǫ(N,M)〉 =
1√
2
[
∣
∣S, 12
〉∣
∣N + ǫ2 ,Ω,M
〉
+ ǫ
∣
∣S,− 12
〉∣
∣N + ǫ2 ,−Ω,M
〉
]
, (16)
with ǫ = ±1.
The eigenvalues corresponding to states F1 and F2 are given by
E+
(
N + 12 ,M ;F1
)
= BN(N + 1) + γ
2
N (17)
E−
(
N − 12 ,M ;F2
)
= BN(N + 1)− γ
2
(N + 1), (18)
whence we see that the spin-rotation interaction splits each rotational level into a doublet separated
by ∆E ≡ E+ − E− = γ(N + 12 ).
In a static magnetic field, H , directed along the space-fixedZ axis, the Hamiltonian acquires
a magnetic dipole potential which is proportional to the projection, SZ , of S on the Z axis
Vm = SZωmB, (19)
with
ωm ≡
gSµBH
2B (20)
a dimensionless interaction parameter involving the electron gyromagnetic ratio gS ≃ 2.0023, the
Bohr magneton µB , and the rotational constant B.
MAGNETIC FIELD
A. A 2Σ molecule in a magnetic field
The field-free Hamiltonian of a rigid 2Σ molecule
H0 = BN2 + γN · S (13)
is represented by a 2 × 2 matrix, diagonal in the Hund’s case (b) basis, |N, J,M〉. Here N and S
are the rotational and (electronic) spin angular momenta, B is the rotational constant and γ the
spin-rotation constant. Its eigenfunctions
Ψ±(J,M) =
1√
2
[
∣
∣S, 12
〉∣
∣J,Ω,M
〉
±
∣
∣S,− 12
〉∣
∣J,−Ω,M
〉
]
, (14)
are combinations of (electronic) spin functions |S,MS〉 with Hund’s case (a) (i.e., symmetric top)
functions |J,Ω,M〉 pertaining to the total angular momentum J = N+S, whose projections on the
space- and body-fixed axes are M and Ω = ± 12 , respectively. The Hund’s case (a) wavefunctions
are given by:
|J,M,Ω〉 =
√
2J + 1
4π D
J∗
MΩ(ϕ, θ, γ = 0) (15)
The Ψ+ and Ψ− states are conventionally designated as F1 and F2 states, for which the
rotational quantum number N = J − 12 and N = J + 12 , respectively. Equation (14) can be recast
in terms of N instead of J :
|Ψǫ(N,M)〉 =
1√
2
[
∣
∣S, 12
〉∣
∣N + ǫ2 ,Ω,M
〉
+ ǫ
∣
∣S,− 12
〉∣
∣N + ǫ2 ,−Ω,M
〉
]
, (16)
with ǫ = ±1.
The eigenvalues corresponding to states F1 and F2 are given by
E+
(
N + 12 ,M ;F1
)
= BN(N + 1) + γ
2
N (17)
E−
(
N − 12 ,M ;F2
)
= BN(N + 1)− γ
2
(N + 1), (18)
whence we see that the spin-rotation interaction splits each rotational level into a doublet separated
by ∆E ≡ E+ − E− = γ(N + 12 ).
In a static magnetic field, H , directed along the space-fixedZ axis, the Hamiltonian acquires
a magnetic dipole potential which is proportional to the projection, SZ , of S on the Z axis
Vm = SZωmB, (19)
with
ωm ≡
gSµBH
2B (20)
a dimensionless interaction parameter involving the electron gyromagnetic ratio gS ≃ 2.0023, the
Bohr magneton µB , and the rotational constant B.
Page 5
5The Zeeman eigenproperties of a 2Σ molecule can be readily obtained in closed form, since
the Vm operator couples states that differ in N by 0 or ±2 and, therefore, the Hamiltonian matrix,
H = H0 + Vm, factors into 2× 2 blocks for each N :
H = −ωmB
− M2N+1 +
E−
ωmB
1
2 [1− M
2
(N+1/2)2 ]
1
2
1
2 [1− M
2
(N+1/2)2 ]
1
2 M
2N+1 +
E+
ωmB
(21)
As a result, the Zeeman eigenfunctions of a 2Σ molecule are given by a linear combination of the
field-free wavefunctions (16),
ψ(N˜ , J˜,M ;ωm) = a(ωm)
∣
∣Ψ−(N,M)
〉
+ b(ωm)
∣
∣Ψ+(N,M)
〉
, (22)
with the hybridization coefficients a(ωm) and b(ωm) obtained by diagonalizing Hamiltonian (21).
Although N and J are no longer good quantum numbers in the magnetic field, they can be employed
as adiabatic labels of the states: we use N˜ and J˜ to denote the angular momentum quantum numbers
of the field-free state that adiabatically correlates with the given state in the field. Since the Zeeman
eigenfunction comprises rotational states with either N even or N odd, the parity of the eigenstates
remains definite even in the presence of the magnetic field; it is given by (−1)N˜ .
The degree of mixing of the Hund’s case (b) states that make up a 2Σ Zeeman eigenfunction
is determined by the splitting of the rotational levels measured in terms of the rotational constant,
∆E/B: for ωm ≤ ∆E/B the mixing (hybridization) is incomplete, while it is perfect in the high-
field limit, ωm ≫ ∆E/B. We note that in the high-field limit, the eignevectors can be found
from matrix (21) with E±/ωmB → 0. As an example, Table I lists the values of the hybridization
coefficients a(ωm) and b(ωm) for the N = 2, J = 52 ,M states of the CaH molecule in the high-field
limit, which is attained at ωm ≫ 0.025.
The degree of molecular axis alignment is given by the alignment cosine, 〈cos2 θ〉, which,
in the 2Σ case, can be obtained in closed form. To the best of our knowledge, this result has not
been presented in the literature before; therefore, we give it in Appendix C. The dependence of the
alignment cosine on the magnetic field strength parameter ωm is shown in Fig. 2 for the two lowest
N states of the CaH molecule. One can see that for ωm ≫ ∆E/B, the alignment cosine smoothly
approaches a constant value, corresponding to as good an alignment as the uncertainty principle
allows.
B. The field-dependent scattering amplitude
In what follows, we consider scattering from the N = 0, J = 1/2 state to some N ′, J ′ state
in a magnetic field. Since the N = 0 state of a 2Σ molecule is not aligned, the effects of the magnetic
field on the scattering arise solely from the alignment of the final state.
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. By making use of the relation
DJ∗MΩ(ϕ♯, θ♯, 0) =
∑
ξ
DJξM (ϕε, θε, 0)DJ∗ξΩ(ϕ, θ, 0) (23)
we transform the wavefunctions (22) to the space-fixed frame. For the initial and the final states we
the Vm operator couples states that differ in N by 0 or ±2 and, therefore, the Hamiltonian matrix,
H = H0 + Vm, factors into 2× 2 blocks for each N :
H = −ωmB
− M2N+1 +
E−
ωmB
1
2 [1− M
2
(N+1/2)2 ]
1
2
1
2 [1− M
2
(N+1/2)2 ]
1
2 M
2N+1 +
E+
ωmB
(21)
As a result, the Zeeman eigenfunctions of a 2Σ molecule are given by a linear combination of the
field-free wavefunctions (16),
ψ(N˜ , J˜,M ;ωm) = a(ωm)
∣
∣Ψ−(N,M)
〉
+ b(ωm)
∣
∣Ψ+(N,M)
〉
, (22)
with the hybridization coefficients a(ωm) and b(ωm) obtained by diagonalizing Hamiltonian (21).
Although N and J are no longer good quantum numbers in the magnetic field, they can be employed
as adiabatic labels of the states: we use N˜ and J˜ to denote the angular momentum quantum numbers
of the field-free state that adiabatically correlates with the given state in the field. Since the Zeeman
eigenfunction comprises rotational states with either N even or N odd, the parity of the eigenstates
remains definite even in the presence of the magnetic field; it is given by (−1)N˜ .
The degree of mixing of the Hund’s case (b) states that make up a 2Σ Zeeman eigenfunction
is determined by the splitting of the rotational levels measured in terms of the rotational constant,
∆E/B: for ωm ≤ ∆E/B the mixing (hybridization) is incomplete, while it is perfect in the high-
field limit, ωm ≫ ∆E/B. We note that in the high-field limit, the eignevectors can be found
from matrix (21) with E±/ωmB → 0. As an example, Table I lists the values of the hybridization
coefficients a(ωm) and b(ωm) for the N = 2, J = 52 ,M states of the CaH molecule in the high-field
limit, which is attained at ωm ≫ 0.025.
The degree of molecular axis alignment is given by the alignment cosine, 〈cos2 θ〉, which,
in the 2Σ case, can be obtained in closed form. To the best of our knowledge, this result has not
been presented in the literature before; therefore, we give it in Appendix C. The dependence of the
alignment cosine on the magnetic field strength parameter ωm is shown in Fig. 2 for the two lowest
N states of the CaH molecule. One can see that for ωm ≫ ∆E/B, the alignment cosine smoothly
approaches a constant value, corresponding to as good an alignment as the uncertainty principle
allows.
B. The field-dependent scattering amplitude
In what follows, we consider scattering from the N = 0, J = 1/2 state to some N ′, J ′ state
in a magnetic field. Since the N = 0 state of a 2Σ molecule is not aligned, the effects of the magnetic
field on the scattering arise solely from the alignment of the final state.
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. By making use of the relation
DJ∗MΩ(ϕ♯, θ♯, 0) =
∑
ξ
DJξM (ϕε, θε, 0)DJ∗ξΩ(ϕ, θ, 0) (23)
we transform the wavefunctions (22) to the space-fixed frame. For the initial and the final states we
Page 6
6have:
|i(N,M)〉 = 1√
4π
∑
ξ
{
a(ωm)
√
NDN−
1
2
ξM (ϕε, θε, 0)
[
DN−
1
2 ∗
ξΩ (ϕ, θ, 0)−D
N−12∗
ξ−Ω (ϕ, θ, 0)
]
+ b(ωm)
√
N + 1DN+
1
2
ξM (ϕε, θε, 0)
[
DN+
1
2∗
ξΩ (ϕ, θ, 0)−D
N+12∗
ξ−Ω (ϕ, θ, 0)
]
}
(24)
〈f(N ′,M ′)| = 1√
4π
∑
ξ′
{
a′(ωm)
√
N ′DN
′− 12
ξ′M ′ (ϕε, θε, 0)
[
DN
′− 12∗
ξ′Ω (ϕ, θ, 0)−D
N ′−12 ∗
ξ′−Ω (ϕ, θ, 0)
]
+ b′(ωm)
√
N ′ + 1DN
′+12
ξ′M ′ (ϕε, θε, 0)
[
DN
′+12∗
ξ′Ω (ϕ, θ, 0)−D
N ′+12∗
ξ′−Ω (ϕ, θ, 0)
]
}
(25)
where Ω = 12 for a
2Σ molecule.
By substituting from Eqs. (24) and (25) into Eq. (11), we finally obtain the scattering
amplitude for inelastic collisions of 2Σ molecules with closed-shell atoms in a magnetic field:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
[
(−1)κ + (−1)∆N
]
×
{
a(ωm)a′(ωm)
√
N
N ′C
(
N − 12 , κ,N
′ − 12 ; Ω0Ω
)
C
(
N − 12 , κ,N
′ − 12 ;M∆MM
′
)
+ a(ωm)b′(ωm)
√
N
N ′ + 1C
(
N − 12 , κ,N
′ + 12 ; Ω0Ω
)
C
(
N − 12 , κ,N
′ + 12 ;M∆MM
′
)
+ a′(ωm)b(ωm)
√
N + 1
N ′ C
(
N + 12 , κ,N
′ − 12 ; Ω0Ω
)
C
(
N + 12 , κ,N
′ − 12 ;M∆MM
′
)
+ b(ωm)b′(ωm)
√
N + 1
N ′ + 1C
(
N + 12 , κ,N
′ + 12 ; Ω0Ω
)
C
(
N + 12 , κ,N
′ + 12 ;M∆MM
′
)
}
(26)
As noted above, there is no hybridization of the initial state for the N = 0, J = 12 → N ′, J ′ collisions,
i.e., a(ωm) = 0, b(ωm) = 1 in Eq. (26). By making use of the properties of the Clebsch-Gordan
coefficients [25],[26], the expression for the scattering amplitude from the N = 0, J = 12 ,M = ± 12
state to an N ′, J ′,M ′ state simplifies to
fωm
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1
∑
ρ
ρ+N ′even
dN ′−ρ,∆M (θε)FN ′ρJ|ρ|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(27)
The amplitude is seen to be directly proportional to the ΞN ′0 Legendre moment. We note that
the cross section for the N, J,M → N ′, J ′,M ′ transition differs from that for the N, J,−M →
N ′, J ′,−M ′ scattering. This is because the magnetic field completely lifts the degeneracy of the M
states, in contrast to the electric field case [9].
|i(N,M)〉 = 1√
4π
∑
ξ
{
a(ωm)
√
NDN−
1
2
ξM (ϕε, θε, 0)
[
DN−
1
2 ∗
ξΩ (ϕ, θ, 0)−D
N−12∗
ξ−Ω (ϕ, θ, 0)
]
+ b(ωm)
√
N + 1DN+
1
2
ξM (ϕε, θε, 0)
[
DN+
1
2∗
ξΩ (ϕ, θ, 0)−D
N+12∗
ξ−Ω (ϕ, θ, 0)
]
}
(24)
〈f(N ′,M ′)| = 1√
4π
∑
ξ′
{
a′(ωm)
√
N ′DN
′− 12
ξ′M ′ (ϕε, θε, 0)
[
DN
′− 12∗
ξ′Ω (ϕ, θ, 0)−D
N ′−12 ∗
ξ′−Ω (ϕ, θ, 0)
]
+ b′(ωm)
√
N ′ + 1DN
′+12
ξ′M ′ (ϕε, θε, 0)
[
DN
′+12∗
ξ′Ω (ϕ, θ, 0)−D
N ′+12∗
ξ′−Ω (ϕ, θ, 0)
]
}
(25)
where Ω = 12 for a
2Σ molecule.
By substituting from Eqs. (24) and (25) into Eq. (11), we finally obtain the scattering
amplitude for inelastic collisions of 2Σ molecules with closed-shell atoms in a magnetic field:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
[
(−1)κ + (−1)∆N
]
×
{
a(ωm)a′(ωm)
√
N
N ′C
(
N − 12 , κ,N
′ − 12 ; Ω0Ω
)
C
(
N − 12 , κ,N
′ − 12 ;M∆MM
′
)
+ a(ωm)b′(ωm)
√
N
N ′ + 1C
(
N − 12 , κ,N
′ + 12 ; Ω0Ω
)
C
(
N − 12 , κ,N
′ + 12 ;M∆MM
′
)
+ a′(ωm)b(ωm)
√
N + 1
N ′ C
(
N + 12 , κ,N
′ − 12 ; Ω0Ω
)
C
(
N + 12 , κ,N
′ − 12 ;M∆MM
′
)
+ b(ωm)b′(ωm)
√
N + 1
N ′ + 1C
(
N + 12 , κ,N
′ + 12 ; Ω0Ω
)
C
(
N + 12 , κ,N
′ + 12 ;M∆MM
′
)
}
(26)
As noted above, there is no hybridization of the initial state for the N = 0, J = 12 → N ′, J ′ collisions,
i.e., a(ωm) = 0, b(ωm) = 1 in Eq. (26). By making use of the properties of the Clebsch-Gordan
coefficients [25],[26], the expression for the scattering amplitude from the N = 0, J = 12 ,M = ± 12
state to an N ′, J ′,M ′ state simplifies to
fωm
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1
∑
ρ
ρ+N ′even
dN ′−ρ,∆M (θε)FN ′ρJ|ρ|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(27)
The amplitude is seen to be directly proportional to the ΞN ′0 Legendre moment. We note that
the cross section for the N, J,M → N ′, J ′,M ′ transition differs from that for the N, J,−M →
N ′, J ′,−M ′ scattering. This is because the magnetic field completely lifts the degeneracy of the M
states, in contrast to the electric field case [9].
Page 7
7C. Results for He – CaH(X2Σ, J = 1/2 → J ′) scattering in a magnetic field
Here we apply the analytic model scattering to the He – CaH(2Σ+, J = 12 → J ′) collision
system. The CaH molecule, employed previously in thermalization experiments with a He buffer
gas [15], [16], has a rotational constant B = 4.2766 cm−1 and a spin-rotational interaction parameter
γ = 0.0430 cm−1 [27]. Such values of molecular constants result in an essentially perfect mixing
(and alignment) of the molecular states for field strengths H ≥ 0.1 Tesla, see Sec. III A.
According to Ref. [28], the ground-state He–CaH potential energy surface has a global
minimum of −10.6 cm−1. Such a weak attractive well can be neglected at a collision energy as low
as 200 cm−1 (which corresponds to a wave number k = 6.58 A˚−1). The corresponding value of the
Massey parameter, ξ ≈ 0.5, warrants the validity of the sudden approximation to the He – CaH
collision system from this collision energy on. The “hard shell” of the potential energy surface was
found by a fit to Eq. (8) for κ ≤ 8, and is shown in Fig. 3. The coefficients Ξκ0 obtained from the
fit are listed in Table II. According to Eq. (9), the Ξ00 coefficient determines the hard-sphere radius
R0, which is responsible for elastic scattering.
1. Differential cross sections
The state-to-state differential cross sections for scattering in a field parallel (‖) and perpen-
dicular (⊥) to k are given by
Iωm,(‖,⊥)0→J′ (ϑ) =
∑
M ′
Iωm,(‖,⊥)0,0→J′,M ′(ϑ) (28)
with
Iωm,(‖,⊥)0,0→J′,M ′(ϑ) =
∣
∣
∣fωm,(‖,⊥)0,0→J˜′,M ′(ϑ)
∣
∣
∣
2
(29)
They are presented in Figs. 4, 5 for He–CaH collisions at zero field, ωm = 0, as well as at high field,
ωm = 0.3 (corresponding to H =2.75 T for CaH), where the hybridization and alignment are as
complete as they can get.
From Eq. (27) for the scattering amplitude, we see that the differential cross section for the
N = 0 → N ′ transitions is proportional to the ΞN ′0 Legendre moment. According to Table II, the
Legendre expansion of the He–CaH potential energy surface is dominated by Ξ20. Therefore, the
transition N = 0 → N ′ = 2 provides the largest contribution to the cross section.
The field dependence of the scattering amplitude, Eq. (27), is encapsulated in the coefficients
a′(ωm) and b′(ωm), whose values cannot affect the angular dependence, as this is determined solely
by the Bessel functions, J|ρ|(kR0ϑ). Furthermore, the summation in Eq. (27) includes only even ρ
for even N ′, and odd ρ for odd N ′. From the asymptotic properties of Bessel functions [29], we have
for large angles such that ϑ≫ πρ/2kR0:
J|ρ|(kR0ϑ) ∼
cos
(
kR0ϑ− π4
)
for ρ even
sin
(
kR0ϑ− π4
)
for ρ odd
(30)
For the He – CaH system, the phase shift between the J0 and J2 Bessel functions, which contribute
to the N = 0 → N ′ = 1, 2 transitions, is negligibly small at angles up to about 30◦. Therefore there
is no field-induced phase shift, neither in the parallel nor in the perpendicular case, as illustrated by
Figs. 4, 5.
Here we apply the analytic model scattering to the He – CaH(2Σ+, J = 12 → J ′) collision
system. The CaH molecule, employed previously in thermalization experiments with a He buffer
gas [15], [16], has a rotational constant B = 4.2766 cm−1 and a spin-rotational interaction parameter
γ = 0.0430 cm−1 [27]. Such values of molecular constants result in an essentially perfect mixing
(and alignment) of the molecular states for field strengths H ≥ 0.1 Tesla, see Sec. III A.
According to Ref. [28], the ground-state He–CaH potential energy surface has a global
minimum of −10.6 cm−1. Such a weak attractive well can be neglected at a collision energy as low
as 200 cm−1 (which corresponds to a wave number k = 6.58 A˚−1). The corresponding value of the
Massey parameter, ξ ≈ 0.5, warrants the validity of the sudden approximation to the He – CaH
collision system from this collision energy on. The “hard shell” of the potential energy surface was
found by a fit to Eq. (8) for κ ≤ 8, and is shown in Fig. 3. The coefficients Ξκ0 obtained from the
fit are listed in Table II. According to Eq. (9), the Ξ00 coefficient determines the hard-sphere radius
R0, which is responsible for elastic scattering.
1. Differential cross sections
The state-to-state differential cross sections for scattering in a field parallel (‖) and perpen-
dicular (⊥) to k are given by
Iωm,(‖,⊥)0→J′ (ϑ) =
∑
M ′
Iωm,(‖,⊥)0,0→J′,M ′(ϑ) (28)
with
Iωm,(‖,⊥)0,0→J′,M ′(ϑ) =
∣
∣
∣fωm,(‖,⊥)0,0→J˜′,M ′(ϑ)
∣
∣
∣
2
(29)
They are presented in Figs. 4, 5 for He–CaH collisions at zero field, ωm = 0, as well as at high field,
ωm = 0.3 (corresponding to H =2.75 T for CaH), where the hybridization and alignment are as
complete as they can get.
From Eq. (27) for the scattering amplitude, we see that the differential cross section for the
N = 0 → N ′ transitions is proportional to the ΞN ′0 Legendre moment. According to Table II, the
Legendre expansion of the He–CaH potential energy surface is dominated by Ξ20. Therefore, the
transition N = 0 → N ′ = 2 provides the largest contribution to the cross section.
The field dependence of the scattering amplitude, Eq. (27), is encapsulated in the coefficients
a′(ωm) and b′(ωm), whose values cannot affect the angular dependence, as this is determined solely
by the Bessel functions, J|ρ|(kR0ϑ). Furthermore, the summation in Eq. (27) includes only even ρ
for even N ′, and odd ρ for odd N ′. From the asymptotic properties of Bessel functions [29], we have
for large angles such that ϑ≫ πρ/2kR0:
J|ρ|(kR0ϑ) ∼
cos
(
kR0ϑ− π4
)
for ρ even
sin
(
kR0ϑ− π4
)
for ρ odd
(30)
For the He – CaH system, the phase shift between the J0 and J2 Bessel functions, which contribute
to the N = 0 → N ′ = 1, 2 transitions, is negligibly small at angles up to about 30◦. Therefore there
is no field-induced phase shift, neither in the parallel nor in the perpendicular case, as illustrated by
Figs. 4, 5.
Page 8
8Figs. 4 and 5 show that the magnetic field induces only small changes in the amplitudes of
the cross sections, without shifting their oscillations. The amplitude variation is so small because
the magnetic field fails to mix contributions from the different Ξκ,0 Legendre moments, in contrast
to scattering in electrostatic [9] and radiative [10] fields. The changes in the amplitudes of the
differential cross sections are closely related to the field dependence of the partial integral cross
sections, which are analyzed next.
2. Integral cross sections
The angular range, ϑ . 30◦, where the Fraunhofer approximation applies the best, comprises
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 approximation, by integrating
the Fraunhofer differential cross sections, Eq. (28) and (29), over the solid angle sinϑdϑdϕ, with
0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ 2π.
The integral cross-sections thus obtained for the magnetic field oriented parallel and per-
pendicular to the initial wave vector are presented in Figs. 6 and 7. A prominent feature of the
cross sections for the N = 0, J = 12 → N ′, J ′ transitions is that, in the parallel field geometry, they
increase for the F1 final states and decrease for the F2 states, while it is the other way around for
the perpendicular geometry.
In order to make sense of these trends in the field dependence of the M -averaged
cross sections, let us take a closer look at the partial, M -resolved cross sections for the
N = 0, J = 12 ,M → N ′, J ′,M ′ channels and the two field geometries, also shown in Figs. 6 and 7.
(i) Magnetic field parallel to the initial wave vector, H ‖ k. In this case, the real Wigner
matrices reduce to the Kronecker delta functions, dN ′−ρ,∆M (0) = δ−ρ,∆M , and the scattering
amplitude (27) becomes:
fωm,‖
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1FN ′,−∆MJ|∆M|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(31)
Eq. (31) allows to readily interpret the dependences presented in Fig. 6. First, we see that the
FN ′,−∆M coefficients, defined by eq. (12), lead to a selection rule, namely that the cross sections
vanish for N ′+∆M odd. Therefore, the partial cross sections for such combinations of N ′ and ∆M
do not contribute anything to the trends seen in Fig. 6 that we wish to explain. Equally absent are
contributions from the transitions leading to the F1 states with M = ±J , since these states exhibit
no alignment, see Fig. 2 (a), (c).
As we can see from Fig. 6, the field-dependence of the cross section for the N = 0, J = 1/2 →
N ′, J ′ transitions is a result of a competition among the partialM -resolved cross sections. Therefore
we need to account for the relative magnitudes of the non-vanishing M -dependent cross sections.
Let us do it for the scattering channel N = 0, J = 12 ,M → N ′ = 2, J ′ = 52 ,M ′. Substituting
the coefficients from Table I into Eq. (31), we see that the term in the square brackets vanishes
for M = − 12 ,M ′ = − 12 and for M = − 12 ,M ′ = 32 , but equals
√
5 for M = 12 ,M ′ = 12 and
M = 12 ,M ′ = − 32 . In addition, taking into account that F2,2 > F2,0, we see that the M -averaged
cross section must go up with increasing field strength.
More generally, the dependence of the cross sections on the magnetic field is contained
the cross sections, without shifting their oscillations. The amplitude variation is so small because
the magnetic field fails to mix contributions from the different Ξκ,0 Legendre moments, in contrast
to scattering in electrostatic [9] and radiative [10] fields. The changes in the amplitudes of the
differential cross sections are closely related to the field dependence of the partial integral cross
sections, which are analyzed next.
2. Integral cross sections
The angular range, ϑ . 30◦, where the Fraunhofer approximation applies the best, comprises
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 approximation, by integrating
the Fraunhofer differential cross sections, Eq. (28) and (29), over the solid angle sinϑdϑdϕ, with
0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ 2π.
The integral cross-sections thus obtained for the magnetic field oriented parallel and per-
pendicular to the initial wave vector are presented in Figs. 6 and 7. A prominent feature of the
cross sections for the N = 0, J = 12 → N ′, J ′ transitions is that, in the parallel field geometry, they
increase for the F1 final states and decrease for the F2 states, while it is the other way around for
the perpendicular geometry.
In order to make sense of these trends in the field dependence of the M -averaged
cross sections, let us take a closer look at the partial, M -resolved cross sections for the
N = 0, J = 12 ,M → N ′, J ′,M ′ channels and the two field geometries, also shown in Figs. 6 and 7.
(i) Magnetic field parallel to the initial wave vector, H ‖ k. In this case, the real Wigner
matrices reduce to the Kronecker delta functions, dN ′−ρ,∆M (0) = δ−ρ,∆M , and the scattering
amplitude (27) becomes:
fωm,‖
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1FN ′,−∆MJ|∆M|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(31)
Eq. (31) allows to readily interpret the dependences presented in Fig. 6. First, we see that the
FN ′,−∆M coefficients, defined by eq. (12), lead to a selection rule, namely that the cross sections
vanish for N ′+∆M odd. Therefore, the partial cross sections for such combinations of N ′ and ∆M
do not contribute anything to the trends seen in Fig. 6 that we wish to explain. Equally absent are
contributions from the transitions leading to the F1 states with M = ±J , since these states exhibit
no alignment, see Fig. 2 (a), (c).
As we can see from Fig. 6, the field-dependence of the cross section for the N = 0, J = 1/2 →
N ′, J ′ transitions is a result of a competition among the partialM -resolved cross sections. Therefore
we need to account for the relative magnitudes of the non-vanishing M -dependent cross sections.
Let us do it for the scattering channel N = 0, J = 12 ,M → N ′ = 2, J ′ = 52 ,M ′. Substituting
the coefficients from Table I into Eq. (31), we see that the term in the square brackets vanishes
for M = − 12 ,M ′ = − 12 and for M = − 12 ,M ′ = 32 , but equals
√
5 for M = 12 ,M ′ = 12 and
M = 12 ,M ′ = − 32 . In addition, taking into account that F2,2 > F2,0, we see that the M -averaged
cross section must go up with increasing field strength.
More generally, the dependence of the cross sections on the magnetic field is contained
Page 9
9in the two hybridization coefficients a′(ωm) and b′(ωm). In the field-free case, a′(ωm) = 0 and
b′(ωm) = 1 for collisions leading to F1 states, whereas a′(ωm) = 1, b′(ωm) = 0 for collisions that
produce F2 states. In a magnetic field, the a′(ωm) and b′(ωm) coefficients assume values ranging
between −1 and +1. As a consequence, the a′(ωm) and b′(ωm) coefficients have the same signs for
the F1 states and opposite signs for the F2 states. Clearly, then, for an F1 state, |a′(ωm)| increases
with the field strength, while |b′(ωm)| decreases. Hence the factor in the square brackets of Eq. (31)
increases with ωm for M = 12 and decreases for M = − 12 , because of the opposite sign of the a′(ωm)
coefficient.This is reversed for the final F2 states, i.e., the cross sections increase for M = − 12 , and
decrease for M = 12 .
(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k. In this case, Eq. (27)
takes the form:
fωm,⊥
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1
∑
ρ
ρ+N ′even
dN ′−ρ,∆M (π2 )FN ′ρJ|ρ|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(32)
which is more involved than for the parallel case, although the field-dependent coefficients a′(ωm)
and b′(ωm) remain outside of the summation. The difference between the parallel and perpendicular
cases is related to the values of the real Wigner d-matrices mixed by eq. (32). For instance, an
inspection of the coefficients from Table I, along with the dN ′−ρ,∆M matrices, reveals that the M -
averaged cross sections for the N = 0, J = 12 → N ′ = 2, J ′ = 52 transition (i.e., transition to an F1
final state) will decrease with ωm, in contrast to the parallel case.
3. Frontal-versus-lateral steric asymmetry
As in our previous work [9], [10], we define a frontal-versus-lateral steric asymmetry by the
expression
Si→f =
σ‖ − σ⊥
σ‖ + σ⊥
(33)
where the integral cross sections σ‖,⊥ correspond, respectively, to H ‖ k and H ⊥ k. The field
dependence of the steric asymmetry for the He – CaH collisions is presented in Fig. 8. One can
see that a particularly pronounced asymmetry obtains for the scattering into the N ′ = 1, J ′ = 12 , 32
final states, while it is smaller for the N = 0 → N ′ = 2 channels. Moreover, the steric asymmetry
exhibits a sign alternation: it is positive for F1 final states and negative for F2 final states. This
behavior is a reflection of the alternation in the trends of the integral cross sections for the F1 and
F2 final states, discussed in the previous subsection, cf. the corresponding M -dependent integral
cross sections, Figs. 6 and 7.
We note that within the Fraunhofer model, elastic collisions do not exhibit any steric asym-
metry. This follows from the isotropy of the elastic scattering amplitude, Eq. (6), which depends
on the radius R0 only: a sphere looks the same from any direction.
b′(ωm) = 1 for collisions leading to F1 states, whereas a′(ωm) = 1, b′(ωm) = 0 for collisions that
produce F2 states. In a magnetic field, the a′(ωm) and b′(ωm) coefficients assume values ranging
between −1 and +1. As a consequence, the a′(ωm) and b′(ωm) coefficients have the same signs for
the F1 states and opposite signs for the F2 states. Clearly, then, for an F1 state, |a′(ωm)| increases
with the field strength, while |b′(ωm)| decreases. Hence the factor in the square brackets of Eq. (31)
increases with ωm for M = 12 and decreases for M = − 12 , because of the opposite sign of the a′(ωm)
coefficient.This is reversed for the final F2 states, i.e., the cross sections increase for M = − 12 , and
decrease for M = 12 .
(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k. In this case, Eq. (27)
takes the form:
fωm,⊥
0,12 ,±
1
2→N
′,J′,M ′
(ϑ) = ikR0
2π
ΞN ′0
2N ′ + 1
∑
ρ
ρ+N ′even
dN ′−ρ,∆M (π2 )FN ′ρJ|ρ|(kR0ϑ)
×
[
±a′(ωm)
√
N ′ ∓M ′ + 12 + b
′(ωm)
√
N ′ ±M ′ + 12
]
(32)
which is more involved than for the parallel case, although the field-dependent coefficients a′(ωm)
and b′(ωm) remain outside of the summation. The difference between the parallel and perpendicular
cases is related to the values of the real Wigner d-matrices mixed by eq. (32). For instance, an
inspection of the coefficients from Table I, along with the dN ′−ρ,∆M matrices, reveals that the M -
averaged cross sections for the N = 0, J = 12 → N ′ = 2, J ′ = 52 transition (i.e., transition to an F1
final state) will decrease with ωm, in contrast to the parallel case.
3. Frontal-versus-lateral steric asymmetry
As in our previous work [9], [10], we define a frontal-versus-lateral steric asymmetry by the
expression
Si→f =
σ‖ − σ⊥
σ‖ + σ⊥
(33)
where the integral cross sections σ‖,⊥ correspond, respectively, to H ‖ k and H ⊥ k. The field
dependence of the steric asymmetry for the He – CaH collisions is presented in Fig. 8. One can
see that a particularly pronounced asymmetry obtains for the scattering into the N ′ = 1, J ′ = 12 , 32
final states, while it is smaller for the N = 0 → N ′ = 2 channels. Moreover, the steric asymmetry
exhibits a sign alternation: it is positive for F1 final states and negative for F2 final states. This
behavior is a reflection of the alternation in the trends of the integral cross sections for the F1 and
F2 final states, discussed in the previous subsection, cf. the corresponding M -dependent integral
cross sections, Figs. 6 and 7.
We note that within the Fraunhofer model, elastic collisions do not exhibit any steric asym-
metry. This follows from the isotropy of the elastic scattering amplitude, Eq. (6), which depends
on the radius R0 only: a sphere looks the same from any direction.
Page 10
10
IV. SCATTERING OF 3Σ MOLECULES BY CLOSED-SHELL ATOMS IN A
MAGNETIC FIELD
A. A 3Σ molecule in a magnetic field
The field-free Hamiltonian of a 3Σ electronic state consists of rotational, spin-rotation, and
spin-spin terms
H0 = BN2 + γN · S + 23λ(3S
2
z − S2) (34)
where γ and λ are the spin-rotation and spin-spin constants, respectively. In the Hund’s case (b)
basis, the field-free Hamiltonian (34) consists of 3 × 3 matrices pertaining to different J values
(except for J=0). The matrix elements of Hamiltonian (34) can be found, e.g., in ref. [30] (see
also [14] and [24]). The eigenenergies of H0 are (in units of the rotational constant B):
E1(J)/B = J(J + 1) + 1−
3γ′
2
− λ
′
3
−X
E2(J)/B = J(J + 1)− γ′ +
2λ′
3
E3(J)/B = J(J + 1) + 1−
3γ′
2
− λ
′
3
+X
(35)
with
X ≡
[
J(J + 1)(γ′ − 2)2 +
(γ′ + 2λ′ − 2
2
)2
]1/2
γ′ ≡ γ/B
λ′ ≡ λ/B
The eigenenergies (35) correspond to the three ways of combining rotational and electronic spin
angular momenta N and S for S = 1 into a total angular momentum J; the total angular momentum
quantum number takes values J = N +1, J = N , and J = N −1 for states which are conventionally
designated as F1, F2, and F3, respectively. For the case when N = 1, J = 0, the sign of the X term
should be reversed [32]. The parity of the states is (−1)N .
The interaction of a 3Σ molecule with a magnetic field H is given by:
Vm = SZωmB, (36)
where
ωm ≡
gSµBH
B (37)
is a dimensionless parameter characterizing the strength of the Zeeman interaction, cf. Eq. (20).
We evaluated the Zeeman effect in Hund’s case (b) basis
|N, J,M〉 = c1NJ |J, 1,M〉+ c0NJ |J, 0,M〉+ c−1NJ |J,−1,M〉 (38)
using the matrix elements of the SZ operator given in Appendix A.
IV. SCATTERING OF 3Σ MOLECULES BY CLOSED-SHELL ATOMS IN A
MAGNETIC FIELD
A. A 3Σ molecule in a magnetic field
The field-free Hamiltonian of a 3Σ electronic state consists of rotational, spin-rotation, and
spin-spin terms
H0 = BN2 + γN · S + 23λ(3S
2
z − S2) (34)
where γ and λ are the spin-rotation and spin-spin constants, respectively. In the Hund’s case (b)
basis, the field-free Hamiltonian (34) consists of 3 × 3 matrices pertaining to different J values
(except for J=0). The matrix elements of Hamiltonian (34) can be found, e.g., in ref. [30] (see
also [14] and [24]). The eigenenergies of H0 are (in units of the rotational constant B):
E1(J)/B = J(J + 1) + 1−
3γ′
2
− λ
′
3
−X
E2(J)/B = J(J + 1)− γ′ +
2λ′
3
E3(J)/B = J(J + 1) + 1−
3γ′
2
− λ
′
3
+X
(35)
with
X ≡
[
J(J + 1)(γ′ − 2)2 +
(γ′ + 2λ′ − 2
2
)2
]1/2
γ′ ≡ γ/B
λ′ ≡ λ/B
The eigenenergies (35) correspond to the three ways of combining rotational and electronic spin
angular momenta N and S for S = 1 into a total angular momentum J; the total angular momentum
quantum number takes values J = N +1, J = N , and J = N −1 for states which are conventionally
designated as F1, F2, and F3, respectively. For the case when N = 1, J = 0, the sign of the X term
should be reversed [32]. The parity of the states is (−1)N .
The interaction of a 3Σ molecule with a magnetic field H is given by:
Vm = SZωmB, (36)
where
ωm ≡
gSµBH
B (37)
is a dimensionless parameter characterizing the strength of the Zeeman interaction, cf. Eq. (20).
We evaluated the Zeeman effect in Hund’s case (b) basis
|N, J,M〉 = c1NJ |J, 1,M〉+ c0NJ |J, 0,M〉+ c−1NJ |J,−1,M〉 (38)
using the matrix elements of the SZ operator given in Appendix A.
Page 11
11
The Zeeman eigenfunctions are hybrids of the Hund’s case (b) basis functions (38):
∣
∣
∣N˜, J˜ ,M ;ωm
〉
=
∑
NJ
aN˜J˜NJ(ωm) |N, J,M〉 (39)
and are labeled by N˜ and J˜ , which are the angular momentum quantum numbers of the field-free
state that adiabatically correlates with a given state in the field. Since Vm couples Hund’s case (b)
states that differ in N by 0 or ±2, the parity remains definite in the presence of a magnetic field,
and is given by (−1)N˜ . However, the Zeeman matrix for a 3Σ molecule is no longer finite, unlike
the 2× 2 Zeeman matrix for a 2Σ state.
Using the Hund’s case (b) rather than Hund’s case (a) basis set makes it possible to directly
relate the field-free states and the Zeeman states, via the hybridization coefficients aN˜J˜NJ .
The alignment cosine, 〈cos2 θ〉, of the Zeeman states can be evaluated from the matrix
elements of Appendix B. The dependence of 〈cos2 θ〉 on the magnetic field strength parameter ωm
is exemplified in Fig. 9 for N˜ = 1, 3 Zeeman states of the 16O2 molecule.
B. The field-dependent scattering amplitude
We consider scattering from an initial N, J state to a final N ′, J ′ state. We transform the
wavefunctions (39) to the space-fixed frame by making use of Eq. (23) – cf. Section III B. As a
result, the initial and final states become:
|i〉 ≡
∣
∣
∣N˜ , J˜ ,M, ωm
〉
=
∑
NJ
√
2J + 1
4π a
N˜J˜
NJ(ωm)
∑
Ω
cΩNJ
∑
ξ
DJξM (ϕε, θε, 0)DJ∗ξΩ(ϕ, θ, 0) (40)
〈f| ≡ 〈N˜ ′, J˜ ′,M ′, ωm| =
∑
N ′J′
√
2J ′ + 1
4π b
N˜ ′J˜′
N ′J′(ωm)
∑
Ω′
cΩ′N ′J′
∑
ξ′
DJ′ξ′M ′(ϕε, θε, 0)DJ
′∗
ξ′Ω′(ϕ, θ, 0) (41)
On substituting from Eqs. (40) and (41) into Eq. (11) and some angular momentum algebra, we
obtain a general expression for the scattering amplitude:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
N˜J˜
NJ (ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′;M∆MM ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (42)
C. Results for He–O2(X3Σ, N = 0, J = 1 → N ′, J ′) scattering in a magnetic field
The 16O2(3Σ−) molecule has a rotational constant B = 1.4377 cm−1, a spin-rotation con-
stant γ = −0.0084 cm−1, and a spin-spin constant λ = 1.9848 cm−1 [33]. According to Ref. [34],
the ground state He – O2 potential energy surface has a global minimum of −27.90 cm−1, which
can be neglected at a collision energy 200 cm−1 (corresponding to a wave number k = 6.49 A˚−1).
A small value of the Massey parameter, ξ ≈ 0.1, ensures the validity of the sudden approximation.
The “hard shell” of the potential energy surface at this collision energy is shown in Fig. 3, and the
The Zeeman eigenfunctions are hybrids of the Hund’s case (b) basis functions (38):
∣
∣
∣N˜, J˜ ,M ;ωm
〉
=
∑
NJ
aN˜J˜NJ(ωm) |N, J,M〉 (39)
and are labeled by N˜ and J˜ , which are the angular momentum quantum numbers of the field-free
state that adiabatically correlates with a given state in the field. Since Vm couples Hund’s case (b)
states that differ in N by 0 or ±2, the parity remains definite in the presence of a magnetic field,
and is given by (−1)N˜ . However, the Zeeman matrix for a 3Σ molecule is no longer finite, unlike
the 2× 2 Zeeman matrix for a 2Σ state.
Using the Hund’s case (b) rather than Hund’s case (a) basis set makes it possible to directly
relate the field-free states and the Zeeman states, via the hybridization coefficients aN˜J˜NJ .
The alignment cosine, 〈cos2 θ〉, of the Zeeman states can be evaluated from the matrix
elements of Appendix B. The dependence of 〈cos2 θ〉 on the magnetic field strength parameter ωm
is exemplified in Fig. 9 for N˜ = 1, 3 Zeeman states of the 16O2 molecule.
B. The field-dependent scattering amplitude
We consider scattering from an initial N, J state to a final N ′, J ′ state. We transform the
wavefunctions (39) to the space-fixed frame by making use of Eq. (23) – cf. Section III B. As a
result, the initial and final states become:
|i〉 ≡
∣
∣
∣N˜ , J˜ ,M, ωm
〉
=
∑
NJ
√
2J + 1
4π a
N˜J˜
NJ(ωm)
∑
Ω
cΩNJ
∑
ξ
DJξM (ϕε, θε, 0)DJ∗ξΩ(ϕ, θ, 0) (40)
〈f| ≡ 〈N˜ ′, J˜ ′,M ′, ωm| =
∑
N ′J′
√
2J ′ + 1
4π b
N˜ ′J˜′
N ′J′(ωm)
∑
Ω′
cΩ′N ′J′
∑
ξ′
DJ′ξ′M ′(ϕε, θε, 0)DJ
′∗
ξ′Ω′(ϕ, θ, 0) (41)
On substituting from Eqs. (40) and (41) into Eq. (11) and some angular momentum algebra, we
obtain a general expression for the scattering amplitude:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
N˜J˜
NJ (ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′;M∆MM ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (42)
C. Results for He–O2(X3Σ, N = 0, J = 1 → N ′, J ′) scattering in a magnetic field
The 16O2(3Σ−) molecule has a rotational constant B = 1.4377 cm−1, a spin-rotation con-
stant γ = −0.0084 cm−1, and a spin-spin constant λ = 1.9848 cm−1 [33]. According to Ref. [34],
the ground state He – O2 potential energy surface has a global minimum of −27.90 cm−1, which
can be neglected at a collision energy 200 cm−1 (corresponding to a wave number k = 6.49 A˚−1).
A small value of the Massey parameter, ξ ≈ 0.1, ensures the validity of the sudden approximation.
The “hard shell” of the potential energy surface at this collision energy is shown in Fig. 3, and the
Page 12
12
Legendre moments Ξκ0, obtained from a fit to the potential energy surface of Ref. [34], are listed in
Table II. Since the He – O2 potential is of D2h symmetry, only even Legendre moments are nonzero.
Furthermore, since the nuclear spin of 16O is zero and the electronic ground state antisym-
metric (a 3Σ−g state), only rotational states with an odd rotational quantum number N are allowed.
We will assume that the O2 molecule is initially in its rotational ground state, |N = 1, J = 0,M = 0〉.
Expression (42) for the scattering amplitude further simplifies for particular geometries. In
what follows, we will consider two such geometries.
(i) Magnetic field parallel to the initial wave vector, H ‖ k, in which case the scattering
amplitude becomes:
fωm,‖1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π J|M ′|(kR0ϑ)
∑
κ 6=0
κ even
Ξκ0FκM ′
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
10
NJ(ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′; 0M ′M ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (43)
(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k, in which case Eq. (42)
simplifies to:
fωm,⊥1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π
∑
κ,ρ even
κ 6=0
Ξκ0 dκ−ρ,M ′(π2 )FκρJ|ρ|(kR0ϑ)
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
10
NJ(ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′; 0M ′M ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (44)
Eqs. (43) and (44) imply that for either geometry, only partial cross sections for the N = 1, J =
0,M = 0 → N ′, J ′,M ′ collisions with M ′ even can contribute to the scattering. This is particularly
easy to see in the H ‖ k case, where the FκM ′ coefficients vanish for M ′ odd as Fκρ vanishes for
odd κ + ρ. In the H ⊥ k case, a summation over ρ arises. Since for κ even and M ′ odd the real
Wigner matrices obey the relation dκ−ρ,M ′
(π
2
)
= −dκρ,M ′
(π
2
)
, the sum over ρ vanishes and so do the
partial cross sections for M ′ odd.
1. Differential cross sections
The differential cross sections of the He – O2 (N = 1, J = 0 → N ′, J ′) collisions, calculated
from Eqs. (28) and (29), are presented in Figs. 10 and 11. Also shown is the elastic cross section,
obtained from the scattering amplitude (6). The differential cross sections are shown for the field-free
case, ωm = 0, as well as for ωm = 5, which for O2 corresponds to a magnetic field H =7.7 T.
The angular dependence of the differential cross sections is determined by the Bessel func-
tions appearing in the scattering amplitudes (43) and (44). In the parallel case, the angular de-
pendence is given expressly by J|M ′|(kR0ϑ), and is not affected by the magnetic field. Since only
even-κ terms contribute to the sum in the scattering amplitude and the coefficients Fκρ vanish for
κ+ ρ odd, the differential cross sections are given solely by even Bessel functions. This is the case
for both parallel and perpendicular geometries. As the elastic scattering amplitude is given by an
odd Bessel function, Eq. (6), the elastic and rotationally inelastic differential cross sections oscillate
with an opposite phase.
Legendre moments Ξκ0, obtained from a fit to the potential energy surface of Ref. [34], are listed in
Table II. Since the He – O2 potential is of D2h symmetry, only even Legendre moments are nonzero.
Furthermore, since the nuclear spin of 16O is zero and the electronic ground state antisym-
metric (a 3Σ−g state), only rotational states with an odd rotational quantum number N are allowed.
We will assume that the O2 molecule is initially in its rotational ground state, |N = 1, J = 0,M = 0〉.
Expression (42) for the scattering amplitude further simplifies for particular geometries. In
what follows, we will consider two such geometries.
(i) Magnetic field parallel to the initial wave vector, H ‖ k, in which case the scattering
amplitude becomes:
fωm,‖1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π J|M ′|(kR0ϑ)
∑
κ 6=0
κ even
Ξκ0FκM ′
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
10
NJ(ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′; 0M ′M ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (43)
(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k, in which case Eq. (42)
simplifies to:
fωm,⊥1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π
∑
κ,ρ even
κ 6=0
Ξκ0 dκ−ρ,M ′(π2 )FκρJ|ρ|(kR0ϑ)
×
∑
NJ
N ′J′
√
2J + 1
2J ′ + 1a
10
NJ(ωm)bN˜
′J˜′
N ′J′(ωm)C(JκJ ′; 0M ′M ′)
∑
Ω
cΩNJcΩN ′J′C(JκJ ′; Ω0Ω) (44)
Eqs. (43) and (44) imply that for either geometry, only partial cross sections for the N = 1, J =
0,M = 0 → N ′, J ′,M ′ collisions with M ′ even can contribute to the scattering. This is particularly
easy to see in the H ‖ k case, where the FκM ′ coefficients vanish for M ′ odd as Fκρ vanishes for
odd κ + ρ. In the H ⊥ k case, a summation over ρ arises. Since for κ even and M ′ odd the real
Wigner matrices obey the relation dκ−ρ,M ′
(π
2
)
= −dκρ,M ′
(π
2
)
, the sum over ρ vanishes and so do the
partial cross sections for M ′ odd.
1. Differential cross sections
The differential cross sections of the He – O2 (N = 1, J = 0 → N ′, J ′) collisions, calculated
from Eqs. (28) and (29), are presented in Figs. 10 and 11. Also shown is the elastic cross section,
obtained from the scattering amplitude (6). The differential cross sections are shown for the field-free
case, ωm = 0, as well as for ωm = 5, which for O2 corresponds to a magnetic field H =7.7 T.
The angular dependence of the differential cross sections is determined by the Bessel func-
tions appearing in the scattering amplitudes (43) and (44). In the parallel case, the angular de-
pendence is given expressly by J|M ′|(kR0ϑ), and is not affected by the magnetic field. Since only
even-κ terms contribute to the sum in the scattering amplitude and the coefficients Fκρ vanish for
κ+ ρ odd, the differential cross sections are given solely by even Bessel functions. This is the case
for both parallel and perpendicular geometries. As the elastic scattering amplitude is given by an
odd Bessel function, Eq. (6), the elastic and rotationally inelastic differential cross sections oscillate
with an opposite phase.
Page 13
13
According to the general properties of Bessel functions [29], at large angles the phase shift
between different even Bessel functions disappears and their asymptotic form is given by Eq. (30).
For the system under consideration, the phase shift between J0(kR0ϑ) and J2(kR0ϑ) functions
becomes negligibly small at angles about 40◦, while the shift between J4(kR0ϑ) and either J0(kR0ϑ)
or J2(kR0ϑ) can only be neglected at angles about 120◦. Therefore, if the cross section is comprised
only of J0 and J2 contributions, it will not be shifted with respect to the field-free case, while
there may appear a field-iduced phase shift if the J4 Bessel function also contributes. Indeed, in the
parallel case, the 1, 0 → 3, 4 cross section, presented in Fig. 10 (e), exhibits a slight phase shift. In the
perpendicular case, the Bessel functions J|ρ|(kR0ϑ) for a range of ρ’s are mixed, see eq. (44), which
results in a field-induced phase shift for both the 1, 0 → 1, 2 and 1, 0 → 3, 4 transitions. Figs. 10
and 11 show the differential cross sections only up to 60◦, since this angular range dominates the
integral cross sections. However, the phase shifts disappear only at larger angles, of about 120◦.
The most dramatic feature of the magnetic field dependence of the differential cross sections
is the onset of inelastic scattering for channels that are closed in the absence of the field: these involve
the transitions N = 1, J = 0 → N ′, J ′ with J ′ = N ′. That these channels are closed in the field-free
case can be gleaned from the scattering amplitude (43) for ωm = 0, which reduces to
fFF1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π J|M ′|(kR0ϑ)
ΞJ′0√
2J ′ + 1
FJ′M ′c0N ′J′ (45)
This field-free amplitude vanishes because the c0N ′J′ coefficients are zero for N ′ = J ′, as can be shown
by the diagonalization of the field-free Hamiltonian (34). As a result, the field-free cross sections
for the transitions to 1, 1 and 3, 3 states vanish. The hybridization by a magnetic field brings in
coefficients c±1N ′J′ which are nonvanishing for N ′ = J ′. The feature manifests itself in the integral
cross sections as well.
2. Integral cross sections
The integral cross sections for the He – O2 (N = 1, J = 0,M = 0 → N ′, J ′,M ′) scattering
are shown in Figs. 12 and 13 for H ‖ k and H ⊥ k, respectively.
First we note that since the expansion of the He – O2 potential is dominated by Ξκ0, see
Table II, the sum in Eqs. (43) and (44) over κ can be approximated by the κ = 2 term. In that case
the Clebsch-Gordan coefficient C(J, κ, J ′, 000) imposes a selection rule which limits the summation
over J and J ′ to terms with J ′ = J ; J ± 2. In the field-free case, this selection rule is only satisfied
for scattering from J = 0 → J ′ = 2 and, therefore, only N = 1, J = 0 → N ′ = 1, J ′ = 2 and
N = 1, J = 0 → N ′ = 3, J ′ = 2 cross sections can be expected to be sizable at zero field. Figs. 12
and 13 corroborate that this is indeed the case.
The field dependence of the N = 1, J = 0 → N ′, J ′ cross sections can be traced to the
variation of the partial N = 1, J = 0,M = 0 → N ′, J ′,M ′ contributions. Therefore, we will
take a look at how the M -resolved integral cross sections vary with the magnetic field. The field
dependence of the partial N = 1, J = 0,M = 0 → M ′, J ′,M ′ cross sections is contained in the
hybridization coefficients a10NJ(ωm) and bN˜
′J˜′
N ′J′(ωm), both for H ‖ k and H ⊥ k. The a10NJ and bN˜
′J˜′
N ′J′
coefficients for various values of N˜ ′, J˜ ′ are presented in Figs. 14, 15, and 16 for M = 0, 2, and −2,
respectively. At zero field, these coefficients are equal to unity for N = N˜ , J = J˜ and are zero
otherwise, see Figs. 14–16 (a). Once the field is applied, the “distributions” of the a10NJ(ωm) and
bN˜ ′J˜′N ′J′(ωm) coefficients undergo a broadening, as presented in Figs. 14–16 (b)-(d).
Such a broadening enhances the mixing of the 1, 1, 0 state with the 1, 2, 0 and 3, 2, 0 states,
According to the general properties of Bessel functions [29], at large angles the phase shift
between different even Bessel functions disappears and their asymptotic form is given by Eq. (30).
For the system under consideration, the phase shift between J0(kR0ϑ) and J2(kR0ϑ) functions
becomes negligibly small at angles about 40◦, while the shift between J4(kR0ϑ) and either J0(kR0ϑ)
or J2(kR0ϑ) can only be neglected at angles about 120◦. Therefore, if the cross section is comprised
only of J0 and J2 contributions, it will not be shifted with respect to the field-free case, while
there may appear a field-iduced phase shift if the J4 Bessel function also contributes. Indeed, in the
parallel case, the 1, 0 → 3, 4 cross section, presented in Fig. 10 (e), exhibits a slight phase shift. In the
perpendicular case, the Bessel functions J|ρ|(kR0ϑ) for a range of ρ’s are mixed, see eq. (44), which
results in a field-induced phase shift for both the 1, 0 → 1, 2 and 1, 0 → 3, 4 transitions. Figs. 10
and 11 show the differential cross sections only up to 60◦, since this angular range dominates the
integral cross sections. However, the phase shifts disappear only at larger angles, of about 120◦.
The most dramatic feature of the magnetic field dependence of the differential cross sections
is the onset of inelastic scattering for channels that are closed in the absence of the field: these involve
the transitions N = 1, J = 0 → N ′, J ′ with J ′ = N ′. That these channels are closed in the field-free
case can be gleaned from the scattering amplitude (43) for ωm = 0, which reduces to
fFF1,0,0→N ′,J′,M ′(ϑ) =
ikR0
4π J|M ′|(kR0ϑ)
ΞJ′0√
2J ′ + 1
FJ′M ′c0N ′J′ (45)
This field-free amplitude vanishes because the c0N ′J′ coefficients are zero for N ′ = J ′, as can be shown
by the diagonalization of the field-free Hamiltonian (34). As a result, the field-free cross sections
for the transitions to 1, 1 and 3, 3 states vanish. The hybridization by a magnetic field brings in
coefficients c±1N ′J′ which are nonvanishing for N ′ = J ′. The feature manifests itself in the integral
cross sections as well.
2. Integral cross sections
The integral cross sections for the He – O2 (N = 1, J = 0,M = 0 → N ′, J ′,M ′) scattering
are shown in Figs. 12 and 13 for H ‖ k and H ⊥ k, respectively.
First we note that since the expansion of the He – O2 potential is dominated by Ξκ0, see
Table II, the sum in Eqs. (43) and (44) over κ can be approximated by the κ = 2 term. In that case
the Clebsch-Gordan coefficient C(J, κ, J ′, 000) imposes a selection rule which limits the summation
over J and J ′ to terms with J ′ = J ; J ± 2. In the field-free case, this selection rule is only satisfied
for scattering from J = 0 → J ′ = 2 and, therefore, only N = 1, J = 0 → N ′ = 1, J ′ = 2 and
N = 1, J = 0 → N ′ = 3, J ′ = 2 cross sections can be expected to be sizable at zero field. Figs. 12
and 13 corroborate that this is indeed the case.
The field dependence of the N = 1, J = 0 → N ′, J ′ cross sections can be traced to the
variation of the partial N = 1, J = 0,M = 0 → N ′, J ′,M ′ contributions. Therefore, we will
take a look at how the M -resolved integral cross sections vary with the magnetic field. The field
dependence of the partial N = 1, J = 0,M = 0 → M ′, J ′,M ′ cross sections is contained in the
hybridization coefficients a10NJ(ωm) and bN˜
′J˜′
N ′J′(ωm), both for H ‖ k and H ⊥ k. The a10NJ and bN˜
′J˜′
N ′J′
coefficients for various values of N˜ ′, J˜ ′ are presented in Figs. 14, 15, and 16 for M = 0, 2, and −2,
respectively. At zero field, these coefficients are equal to unity for N = N˜ , J = J˜ and are zero
otherwise, see Figs. 14–16 (a). Once the field is applied, the “distributions” of the a10NJ(ωm) and
bN˜ ′J˜′N ′J′(ωm) coefficients undergo a broadening, as presented in Figs. 14–16 (b)-(d).
Such a broadening enhances the mixing of the 1, 1, 0 state with the 1, 2, 0 and 3, 2, 0 states,
Page 14
14
leading to an increase of the N = 1, J = 0 → N ′ = 1, J ′ = 1 cross section, see Figs. 12 and 13 (b). On
the other hand, the broadening of the “distribution” of the coefficients of the 1, 2, 0 and 3, 2, 0 final
states reduces the overlap with the initial state’s coefficients and thus diminishes the 1, 0, 0 → 1, 2, 0
and 1, 0, 0 → 3, 2, 0 cross sections.
The cross sections for the 3, 3, 0 final state are distinctly non-monotonous, exhibiting max-
ima. These arise because at zero field the overlap of the hybridization coefficients for the 1, 0, 0 and
3, 3, 0 states is zero (if only κ = 2 is taken into account), but turning on the field enhances the
mixing of the 3, 3, 0 and 3, 2, 0 states, causing the corresponding cross section to increase. At higher
ωm, the spread of coefficients becomes so large that the products a10NJb33N ′J′ , corresponding to the
selection rule J ′ = J ; J ± 2, become very small, cf. Fig. 14 (d). As a result, the cross section for the
1, 0, 0 → 3, 3, 0 channel decreases again.
Figures 12 and 13 (a),(c)–(e) show that the cross sections corresponding to the final states
with M ′ = ±2 exhibit maxima or minima, depending on the sign of M ′. These are due to the
changing overlap of the hybridization coefficients, just as for the 3, 3, 0 final state, see Figs. 15
and 16.
Indeed, the overlaps of the coefficients corresponding to M ′ = 2 and M ′ = −2 exhibit
a different field dependence. For instance, the mixing of the 3, 3,−2 and 3, 2,−2 states, which
increases with field strength, see Fig. 16 (b), results in an increase of the 1, 0, 0 → 1, 2,−2 cross
section, presented in Fig. 12 (a). For M = 2, there is little mixing of the 3, 4, 2 and 3, 3, 2 states
with the 3, 2, 2 state, see Fig. 15 (b), (c), which results in a decreasing 1, 0, 0 → 1, 2, 2 cross section.
Once the coefficients’ overlap increases at high fields, the corresponding cross section goes up again.
We note that both field-free and field-dressed transitions to the states with M ′ = ±4 are
negligibly small, since their cross sections are dominated by other than the (dominant) Ξ20 moment.
The difference between the parallel and perpendicular geometries is due to the real d-
matrices, appearing in Eq. (44). For instance, an inspection of equations (43) and (44) reveals
that the integral cross sections for the 1, 0 → 1, 1 transition will be larger for H ⊥ k due to the
coefficients dκ−ρ,0(π2 ). A similar argument holds in the case of scattering in electrostatic [9] and
radiative [10] fields.
3. Frontal-versus-lateral steric asymmetry
Fig. 17 shows the steric asymmetry for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions as a
function of the magnetic field. We see that the asymmetry is most pronounced for the N = 1, J =
0 → N = 1, J = 2 and N = 1, J = 0 → N = 3, J = 4 channels. This has its origin in the field
dependence of the integral cross sections, see Figs. 12 and 13.
V. SCATTERING OF 2Π MOLECULES BY CLOSED-SHELL ATOMS IN MAGNETIC
FIELDS
A. The 2Π molecule in magnetic field
In this Section, we consider a Hund’s case molecule, equivalent to a linear symmetric top. A
good example of such a molecules is the OH radical in its electronic ground state, X2ΠΩ, whose elec-
tronic spin and orbital angular momenta are strongly coupled to the molecular axis. Each rotational
state within the 2ΠΩ ground state is equivalent to a symmetric-top state |J,Ω,M〉 with projections
leading to an increase of the N = 1, J = 0 → N ′ = 1, J ′ = 1 cross section, see Figs. 12 and 13 (b). On
the other hand, the broadening of the “distribution” of the coefficients of the 1, 2, 0 and 3, 2, 0 final
states reduces the overlap with the initial state’s coefficients and thus diminishes the 1, 0, 0 → 1, 2, 0
and 1, 0, 0 → 3, 2, 0 cross sections.
The cross sections for the 3, 3, 0 final state are distinctly non-monotonous, exhibiting max-
ima. These arise because at zero field the overlap of the hybridization coefficients for the 1, 0, 0 and
3, 3, 0 states is zero (if only κ = 2 is taken into account), but turning on the field enhances the
mixing of the 3, 3, 0 and 3, 2, 0 states, causing the corresponding cross section to increase. At higher
ωm, the spread of coefficients becomes so large that the products a10NJb33N ′J′ , corresponding to the
selection rule J ′ = J ; J ± 2, become very small, cf. Fig. 14 (d). As a result, the cross section for the
1, 0, 0 → 3, 3, 0 channel decreases again.
Figures 12 and 13 (a),(c)–(e) show that the cross sections corresponding to the final states
with M ′ = ±2 exhibit maxima or minima, depending on the sign of M ′. These are due to the
changing overlap of the hybridization coefficients, just as for the 3, 3, 0 final state, see Figs. 15
and 16.
Indeed, the overlaps of the coefficients corresponding to M ′ = 2 and M ′ = −2 exhibit
a different field dependence. For instance, the mixing of the 3, 3,−2 and 3, 2,−2 states, which
increases with field strength, see Fig. 16 (b), results in an increase of the 1, 0, 0 → 1, 2,−2 cross
section, presented in Fig. 12 (a). For M = 2, there is little mixing of the 3, 4, 2 and 3, 3, 2 states
with the 3, 2, 2 state, see Fig. 15 (b), (c), which results in a decreasing 1, 0, 0 → 1, 2, 2 cross section.
Once the coefficients’ overlap increases at high fields, the corresponding cross section goes up again.
We note that both field-free and field-dressed transitions to the states with M ′ = ±4 are
negligibly small, since their cross sections are dominated by other than the (dominant) Ξ20 moment.
The difference between the parallel and perpendicular geometries is due to the real d-
matrices, appearing in Eq. (44). For instance, an inspection of equations (43) and (44) reveals
that the integral cross sections for the 1, 0 → 1, 1 transition will be larger for H ⊥ k due to the
coefficients dκ−ρ,0(π2 ). A similar argument holds in the case of scattering in electrostatic [9] and
radiative [10] fields.
3. Frontal-versus-lateral steric asymmetry
Fig. 17 shows the steric asymmetry for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions as a
function of the magnetic field. We see that the asymmetry is most pronounced for the N = 1, J =
0 → N = 1, J = 2 and N = 1, J = 0 → N = 3, J = 4 channels. This has its origin in the field
dependence of the integral cross sections, see Figs. 12 and 13.
V. SCATTERING OF 2Π MOLECULES BY CLOSED-SHELL ATOMS IN MAGNETIC
FIELDS
A. The 2Π molecule in magnetic field
In this Section, we consider a Hund’s case molecule, equivalent to a linear symmetric top. A
good example of such a molecules is the OH radical in its electronic ground state, X2ΠΩ, whose elec-
tronic spin and orbital angular momenta are strongly coupled to the molecular axis. Each rotational
state within the 2ΠΩ ground state is equivalent to a symmetric-top state |J,Ω,M〉 with projections
Page 15
15
Ω and M of the total angular momentum J on the body- and space-fixed axes, respectively. Due
to a coupling of the Π state with a nearby Σ state [37], the levels with the same Ω are split into
nearly-degenerate doublets whose members have opposite parities. The Ω doubling of the 2Π 3
2
state
of OH increases as J3, whereas that of the 2Π 1
2
state increases linearly with J [25]. In our study,
we used the values of the Ω doubling listed in Table 8.24 of ref. [37].
The definite-parity rotational states of a Hund’s case (a) molecule can be written as
|J,M,Ω, ǫ〉 = 1√
2
[
|J,M,Ω〉+ ǫ|J,M,−Ω〉
]
(46)
where the symmetry index ǫ distinguishes between the members of a given Ω doublet. Here and
below we use the definition Ω ≡ |Ω|. The symmetry index takes the value of +1 or −1 for e or
f levels, respectively. The parity of wave function (46) is equal to ǫ(−1)J− 12 [35]. The rotational
energy level structure of the OH radical in its X2ΠΩ state is reviewed in Sec. 2.1.4 of Ref. [38].
When subject to a magnetic field, a Hund’s case (a) molecule acquires a Zeeman potential
Vm = JZωmB (47)
with JZ the Z component of the total angular momentum (apart from nuclear spin), J, and
ωm ≡ (gLΛ + gSΣ)µBH /B (48)
Here Λ and Σ are projections of the orbital, L, and spin, S, electronic angular momenta on the
molecular axis, gL = 1 and gS ≃ 2.0023 are the electronic orbital and spin gyromagnetic ratios, µB
is the Bohr magneton, H is the magnetic field strength, and B is the rotational constant, cf. Eqs.
(20) and (37). The matrix elements of Hamiltonian (47) in the definite-parity basis (46) are
〈J ′M ′ǫ′|Vm|JMǫ〉 = ωmB
(
1 + ǫǫ′(−1)J+J′+2Ω
2
)
(−1)J+J′+M−1/2
×
√
(2J + 1)(2J ′ + 1)
(
j 1 j′
−Ω 0 Ω′
)(
j′ 1 j
M 0 −M ′
)
(49)
where the last two factors are 3j-symbols [25], [26]. The matrix elements (49) are a generalization of
Eqs. (A1)–(A6), and were presented, e.g., in Ref. [39]. For an OH molecule in its ground 2Π3/2 state,
the parity factor, (1 + ǫǫ′(−1)J+J′+2Ω)/2, reduces to δǫǫ′ , which means that the Zeeman interaction
preserves parity. The Zeeman eigenstates are hybrids of the field-free states (46)
∣
∣
∣J˜ ,M,Ω, ǫ;ωm
〉
=
∑
J
aJ˜JM (ωm)|J,M,Ω, ǫ〉 (50)
where J˜ designates the angular momentum quantum number of the field-free state that adiabati-
cally correlates with a given state in the field. The coefficients aJ˜JM (ωm) can be obtained by the
diagonalization of the Hamiltonian (47) in the basis (46).
The dependence of the alignment cosine, 〈cos2 θ〉, on the field strength parameter ωm is
shown in Fig. 18 for the 3/2, f and 5/2, f states of the OH molecule. The matrix elements of the
〈cos2 θ〉 operator are listed in Appendix B.
B. The field-dependent scattering amplitude
We consider scattering from the initial J = 3/2, e state to some J ′, e/f state. As in the
previous Sections, we use Eq. (23) to transform the wavefunctions (46). Considering only the Ω-
Ω and M of the total angular momentum J on the body- and space-fixed axes, respectively. Due
to a coupling of the Π state with a nearby Σ state [37], the levels with the same Ω are split into
nearly-degenerate doublets whose members have opposite parities. The Ω doubling of the 2Π 3
2
state
of OH increases as J3, whereas that of the 2Π 1
2
state increases linearly with J [25]. In our study,
we used the values of the Ω doubling listed in Table 8.24 of ref. [37].
The definite-parity rotational states of a Hund’s case (a) molecule can be written as
|J,M,Ω, ǫ〉 = 1√
2
[
|J,M,Ω〉+ ǫ|J,M,−Ω〉
]
(46)
where the symmetry index ǫ distinguishes between the members of a given Ω doublet. Here and
below we use the definition Ω ≡ |Ω|. The symmetry index takes the value of +1 or −1 for e or
f levels, respectively. The parity of wave function (46) is equal to ǫ(−1)J− 12 [35]. The rotational
energy level structure of the OH radical in its X2ΠΩ state is reviewed in Sec. 2.1.4 of Ref. [38].
When subject to a magnetic field, a Hund’s case (a) molecule acquires a Zeeman potential
Vm = JZωmB (47)
with JZ the Z component of the total angular momentum (apart from nuclear spin), J, and
ωm ≡ (gLΛ + gSΣ)µBH /B (48)
Here Λ and Σ are projections of the orbital, L, and spin, S, electronic angular momenta on the
molecular axis, gL = 1 and gS ≃ 2.0023 are the electronic orbital and spin gyromagnetic ratios, µB
is the Bohr magneton, H is the magnetic field strength, and B is the rotational constant, cf. Eqs.
(20) and (37). The matrix elements of Hamiltonian (47) in the definite-parity basis (46) are
〈J ′M ′ǫ′|Vm|JMǫ〉 = ωmB
(
1 + ǫǫ′(−1)J+J′+2Ω
2
)
(−1)J+J′+M−1/2
×
√
(2J + 1)(2J ′ + 1)
(
j 1 j′
−Ω 0 Ω′
)(
j′ 1 j
M 0 −M ′
)
(49)
where the last two factors are 3j-symbols [25], [26]. The matrix elements (49) are a generalization of
Eqs. (A1)–(A6), and were presented, e.g., in Ref. [39]. For an OH molecule in its ground 2Π3/2 state,
the parity factor, (1 + ǫǫ′(−1)J+J′+2Ω)/2, reduces to δǫǫ′ , which means that the Zeeman interaction
preserves parity. The Zeeman eigenstates are hybrids of the field-free states (46)
∣
∣
∣J˜ ,M,Ω, ǫ;ωm
〉
=
∑
J
aJ˜JM (ωm)|J,M,Ω, ǫ〉 (50)
where J˜ designates the angular momentum quantum number of the field-free state that adiabati-
cally correlates with a given state in the field. The coefficients aJ˜JM (ωm) can be obtained by the
diagonalization of the Hamiltonian (47) in the basis (46).
The dependence of the alignment cosine, 〈cos2 θ〉, on the field strength parameter ωm is
shown in Fig. 18 for the 3/2, f and 5/2, f states of the OH molecule. The matrix elements of the
〈cos2 θ〉 operator are listed in Appendix B.
B. The field-dependent scattering amplitude
We consider scattering from the initial J = 3/2, e state to some J ′, e/f state. As in the
previous Sections, we use Eq. (23) to transform the wavefunctions (46). Considering only the Ω-
Page 16
16
conserving transitions (Ω′ = Ω), the initial and final states are:
|i〉 ≡
∣
∣
∣J˜ ,M,Ω, ǫ, ωm
〉
=
1√
2
∑
J
√
2J + 1
4π a
J˜
JM (ωm)
∑
ξ
DJξM (ϕε, θε, 0)
[
DJ∗ξΩ(ϕ, θ, 0) + ǫDJ∗ξ−Ω(ϕ, θ, 0)
]
(51)
〈f| ≡
〈
J˜ ′,M ′,Ω, ǫ′, ωm
∣
∣
∣ =
1√
2
∑
J′
√
2J ′ + 1
4π b
J˜′
J′M ′(ωm)
∑
ξ′
DJ′∗ξ′M ′(ϕε, θε, 0)
[
DJ′ξ′Ω′(ϕ, θ, 0) + ǫ′DJ
′
ξ′−Ω′(ϕ, θ, 0)
]
(52)
By substituting Eqs. (51) and (52) into Eq. (11), we obtain a closed expression for the scattering
amplitude:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(53)
Eq. (53) simplifies for parallel or perpendicular orientations of the magnetic field with respect to the
relative velocity vector.
(i) For H ‖ k we have
fωm,‖i→f (ϑ) =
ikR0
4π J|∆M|(kR0ϑ)
∑
κ 6=0
κ+∆M even
Ξκ0Fκ∆M
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(54)
(ii) and for H ⊥ k we obtain
fωm,⊥i→f (ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0dκ∗−ρ,∆M (π2 )FκρJ|ρ|(kR0ϑ)
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(55)
C. Results for He–OH(X2Π 3
2
, J = 32 , f → J
′, e/f) scattering in a magnetic field
According to Ref. [36], the ground state He–OH potential energy surface has a global min-
imum of −30.02 cm−1, which could be considered negligible with respect to a collision energy on
the order of 100 cm−1, as for the He – CaH and He – O2 systems treated above. However, the
OH molecule has a large rotational constant, B = 18.5348 cm−1, and so the Massey parameter (2)
becomes significantly smaller than unity only at higher energies. Therefore, in order to ensure
the validity of the sudden approximation, we had to work with a collision energy of 1000 cm−1
(k = 13.86 A˚−1; Massey parameter ξ ≈ 0.5). The corresponding equipotential line of the He – OH
conserving transitions (Ω′ = Ω), the initial and final states are:
|i〉 ≡
∣
∣
∣J˜ ,M,Ω, ǫ, ωm
〉
=
1√
2
∑
J
√
2J + 1
4π a
J˜
JM (ωm)
∑
ξ
DJξM (ϕε, θε, 0)
[
DJ∗ξΩ(ϕ, θ, 0) + ǫDJ∗ξ−Ω(ϕ, θ, 0)
]
(51)
〈f| ≡
〈
J˜ ′,M ′,Ω, ǫ′, ωm
∣
∣
∣ =
1√
2
∑
J′
√
2J ′ + 1
4π b
J˜′
J′M ′(ωm)
∑
ξ′
DJ′∗ξ′M ′(ϕε, θε, 0)
[
DJ′ξ′Ω′(ϕ, θ, 0) + ǫ′DJ
′
ξ′−Ω′(ϕ, θ, 0)
]
(52)
By substituting Eqs. (51) and (52) into Eq. (11), we obtain a closed expression for the scattering
amplitude:
fωmi→f(ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0Dκ∗−ρ,∆M (ϕε, θε, 0)FκρJ|ρ|(kR0ϑ)
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(53)
Eq. (53) simplifies for parallel or perpendicular orientations of the magnetic field with respect to the
relative velocity vector.
(i) For H ‖ k we have
fωm,‖i→f (ϑ) =
ikR0
4π J|∆M|(kR0ϑ)
∑
κ 6=0
κ+∆M even
Ξκ0Fκ∆M
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(54)
(ii) and for H ⊥ k we obtain
fωm,⊥i→f (ϑ) =
ikR0
4π
∑
κρ
κ 6=0
κ+ρ even
Ξκ0dκ∗−ρ,∆M (π2 )FκρJ|ρ|(kR0ϑ)
×
∑
JJ′
√
2J + 1
2J ′ + 1a
J˜
JM (ωm)bJ˜
′
J′M ′(ωm)C(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(55)
C. Results for He–OH(X2Π 3
2
, J = 32 , f → J
′, e/f) scattering in a magnetic field
According to Ref. [36], the ground state He–OH potential energy surface has a global min-
imum of −30.02 cm−1, which could be considered negligible with respect to a collision energy on
the order of 100 cm−1, as for the He – CaH and He – O2 systems treated above. However, the
OH molecule has a large rotational constant, B = 18.5348 cm−1, and so the Massey parameter (2)
becomes significantly smaller than unity only at higher energies. Therefore, in order to ensure
the validity of the sudden approximation, we had to work with a collision energy of 1000 cm−1
(k = 13.86 A˚−1; Massey parameter ξ ≈ 0.5). The corresponding equipotential line of the He – OH
Page 17
17
(2Π) potential energy surface is shown in Fig. 3, and the Legendre moments, Ξκ0, obtained by fitting
the surface, are listed in Table II.
Because of the negative spin-orbit constant, A = −139.051 cm−1 [37], the Ω doublet of
the OH(X2ΠΩ) molecule is inverted, with the paramagnetic 2Π3/2 state as its ground state. Since
|A| ≫ |B|, we can see why the OH molecule can be well described by the Hund’s case (a) coupling
scheme.
In what follows, we consider OH (2Π) radicals prepared in the v = 0,Ω = 32 , J = 32 , f state
by hexapole state selection, like, e.g., in ref. [41]. The molecules enter a magnetic field region where
they collide with 4He atoms. The scattered molecules are state-sensitively detected in a field-free
region.
1. Differential cross sections
We note that due to a large rotational constant, the Zeeman effect in the case of the OH
radical is very weak, and so are the field-induced changes of the scattering. The differential cross
sections for the He – OH collisions, as obtained from Eqs. (28) and (29), are shown in Figs. 19
and 20, together with the elastic scattering cross section obtained from Eq. (6). The differential
cross sections are presented for the field-free case, ωm = 0, as well as for ωm = 5, which for the OH
radical corresponds to an extreme field strength of H =99.2 T. First, let us consider the field-free
scattering amplitude
fw=0i→f (ϑ) =
√
2J + 1
2J ′ + 1J|∆M|(kR0ϑ)
×
∑
κ 6=0
κ+∆M even
Ξκ0Fκ,∆MC(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(56)
We see that the angular dependence of the amplitude is given by the Bessel function J|∆M|.
The term in the square brackets and the Fκ,∆M coefficient provide a selection rule: ∆M +∆J must
be even for parity conserving (f → f) transitions, and odd for parity breaking (f → e) transitions.
The effect of this selection rule can be seen in Figs. 19 and 20. The elastic cross section, Figs. 19(a)
and 20(a), is proportional to an odd Bessel function, cf. Eq. (6). Therefore, it is in phase with
the 3/2, f → 5/2, f and 3/2, f → 7/2, e cross sections, but out of phase with 3/2, f → 5/2, e and
3/2, f → 7/2, f cross sections.
For a magnetic field parallel to the relative velocity, H ‖ k, the angular dependence is given
explicitly by J|∆M|(kR0ϑ), and is seen to be independent of the field, cf. Eq. (54). Therefore, as
Fig. 19 shows, no field-induced phase shift of the differential cross sections takes place. For H ⊥ k,
a mix of Bessel functions, J|ρ|(kR0ϑ), contribute to the sum. However, since the Zeeman effect is
so weak for the OH molecule, it is the aJ˜JM (ωm), bJ˜
′
J′M ′(ωm) hybridization coefficients with J = J˜
which provide the main contribution to the sum, even at ωm ≈ 5. As a result, no contributions from
higher Bessel functions are drawn in, and so no significant field-induced phase shift is observed for
the perpendicular case either.
2. Integral cross sections
The integral cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions are presented
in Fig. 21 for J ′ = 5/2, 7/2 and the two orientations of the magnetic field with respect to the
(2Π) potential energy surface is shown in Fig. 3, and the Legendre moments, Ξκ0, obtained by fitting
the surface, are listed in Table II.
Because of the negative spin-orbit constant, A = −139.051 cm−1 [37], the Ω doublet of
the OH(X2ΠΩ) molecule is inverted, with the paramagnetic 2Π3/2 state as its ground state. Since
|A| ≫ |B|, we can see why the OH molecule can be well described by the Hund’s case (a) coupling
scheme.
In what follows, we consider OH (2Π) radicals prepared in the v = 0,Ω = 32 , J = 32 , f state
by hexapole state selection, like, e.g., in ref. [41]. The molecules enter a magnetic field region where
they collide with 4He atoms. The scattered molecules are state-sensitively detected in a field-free
region.
1. Differential cross sections
We note that due to a large rotational constant, the Zeeman effect in the case of the OH
radical is very weak, and so are the field-induced changes of the scattering. The differential cross
sections for the He – OH collisions, as obtained from Eqs. (28) and (29), are shown in Figs. 19
and 20, together with the elastic scattering cross section obtained from Eq. (6). The differential
cross sections are presented for the field-free case, ωm = 0, as well as for ωm = 5, which for the OH
radical corresponds to an extreme field strength of H =99.2 T. First, let us consider the field-free
scattering amplitude
fw=0i→f (ϑ) =
√
2J + 1
2J ′ + 1J|∆M|(kR0ϑ)
×
∑
κ 6=0
κ+∆M even
Ξκ0Fκ,∆MC(JκJ ′;M∆MM ′)C(JκJ ′; Ω0Ω)
[
1 + ǫǫ′(−1)κ+∆J
]
(56)
We see that the angular dependence of the amplitude is given by the Bessel function J|∆M|.
The term in the square brackets and the Fκ,∆M coefficient provide a selection rule: ∆M +∆J must
be even for parity conserving (f → f) transitions, and odd for parity breaking (f → e) transitions.
The effect of this selection rule can be seen in Figs. 19 and 20. The elastic cross section, Figs. 19(a)
and 20(a), is proportional to an odd Bessel function, cf. Eq. (6). Therefore, it is in phase with
the 3/2, f → 5/2, f and 3/2, f → 7/2, e cross sections, but out of phase with 3/2, f → 5/2, e and
3/2, f → 7/2, f cross sections.
For a magnetic field parallel to the relative velocity, H ‖ k, the angular dependence is given
explicitly by J|∆M|(kR0ϑ), and is seen to be independent of the field, cf. Eq. (54). Therefore, as
Fig. 19 shows, no field-induced phase shift of the differential cross sections takes place. For H ⊥ k,
a mix of Bessel functions, J|ρ|(kR0ϑ), contribute to the sum. However, since the Zeeman effect is
so weak for the OH molecule, it is the aJ˜JM (ωm), bJ˜
′
J′M ′(ωm) hybridization coefficients with J = J˜
which provide the main contribution to the sum, even at ωm ≈ 5. As a result, no contributions from
higher Bessel functions are drawn in, and so no significant field-induced phase shift is observed for
the perpendicular case either.
2. Integral cross sections
The integral cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions are presented
in Fig. 21 for J ′ = 5/2, 7/2 and the two orientations of the magnetic field with respect to the
Page 18
18
relative velocity. We see that no dramatic changes of the cross sections with the field strength take
place. Again, the dependence of the M -averaged integral cross sections (here 3/2, f → J ′, e/f)
on the magnetic field can be traced to the field-dependences of the partial M - and M ′-dependent
contributions. Since for OH (J = 3/2, f), the initial state comprises fourM values and the final state
six or eightM ′ values, a discussion of how the M - andM ′-averaged cross sections come about would
be rather involved. Therefore, we resort to considering an example, namely how the 3/2, f → 5/2, f
cross section arises from the 3/2, f,M → 5/2, f,M ′ components, shown in Fig. 22 for a magnetic
field parallel to the relative velocity. From Table II, we see that the He – OH potential is dominated
by odd Ξκ0 terms – in contrast to the He – CaH and He – O2 potentials. Therefore, odd ∆M
values will yield the main contribution to the scattering amplitude (54), because of the selection
rule which dictates that κ + ∆M be even. This effect can be clearly discerned in Fig. 22. In the
field-free case, even ∆M transitions have very small amplitudes, cf. Eq. (56). When the field is on,
the corresponding cross sections increase with ωm due to the increasing overlap of the aJ˜JM (ωm) and
bJ˜′J′M ′(ωm) coefficients. This is a situation analogous to the one described in detail in Sec. IVC2 for
the O2 – He system (see also Figs. 15, 16).
3. Frontal-versus-lateral steric asymmetry
The steric asymmetry for the He – OH (J = 3/2, f → J ′, e/f) collisions, calculated by means
of Eq. (33), is presented in Fig. 23. The most pronounced asymmetry is observed for the 5/2, e and
7/2, f channels, while the asymmetry for the J = 3/2, f → 5/2, f and J = 3/2, f → 7/2, e channels
is almost flat, especially at the feasible magnetic field strengths of up to 20 T. The difference between
the scattering for parallel and perpendicular orientations of the magnetic field with respect to the
initial velocity is contained in the dκ∗−ρ,∆M (π2 ) matrices, appearing in Eq. (55), and the observed
trends can be gleaned from Eqs. (54) and (55).
VI. CONCLUSIONS
We extended the Fraunhofer theory of matter wave scattering to tackle rotationally inelastic
collisions of paramagnetic, open shell molecules with closed-shell atoms in magnetic fields. The
description is inherently quantum and, therefore, capable of accounting for interference and other
non-classical effects. The effect of the magnetic field enters the model via the directional properties
of the molecular states, which exhibit alignment of the molecular axis induced by the magnetic field.
We applied the model to the He–CaH(X2Σ, J = 1/2 → J ′) and He–O2(X3Σ, N = 0, J = 1 → N ′, J ′)
scattering at a collision energy of 200 cm−1, as well as to the He–OH(X2Π, J = 3/2, f → J ′, e/f)
collisions at an energy of 1000 cm−1.
In this Section, we mull over the results for the three collision systems and point out what
they have in common and where they differ.
The CaH molecule, studied in Sec. III, has a non-magnetic N = 0, J = 1 ground state
(taken as the initial state) and, therefore, all the field-induced changes of the He – CaH scattering
are due to the Zeeman effect of the final state. The magnetic eigenproperties of a 2Σ molecule can
be obtained in closed form, by diagonalizing a 2× 2 Hamiltonian matrix. When the magnetic field
strength increases, the hybridization coefficients quickly approach an asymptotic value, see Table I,
as do the alignment cosine, Fig. 2, and the integral cross sections, Figs. 6 and 7. The changes of
the cross sections between zero-field and the high-field limit are quite weak, which results in a small
relative velocity. We see that no dramatic changes of the cross sections with the field strength take
place. Again, the dependence of the M -averaged integral cross sections (here 3/2, f → J ′, e/f)
on the magnetic field can be traced to the field-dependences of the partial M - and M ′-dependent
contributions. Since for OH (J = 3/2, f), the initial state comprises fourM values and the final state
six or eightM ′ values, a discussion of how the M - andM ′-averaged cross sections come about would
be rather involved. Therefore, we resort to considering an example, namely how the 3/2, f → 5/2, f
cross section arises from the 3/2, f,M → 5/2, f,M ′ components, shown in Fig. 22 for a magnetic
field parallel to the relative velocity. From Table II, we see that the He – OH potential is dominated
by odd Ξκ0 terms – in contrast to the He – CaH and He – O2 potentials. Therefore, odd ∆M
values will yield the main contribution to the scattering amplitude (54), because of the selection
rule which dictates that κ + ∆M be even. This effect can be clearly discerned in Fig. 22. In the
field-free case, even ∆M transitions have very small amplitudes, cf. Eq. (56). When the field is on,
the corresponding cross sections increase with ωm due to the increasing overlap of the aJ˜JM (ωm) and
bJ˜′J′M ′(ωm) coefficients. This is a situation analogous to the one described in detail in Sec. IVC2 for
the O2 – He system (see also Figs. 15, 16).
3. Frontal-versus-lateral steric asymmetry
The steric asymmetry for the He – OH (J = 3/2, f → J ′, e/f) collisions, calculated by means
of Eq. (33), is presented in Fig. 23. The most pronounced asymmetry is observed for the 5/2, e and
7/2, f channels, while the asymmetry for the J = 3/2, f → 5/2, f and J = 3/2, f → 7/2, e channels
is almost flat, especially at the feasible magnetic field strengths of up to 20 T. The difference between
the scattering for parallel and perpendicular orientations of the magnetic field with respect to the
initial velocity is contained in the dκ∗−ρ,∆M (π2 ) matrices, appearing in Eq. (55), and the observed
trends can be gleaned from Eqs. (54) and (55).
VI. CONCLUSIONS
We extended the Fraunhofer theory of matter wave scattering to tackle rotationally inelastic
collisions of paramagnetic, open shell molecules with closed-shell atoms in magnetic fields. The
description is inherently quantum and, therefore, capable of accounting for interference and other
non-classical effects. The effect of the magnetic field enters the model via the directional properties
of the molecular states, which exhibit alignment of the molecular axis induced by the magnetic field.
We applied the model to the He–CaH(X2Σ, J = 1/2 → J ′) and He–O2(X3Σ, N = 0, J = 1 → N ′, J ′)
scattering at a collision energy of 200 cm−1, as well as to the He–OH(X2Π, J = 3/2, f → J ′, e/f)
collisions at an energy of 1000 cm−1.
In this Section, we mull over the results for the three collision systems and point out what
they have in common and where they differ.
The CaH molecule, studied in Sec. III, has a non-magnetic N = 0, J = 1 ground state
(taken as the initial state) and, therefore, all the field-induced changes of the He – CaH scattering
are due to the Zeeman effect of the final state. The magnetic eigenproperties of a 2Σ molecule can
be obtained in closed form, by diagonalizing a 2× 2 Hamiltonian matrix. When the magnetic field
strength increases, the hybridization coefficients quickly approach an asymptotic value, see Table I,
as do the alignment cosine, Fig. 2, and the integral cross sections, Figs. 6 and 7. The changes of
the cross sections between zero-field and the high-field limit are quite weak, which results in a small
Page 19
19
frontal-versus-lateral steric asymmetry of the He – CaH scattering, Fig. 8.
In contrast to the 2Σ case, the Hamiltonian matrices of the 3Σ and 2Π molecules in a
magnetic field are in principle infinite. In practice, the Zeeman interaction couples a range of
rotational states which is limited by the strength of the interaction to less than ten for ωm ≤ 5.
The He – O2 collision system exhibits a dramatic feature: in the absence of a magnetic
field, the scattering vanishes for channels leading to the F2 manifold, i.e., final states with J ′ = N ′.
However, in the presence of the field, such transitions become allowed, and, although weak, should be
observable. Also, some of the He – O2 integral cross sections are non-monotonous – unlike the cross
sections of the He – CaH and He – OH systems. They exhibit maxima or minima, which depend
characteristically on the sign of M ′, see Figs. 12 and 13. This contributes, for some channels, to
the strong dependence of the He – O2 cross sections on the orientation of the magnetic field with
respect to the relative velocity, as quantified by the frontal-versus-lateral steric asymmetry, Fig. 17.
The OH molecule has a large rotational constant and is, therefore, only weakly aligned by
the magnetic field, see Fig. 18. As a result, the field-induced changes of the scattering cross sections
are tiny, Figs. 19 – 21, and so is the variation with field of the steric asymmetry, Fig. 23. Unlike the
He – CaH and He – O2 systems, the equipotential line on the He – OH potential energy surface is
dominated by odd Legendre moments, see Table II. This gives rise to scattering features which are
qualitatively different from those of the other systems. For instance, as described in Sec. VC2, it is
the odd ∆M transitions that dominate the He – OH (J = 3/2, f → J ′, e/f) integral cross sections
in a magnetic field parallel to the relative velocity vector. In the other two systems, it is the even
∆M transitions.
In all three systems studied, the field-induced changes of the differential cross sections –
such as angular shifts of their oscillations – were puny. The only exception was found for the
N = 1, J = 0 → N ′ = 1, J ′ = 2 and N = 1, J = 0 → N ′ = 3, J ′ = 4 transitions in the He – O2
system. These occur for scattering in a magnetic field perpendicular to the relative velocity vector,
and are due to a field-induced mixing-in of higher Bessel-functions.
The strength of the analytic model lies in its ability to separate dynamical and geometrical
effects and to qualitatively explain the resulting scattering features. These include the angular
oscillations in the state-to-state differential cross sections or the rotational-state dependent variation
of the integral cross sections as functions of the magnetic field. We hope that the model will inspire
new collisional experiments that make use either of crossed molecular beams or of a combination of
a magnetic trap with a hot beam of atoms.
Acknowledgments
Our special thanks are due to Gerard Meijer for discussions and support, and to Elena
Dashevskaya and Evgueni Nikitin for their most helpful comments. We greatly enjoyed discussions
with Bas van de Meerakker, Ludwig Scharfenberg, Joop Gilijamse, and Steven Hoekstra.
APPENDIX A: MATRIX ELEMENTS OF THE JZ OPERATOR
In general, the Zeeman operator is proportional to the projection, JZ , of the total electronic
angular momentum, J, on the space-fixed field axis, Z, see e.g. eq. (47). In this Appendix we present
the matrix elements of the JZ operator, employed in this work. For Σ electronic states, JZ reduces
to SZ .
frontal-versus-lateral steric asymmetry of the He – CaH scattering, Fig. 8.
In contrast to the 2Σ case, the Hamiltonian matrices of the 3Σ and 2Π molecules in a
magnetic field are in principle infinite. In practice, the Zeeman interaction couples a range of
rotational states which is limited by the strength of the interaction to less than ten for ωm ≤ 5.
The He – O2 collision system exhibits a dramatic feature: in the absence of a magnetic
field, the scattering vanishes for channels leading to the F2 manifold, i.e., final states with J ′ = N ′.
However, in the presence of the field, such transitions become allowed, and, although weak, should be
observable. Also, some of the He – O2 integral cross sections are non-monotonous – unlike the cross
sections of the He – CaH and He – OH systems. They exhibit maxima or minima, which depend
characteristically on the sign of M ′, see Figs. 12 and 13. This contributes, for some channels, to
the strong dependence of the He – O2 cross sections on the orientation of the magnetic field with
respect to the relative velocity, as quantified by the frontal-versus-lateral steric asymmetry, Fig. 17.
The OH molecule has a large rotational constant and is, therefore, only weakly aligned by
the magnetic field, see Fig. 18. As a result, the field-induced changes of the scattering cross sections
are tiny, Figs. 19 – 21, and so is the variation with field of the steric asymmetry, Fig. 23. Unlike the
He – CaH and He – O2 systems, the equipotential line on the He – OH potential energy surface is
dominated by odd Legendre moments, see Table II. This gives rise to scattering features which are
qualitatively different from those of the other systems. For instance, as described in Sec. VC2, it is
the odd ∆M transitions that dominate the He – OH (J = 3/2, f → J ′, e/f) integral cross sections
in a magnetic field parallel to the relative velocity vector. In the other two systems, it is the even
∆M transitions.
In all three systems studied, the field-induced changes of the differential cross sections –
such as angular shifts of their oscillations – were puny. The only exception was found for the
N = 1, J = 0 → N ′ = 1, J ′ = 2 and N = 1, J = 0 → N ′ = 3, J ′ = 4 transitions in the He – O2
system. These occur for scattering in a magnetic field perpendicular to the relative velocity vector,
and are due to a field-induced mixing-in of higher Bessel-functions.
The strength of the analytic model lies in its ability to separate dynamical and geometrical
effects and to qualitatively explain the resulting scattering features. These include the angular
oscillations in the state-to-state differential cross sections or the rotational-state dependent variation
of the integral cross sections as functions of the magnetic field. We hope that the model will inspire
new collisional experiments that make use either of crossed molecular beams or of a combination of
a magnetic trap with a hot beam of atoms.
Acknowledgments
Our special thanks are due to Gerard Meijer for discussions and support, and to Elena
Dashevskaya and Evgueni Nikitin for their most helpful comments. We greatly enjoyed discussions
with Bas van de Meerakker, Ludwig Scharfenberg, Joop Gilijamse, and Steven Hoekstra.
APPENDIX A: MATRIX ELEMENTS OF THE JZ OPERATOR
In general, the Zeeman operator is proportional to the projection, JZ , of the total electronic
angular momentum, J, on the space-fixed field axis, Z, see e.g. eq. (47). In this Appendix we present
the matrix elements of the JZ operator, employed in this work. For Σ electronic states, JZ reduces
to SZ .
Page 20
20
We transform the angular momentum projection operator from the body-fixed to the space-
fixed coordinates using the direction cosines operator, Φ:
JZ = 12
(
Φ+ZJ− +Φ−ZJ+
)
+ΦzZJz (A1)
The matrix elements of the body-fixed spin operator in the Hund’s case (a) basis, |J,Ω,M〉, are
given by the standard relations [42]:
〈J,Ω,M |J± |J,Ω∓ 1,M〉 =
√
(J ± Ω)(J ∓ Ω+ 1) (A2)
〈J,Ω,M |Jz |J,Ω,M〉 = Ω (A3)
The matrix elements of the direction cosine operator can be obtained from Table 6 of [31].
Some of them are also presented in Table 1.1 (p.19) of Ref. [24]. We list here all non-vanishing
matrix elements for M ′ =M :
〈J ′,Ω,M |ΦzZ |J,Ω,M〉 =
ΩM
J(J+1) J ′ = J
√
(J+Ω+1)(J−Ω+1)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
√
(J+Ω)(J−Ω)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A4)
〈J ′,Ω− 1,M |Φ+Z |J,Ω,M〉 =
M
√
(J+Ω)(J−Ω+1)
J(J+1) J ′ = J
√
(J−Ω+1)(J−Ω+2)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
−
√
(J+Ω)(J+Ω−1)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A5)
〈J ′,Ω + 1,M |Φ−Z |J,Ω,M〉 =
M
√
(J−Ω)(J+Ω+1)
J(J+1) J ′ = J
−
√
(J+Ω+1)(J+Ω+2)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
√
(J−Ω)(J−Ω−1)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A6)
APPENDIX B: MATRIX ELEMENTS OF THE (ΦzZ)2 OPERATOR
The matrix elements of the alignment cosine can be reduced to the matrix elements of the
ΦzZ operator by means of the full basis set:
〈cos2 θ〉 = 〈J ′,Ω,M |(ΦzZ)2|J,Ω,M〉 =
∑
J′′
〈J ′,Ω,M |ΦzZ |J ′′,Ω,M〉〈J ′′,Ω,M |ΦzZ|J,Ω,M〉 (B1)
By taking into account that the matrix elements (A4) are nonzero only for ∆J = 0,±1, we obtain
the matrix elements of the direction cosine operator:
〈J,Ω,M |(ΦzZ)2|J,Ω,M〉 = |〈J − 1,Ω,M |ΦzZ|J,Ω,M〉|
2 + |〈J,Ω,M |ΦzZ |J,Ω,M〉|
2
+ |〈J + 1,Ω,M |ΦzZ|J,Ω,M〉|
2 (B2)
We transform the angular momentum projection operator from the body-fixed to the space-
fixed coordinates using the direction cosines operator, Φ:
JZ = 12
(
Φ+ZJ− +Φ−ZJ+
)
+ΦzZJz (A1)
The matrix elements of the body-fixed spin operator in the Hund’s case (a) basis, |J,Ω,M〉, are
given by the standard relations [42]:
〈J,Ω,M |J± |J,Ω∓ 1,M〉 =
√
(J ± Ω)(J ∓ Ω+ 1) (A2)
〈J,Ω,M |Jz |J,Ω,M〉 = Ω (A3)
The matrix elements of the direction cosine operator can be obtained from Table 6 of [31].
Some of them are also presented in Table 1.1 (p.19) of Ref. [24]. We list here all non-vanishing
matrix elements for M ′ =M :
〈J ′,Ω,M |ΦzZ |J,Ω,M〉 =
ΩM
J(J+1) J ′ = J
√
(J+Ω+1)(J−Ω+1)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
√
(J+Ω)(J−Ω)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A4)
〈J ′,Ω− 1,M |Φ+Z |J,Ω,M〉 =
M
√
(J+Ω)(J−Ω+1)
J(J+1) J ′ = J
√
(J−Ω+1)(J−Ω+2)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
−
√
(J+Ω)(J+Ω−1)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A5)
〈J ′,Ω + 1,M |Φ−Z |J,Ω,M〉 =
M
√
(J−Ω)(J+Ω+1)
J(J+1) J ′ = J
−
√
(J+Ω+1)(J+Ω+2)(J+M+1)(J−M+1)
(J+1)
√
(2J+1)(2J+3)
J ′ = J + 1
√
(J−Ω)(J−Ω−1)(J+M)(J−M)
J
√
(2J+1)(2J−1)
J ′ = J − 1
(A6)
APPENDIX B: MATRIX ELEMENTS OF THE (ΦzZ)2 OPERATOR
The matrix elements of the alignment cosine can be reduced to the matrix elements of the
ΦzZ operator by means of the full basis set:
〈cos2 θ〉 = 〈J ′,Ω,M |(ΦzZ)2|J,Ω,M〉 =
∑
J′′
〈J ′,Ω,M |ΦzZ |J ′′,Ω,M〉〈J ′′,Ω,M |ΦzZ|J,Ω,M〉 (B1)
By taking into account that the matrix elements (A4) are nonzero only for ∆J = 0,±1, we obtain
the matrix elements of the direction cosine operator:
〈J,Ω,M |(ΦzZ)2|J,Ω,M〉 = |〈J − 1,Ω,M |ΦzZ|J,Ω,M〉|
2 + |〈J,Ω,M |ΦzZ |J,Ω,M〉|
2
+ |〈J + 1,Ω,M |ΦzZ|J,Ω,M〉|
2 (B2)
Page 21
21
〈J−1,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J−1,Ω,M |ΦzZ|J,Ω,M〉
{
〈J,Ω,M |ΦzZ |J,Ω,M〉+〈J−1,Ω,M |ΦzZ|J−1,Ω,M〉
}
(B3)
〈J+1,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J+1,Ω,M |ΦzZ|J,Ω,M〉
{
〈J,Ω,M |ΦzZ |J,Ω,M〉+〈J+1,Ω,M |ΦzZ|J+1,Ω,M〉
}
(B4)
〈J − 2,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J − 2,Ω,M |ΦzZ |J − 1,Ω,M〉〈J − 1,Ω,M |ΦzZ|J,Ω,M〉 (B5)
〈J + 2,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J + 2,Ω,M |ΦzZ |J + 1,Ω,M〉〈J + 1,Ω,M |ΦzZ|J,Ω,M〉 (B6)
APPENDIX C: THE ALIGNMENT COSINE OF THE 2Σ MOLECULE IN A MAGNETIC
FIELD
Within the Hund’s (b) basis functions, eq. (16), the expectation value of the alignment
cosine takes the form:
〈cos2 θ〉 = a2(ωm)
〈
N − 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N − 12 ,Ω,M
〉
+ b2(ωm)
〈
N + 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N + 12 ,Ω,M
〉
+ 2a(ωm)b(ωm)
〈
N − 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N + 12 ,Ω,M
〉
,
(C1)
where the matrix elements of the cos2 θ operator in the Hund’s case (a) basis can be obtained
from (B2) and (B4):
〈
J,Ω,M
∣
∣ cos2 θ
∣
∣J,Ω,M
〉
=
1
3
+
2
3
[
J(J + 1)− 3M2
] [
J(J + 1)− 3Ω2
]
J(J + 1)(2J − 1)(2J + 3) (C2)
〈
J,Ω,M
∣
∣ cos2 θ
∣
∣J + 1,Ω,M
〉
= 2ΩM
√
[(J + 1)2 −M2] [(J + 1)2 − Ω2]
J(J + 1)(J + 2)
√
(2J + 1)(2J + 3)
(C3)
The coefficients a(ωm) and b(ωm) are given by the solution of the Zeeman problem, Eqs. (21)–(22).
[1] Special issue on cold molecules, Eur. Phys. J. D 31, 149 (2004)
[2] Special issue on coherent control, J. Phys. B 41, 074001 (2008)
[3] Special issue on stereodynamics, Eur. Phys. J. D 38, 3 (2006)
[4] R. V. Krems, A. Dalgarno, J. Chem. Phys 120, 2296 (2004).
[5] H. R. Sadeghpour, J. L. Bohn, M. J. Cavagnero, B. D. Esry, I. I. Fabrikant, J. H. Macek, and
A. R. P. Rau, J. Phys. B. 33, R93 (2000).
[6] S. I. Drozdov, Soviet. Phys. JETP. 1, 591, 588 (1955)
[7] J. S. Blair, in Nuclear Structure Physics, edited by P. D. Kunz et al. (The University of Colorado,
Boulder, 1966), Vol. VII C, 343-444.
[8] M. Faubel, J. Chem. Phys. 81, 5559 (1984).
[9] M. Lemeshko, B. Friedrich, J. Chem. Phys., 129, 024301 (2008).
[10] M. Lemeshko, B. Friedrich, Int. J. Mass. Spec., in press; see arXiv:0804.4845 (2008).
〈J−1,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J−1,Ω,M |ΦzZ|J,Ω,M〉
{
〈J,Ω,M |ΦzZ |J,Ω,M〉+〈J−1,Ω,M |ΦzZ|J−1,Ω,M〉
}
(B3)
〈J+1,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J+1,Ω,M |ΦzZ|J,Ω,M〉
{
〈J,Ω,M |ΦzZ |J,Ω,M〉+〈J+1,Ω,M |ΦzZ|J+1,Ω,M〉
}
(B4)
〈J − 2,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J − 2,Ω,M |ΦzZ |J − 1,Ω,M〉〈J − 1,Ω,M |ΦzZ|J,Ω,M〉 (B5)
〈J + 2,Ω,M |(ΦzZ)2|J,Ω,M〉 = 〈J + 2,Ω,M |ΦzZ |J + 1,Ω,M〉〈J + 1,Ω,M |ΦzZ|J,Ω,M〉 (B6)
APPENDIX C: THE ALIGNMENT COSINE OF THE 2Σ MOLECULE IN A MAGNETIC
FIELD
Within the Hund’s (b) basis functions, eq. (16), the expectation value of the alignment
cosine takes the form:
〈cos2 θ〉 = a2(ωm)
〈
N − 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N − 12 ,Ω,M
〉
+ b2(ωm)
〈
N + 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N + 12 ,Ω,M
〉
+ 2a(ωm)b(ωm)
〈
N − 12 ,Ω,M
∣
∣ cos2 θ
∣
∣N + 12 ,Ω,M
〉
,
(C1)
where the matrix elements of the cos2 θ operator in the Hund’s case (a) basis can be obtained
from (B2) and (B4):
〈
J,Ω,M
∣
∣ cos2 θ
∣
∣J,Ω,M
〉
=
1
3
+
2
3
[
J(J + 1)− 3M2
] [
J(J + 1)− 3Ω2
]
J(J + 1)(2J − 1)(2J + 3) (C2)
〈
J,Ω,M
∣
∣ cos2 θ
∣
∣J + 1,Ω,M
〉
= 2ΩM
√
[(J + 1)2 −M2] [(J + 1)2 − Ω2]
J(J + 1)(J + 2)
√
(2J + 1)(2J + 3)
(C3)
The coefficients a(ωm) and b(ωm) are given by the solution of the Zeeman problem, Eqs. (21)–(22).
[1] Special issue on cold molecules, Eur. Phys. J. D 31, 149 (2004)
[2] Special issue on coherent control, J. Phys. B 41, 074001 (2008)
[3] Special issue on stereodynamics, Eur. Phys. J. D 38, 3 (2006)
[4] R. V. Krems, A. Dalgarno, J. Chem. Phys 120, 2296 (2004).
[5] H. R. Sadeghpour, J. L. Bohn, M. J. Cavagnero, B. D. Esry, I. I. Fabrikant, J. H. Macek, and
A. R. P. Rau, J. Phys. B. 33, R93 (2000).
[6] S. I. Drozdov, Soviet. Phys. JETP. 1, 591, 588 (1955)
[7] J. S. Blair, in Nuclear Structure Physics, edited by P. D. Kunz et al. (The University of Colorado,
Boulder, 1966), Vol. VII C, 343-444.
[8] M. Faubel, J. Chem. Phys. 81, 5559 (1984).
[9] M. Lemeshko, B. Friedrich, J. Chem. Phys., 129, 024301 (2008).
[10] M. Lemeshko, B. Friedrich, Int. J. Mass. Spec., in press; see arXiv:0804.4845 (2008).
Page 22
22
[11] A. I. Maergoiz, E. E. Nikitin, J. Troe, V. G. Ushakov, J. Chem. Phys. 105, 6263 (1996).
[12] E. E. Nikitin, Theory of Elementary Atomic and Molecular Processes in Gases, Clarendon, Oxford
(1974).
[13] B. Friedrich and D. Herschbach, Phys. Chem. Chem. Phys. 2, 419 (2000).
[14] A. Boca, B. Friedrich, J. Chem. Phys. 112, 3609 (2000).
[15] J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich, and J. M. Doyle, Nature(London) 395, 148
(1998).
[16] B. Friedrich, J. D. Weinstein, R. deCarvalho, and J. M. Doyle, J. Chem. Phys. 110, 2376 (1999).
[17] W.C. Cambell and J.M. Doyle, in Cold Molecules: Theory, Experiment, Applications, R. Krems, W.C.
Stwalley, and B. Friedrich (eds.), Taylor & Francis, London (2009).
[18] D. Beck, U. Ross, and W. Schepper, Z. Phys. A 293, 107-117 (1979).
[19] A. Ichimura, M. Nakamura, Phys. Rev. A 69, 022716 (2004);
M. Nakamura, A. Ichimura, Phys. Rev. A 71, 062701 (2005).
[20] A. J. Marks, J. Chem. Soc. Far. Trans. 90, 2857 (1994).
[21] S. Bosanac, Phys. Rev. A 26, 282 (1982).
[22] A. Gijsbertsen, H. Linnartz, C. A. Taatjes, and S. Stolte, J. Am. Chem. Soc 128, 8777 (2006);
C. A. Taatjes, A. Gijsbertsen, M. J. L. de Lange, and S. Stolte, J. Chem. Phys 111, 7631 (2007).
[23] M. Born and E. Wolf, Principles of optics, 7th ed., Cambridge University Press (1997).
[24] H. Lefebvre-Brion, R. W. Field, Perturbations in the spectra of diatomic molecules, Academic Press,
1986; H. Lefebvre-Brion, R. W. Field, The spectra and dynamics of diatomic molecules, Elsevier, 2004.
[25] R. N. Zare, Angular momentum: Understanding spatial aspects in chemistry and physics, Wiley, New
York (1988).
[26] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonski, Quantum theory of angular momentum, World
Scientific Publications, Singapore and Teaneck, N.J (1988)
[27] H. Martin, J. Mol. Spec. 108, 44 (1984).
[28] G. C. Groenenboom, N. Balakrishnan, J. Chem. Phys. 118, 7380 (2003).
[29] G. N. Watson, Theory of Bessel Functions, Cambridge University Press (1922).
[30] C. Amiot, J. Verges, Can. J. Phys. 59, 1391 (1981).
[31] J. T. Hougen, The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic
Molecules, NIST, 1970.
[32] G. Herzberg, Spectra of diatomic molecules, van Nostrand, Princeton, 1950.
[33] L. Tomuta, M. Mizushima, C. J. Howard, K. M. Evenson, Phys. Rev. A 12, 974 (1975).
[34] G. C. Groenenboom, I. M. Struniewicz, J. Chem. Phys. 113, 9562 (2000).
[35] 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).
[36] H.-S. Lee, A. B. McCoy, R. R. Toczy lowski, S. M. Cybulski, J. Chem. Phys. 113, 5736 (2000).
[37] J. Brown, A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press
(2003).
[38] K. Gubbels, Dynamics of Stark Acceleration/Deceleration: Molecules Riding Waves, Master’s thesis,
Radbud University Nijmegen (2006).
[39] C. Ticknor, J. L. Bohn, Phys. Rev. A 71, 022709 (2005).
[40] P. F. Bernath, Spectra of atoms and molecules, Oxford Univ. Press, New York (1995).
[41] M. C. van Beek, J. J. ter Meulen, M. H. Alexander, J. Chem. Phys. 113, 628 (2000).
[42] L. D. Landau, E. M. Lifshitz Quantum Mechanics, Non-Relativistic Theory, Oxford: Pergamon (1977).
[11] A. I. Maergoiz, E. E. Nikitin, J. Troe, V. G. Ushakov, J. Chem. Phys. 105, 6263 (1996).
[12] E. E. Nikitin, Theory of Elementary Atomic and Molecular Processes in Gases, Clarendon, Oxford
(1974).
[13] B. Friedrich and D. Herschbach, Phys. Chem. Chem. Phys. 2, 419 (2000).
[14] A. Boca, B. Friedrich, J. Chem. Phys. 112, 3609 (2000).
[15] J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich, and J. M. Doyle, Nature(London) 395, 148
(1998).
[16] B. Friedrich, J. D. Weinstein, R. deCarvalho, and J. M. Doyle, J. Chem. Phys. 110, 2376 (1999).
[17] W.C. Cambell and J.M. Doyle, in Cold Molecules: Theory, Experiment, Applications, R. Krems, W.C.
Stwalley, and B. Friedrich (eds.), Taylor & Francis, London (2009).
[18] D. Beck, U. Ross, and W. Schepper, Z. Phys. A 293, 107-117 (1979).
[19] A. Ichimura, M. Nakamura, Phys. Rev. A 69, 022716 (2004);
M. Nakamura, A. Ichimura, Phys. Rev. A 71, 062701 (2005).
[20] A. J. Marks, J. Chem. Soc. Far. Trans. 90, 2857 (1994).
[21] S. Bosanac, Phys. Rev. A 26, 282 (1982).
[22] A. Gijsbertsen, H. Linnartz, C. A. Taatjes, and S. Stolte, J. Am. Chem. Soc 128, 8777 (2006);
C. A. Taatjes, A. Gijsbertsen, M. J. L. de Lange, and S. Stolte, J. Chem. Phys 111, 7631 (2007).
[23] M. Born and E. Wolf, Principles of optics, 7th ed., Cambridge University Press (1997).
[24] H. Lefebvre-Brion, R. W. Field, Perturbations in the spectra of diatomic molecules, Academic Press,
1986; H. Lefebvre-Brion, R. W. Field, The spectra and dynamics of diatomic molecules, Elsevier, 2004.
[25] R. N. Zare, Angular momentum: Understanding spatial aspects in chemistry and physics, Wiley, New
York (1988).
[26] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonski, Quantum theory of angular momentum, World
Scientific Publications, Singapore and Teaneck, N.J (1988)
[27] H. Martin, J. Mol. Spec. 108, 44 (1984).
[28] G. C. Groenenboom, N. Balakrishnan, J. Chem. Phys. 118, 7380 (2003).
[29] G. N. Watson, Theory of Bessel Functions, Cambridge University Press (1922).
[30] C. Amiot, J. Verges, Can. J. Phys. 59, 1391 (1981).
[31] J. T. Hougen, The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic
Molecules, NIST, 1970.
[32] G. Herzberg, Spectra of diatomic molecules, van Nostrand, Princeton, 1950.
[33] L. Tomuta, M. Mizushima, C. J. Howard, K. M. Evenson, Phys. Rev. A 12, 974 (1975).
[34] G. C. Groenenboom, I. M. Struniewicz, J. Chem. Phys. 113, 9562 (2000).
[35] 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).
[36] H.-S. Lee, A. B. McCoy, R. R. Toczy lowski, S. M. Cybulski, J. Chem. Phys. 113, 5736 (2000).
[37] J. Brown, A. Carrington, Rotational Spectroscopy of Diatomic Molecules, Cambridge University Press
(2003).
[38] K. Gubbels, Dynamics of Stark Acceleration/Deceleration: Molecules Riding Waves, Master’s thesis,
Radbud University Nijmegen (2006).
[39] C. Ticknor, J. L. Bohn, Phys. Rev. A 71, 022709 (2005).
[40] P. F. Bernath, Spectra of atoms and molecules, Oxford Univ. Press, New York (1995).
[41] M. C. van Beek, J. J. ter Meulen, M. H. Alexander, J. Chem. Phys. 113, 628 (2000).
[42] L. D. Landau, E. M. Lifshitz Quantum Mechanics, Non-Relativistic Theory, Oxford: Pergamon (1977).
Page 23
23
TABLE I: The hybridization coefficients a(ωm) and b(ωm) for the N = 2, J = 52 ,M state in the high-field
limit, ωm ≫ ∆E/B, which arises for ωm ≫ 0.025 for the N = 2 level of the CaH(X2Σ+) molecule. See text.
M a(ωm) b(ωm)
1
2
q
2
5
q
3
5
- 12
q
3
5
q
2
5
3
2
1√
5
2√
5
- 32 2√5
1√
5
± 52 0 1
TABLE I: The hybridization coefficients a(ωm) and b(ωm) for the N = 2, J = 52 ,M state in the high-field
limit, ωm ≫ ∆E/B, which arises for ωm ≫ 0.025 for the N = 2 level of the CaH(X2Σ+) molecule. See text.
M a(ωm) b(ωm)
1
2
q
2
5
q
3
5
- 12
q
3
5
q
2
5
3
2
1√
5
2√
5
- 32 2√5
1√
5
± 52 0 1
Page 24
24
TABLE II: Hard-shell Legendre moments Ξκ0 for He – CaH (X2Σ+) and He – O2 (X3Σ−) potential energy
surfaces at a collision energy of 200 cm−1, and for He – OH (X2Π 3
2
) potential at 1000 cm−1.
Ξκ0 (A˚)
κ He–CaH He–O2 He–OH
0 13.3207 9.5987 7.7941
1 -0.4397 0 0.1380
2 1.0140 0.5672 0.1625
3 0.6147 0 0.0961
4 0.0337 -0.1320 0.01789
5 -0.1475 0 -0.0032
6 -0.0653 0.0250 -0.0034
7 0.0265 0 -0.0008
8 0.0277 -0.0060 0.0002
TABLE II: Hard-shell Legendre moments Ξκ0 for He – CaH (X2Σ+) and He – O2 (X3Σ−) potential energy
surfaces at a collision energy of 200 cm−1, and for He – OH (X2Π 3
2
) potential at 1000 cm−1.
Ξκ0 (A˚)
κ He–CaH He–O2 He–OH
0 13.3207 9.5987 7.7941
1 -0.4397 0 0.1380
2 1.0140 0.5672 0.1625
3 0.6147 0 0.0961
4 0.0337 -0.1320 0.01789
5 -0.1475 0 -0.0032
6 -0.0653 0.0250 -0.0034
7 0.0265 0 -0.0008
8 0.0277 -0.0060 0.0002
Page 25
25
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 azimuthal 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.
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 azimuthal 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.
Page 27
27
0
2
4
0o
30o
60o
90o
120o
210o
240o
270o
300o
330o
R
(Å
)
CaH
OH
O2
FIG. 3: Equipotential lines R(θ) for the He – CaH (X2Σ+) and He – O2 (X3Σ−) potential energy surfaces
at a collision energy of 200 cm−1, and for the He – OH (X2ΠΩ) potential at 1000 cm−1. The Legendre
moments for the potential energy surfaces are listed in Table II.
0
2
4
0o
30o
60o
90o
120o
210o
240o
270o
300o
330o
R
(Å
)
CaH
OH
O2
FIG. 3: Equipotential lines R(θ) for the He – CaH (X2Σ+) and He – O2 (X3Σ−) potential energy surfaces
at a collision energy of 200 cm−1, and for the He – OH (X2ΠΩ) potential at 1000 cm−1. The Legendre
moments for the potential energy surfaces are listed in Table II.
Page 28
28
0.001
0.01
0.1
1
10
100
(a) 0,1/2
0.1
1
(b) 1,3/2
0.1
1 (c) 1,1/2
0.1
1
10
(d) 2,5/2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.1
1
10
0o 10o 20o 30o 40o 50o 60o
(e) 2,3/2
!
FIG. 4: Differential cross sections for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions in a magnetic field
ωm = 0.3 (blue dashed line) parallel to the relative velocity vector, H ‖ k. 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.
0.001
0.01
0.1
1
10
100
(a) 0,1/2
0.1
1
(b) 1,3/2
0.1
1 (c) 1,1/2
0.1
1
10
(d) 2,5/2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.1
1
10
0o 10o 20o 30o 40o 50o 60o
(e) 2,3/2
!
FIG. 4: Differential cross sections for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions in a magnetic field
ωm = 0.3 (blue dashed line) parallel to the relative velocity vector, H ‖ k. 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.
Page 29
29
0.001
0.01
0.1
1
10
100
(a) 0,1/2
0.1
1
(b) 1,3/2
0.1
1
(c) 1,1/2
0.1
1
10
(d) 2,5/2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.1
1
10
0o 10o 20o 30o 40o 50o 60o
(e) 2,3/2
!
FIG. 5: Differential cross sections for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions in a magnetic
field ωm = 0.3 (blue dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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.
0.001
0.01
0.1
1
10
100
(a) 0,1/2
0.1
1
(b) 1,3/2
0.1
1
(c) 1,1/2
0.1
1
10
(d) 2,5/2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.1
1
10
0o 10o 20o 30o 40o 50o 60o
(e) 2,3/2
!
FIG. 5: Differential cross sections for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions in a magnetic
field ωm = 0.3 (blue dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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.
Page 30
30
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5
M=1/2, M'=3/2; M=-1/2, M'=-3/2
N'=1, J'=3/2
(a)
M=1/2, M'=-1/2
M=-1/2, M'=1/2
0
0.2
0.4
0.6
N'=1, J'=1/2
M=1/2, M'=-1/2
M=-1/2, M'=1/2
(b)
0
1
2
3
4
5
M=1/2, M'=-3/2
N'=2, J'=5/2
(c)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
M=1/2, M'=5/2; M=-1/2, M'=-5/2
0
1
2
3
0 0.1 0.2 0.3
M=1/2, M'=-3/2
N'=2, J'=3/2
(d)
M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
!m
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
FIG. 6: Partial integral cross sections for the He – CaH (N = 0, J = 12 ,M → N
′, J ′,M ′) collisions in a
magnetic field parallel to the initial wave vector, H ‖ k. The red solid lines show the M ′-averaged cross
sections for the (N = 0, J = 12 → N
′, J ′) collisions.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5
M=1/2, M'=3/2; M=-1/2, M'=-3/2
N'=1, J'=3/2
(a)
M=1/2, M'=-1/2
M=-1/2, M'=1/2
0
0.2
0.4
0.6
N'=1, J'=1/2
M=1/2, M'=-1/2
M=-1/2, M'=1/2
(b)
0
1
2
3
4
5
M=1/2, M'=-3/2
N'=2, J'=5/2
(c)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
M=1/2, M'=5/2; M=-1/2, M'=-5/2
0
1
2
3
0 0.1 0.2 0.3
M=1/2, M'=-3/2
N'=2, J'=3/2
(d)
M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
!m
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
FIG. 6: Partial integral cross sections for the He – CaH (N = 0, J = 12 ,M → N
′, J ′,M ′) collisions in a
magnetic field parallel to the initial wave vector, H ‖ k. The red solid lines show the M ′-averaged cross
sections for the (N = 0, J = 12 → N
′, J ′) collisions.
Page 31
31
0
0.5
1
1.5 0 0.5 1 1.5 2 2.5
N'=1, J'=3/2
(a)
M=1/2, M'=1/2
M=-1/2, M'=-1/2
0
0.5
1
N'=1, J'=1/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
(b)
0
1
2
3
4
5
6
7
M=1/2, M'=-3/2
N'=2, J'=5/2
(c)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
M=1/2, M'=5/2; M=-1/2, M'=-5/2
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3
M=1/2, M'=-3/2
N'=2, J'=3/2
(d)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
!m
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
FIG. 7: Partial integral cross sections for the He – CaH (N = 0, J = 12 ,M → N
′, J ′,M ′) collisions in a
magnetic field perpendicular to the initial wave vector, H ⊥ k. The red solid lines show the M ′-averaged
cross sections for the (N = 0, J = 12 → N
′, J ′) collisions.
0
0.5
1
1.5 0 0.5 1 1.5 2 2.5
N'=1, J'=3/2
(a)
M=1/2, M'=1/2
M=-1/2, M'=-1/2
0
0.5
1
N'=1, J'=1/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
(b)
0
1
2
3
4
5
6
7
M=1/2, M'=-3/2
N'=2, J'=5/2
(c)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
M=1/2, M'=5/2; M=-1/2, M'=-5/2
0
1
2
3
4
5
6
7
0 0.1 0.2 0.3
M=1/2, M'=-3/2
N'=2, J'=3/2
(d)M=-1/2, M'=3/2
M=1/2, M'=1/2
M=-1/2, M'=-1/2
!m
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
FIG. 7: Partial integral cross sections for the He – CaH (N = 0, J = 12 ,M → N
′, J ′,M ′) collisions in a
magnetic field perpendicular to the initial wave vector, H ⊥ k. The red solid lines show the M ′-averaged
cross sections for the (N = 0, J = 12 → N
′, J ′) collisions.
Page 32
32
-0.4
-0.2
0
0.2
0.4
0 0.1 0.2 0.3
0 1 2
N'=1, J'=3/2
!m
H (Tesla)
1,1/2
2,3/2
2,5/2
St
er
ic
a
sy
m
m
et
ry
FIG. 8: Steric asymmetry, as defined by Eq. (33), for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions.
Curves are labeled by N ′, J ′.
-0.4
-0.2
0
0.2
0.4
0 0.1 0.2 0.3
0 1 2
N'=1, J'=3/2
!m
H (Tesla)
1,1/2
2,3/2
2,5/2
St
er
ic
a
sy
m
m
et
ry
FIG. 8: Steric asymmetry, as defined by Eq. (33), for the He – CaH (N = 0, J = 12 → N
′, J ′) collisions.
Curves are labeled by N ′, J ′.
Page 34
34
0.001
0.1
10
(a) 1,0
0.001
0.01
0.1
(b) 1,2
0.001
0.01
(c) 1,1
0.001
0.01
0.1
(d) 3,2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.0001
0.001
0.01 (e) 3,4
10-6
10-5
0.0001
0.001
0o 10o 20o 30o 40o 50o 60o
(f) 3,3
!
FIG. 10: Differential cross sections for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions in a magnetic field
ωm = 5 (red dashed line) parallel to the relative velocity vector, H ‖ k. 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. The field-free cross sections for the scattering to final states with J ′ = N ′
vanish, see text.
0.001
0.1
10
(a) 1,0
0.001
0.01
0.1
(b) 1,2
0.001
0.01
(c) 1,1
0.001
0.01
0.1
(d) 3,2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.0001
0.001
0.01 (e) 3,4
10-6
10-5
0.0001
0.001
0o 10o 20o 30o 40o 50o 60o
(f) 3,3
!
FIG. 10: Differential cross sections for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions in a magnetic field
ωm = 5 (red dashed line) parallel to the relative velocity vector, H ‖ k. 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. The field-free cross sections for the scattering to final states with J ′ = N ′
vanish, see text.
Page 35
35
0.001
0.1
10
(a) 1,0
0.001
0.01
0.1
(b) 1,2
0.001
0.01
0.1 (c) 1,1
0.001
0.01
0.1
(d) 3,2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.0001
0.001
0.01 (e) 3,4
10-6
10-5
0.0001
0.001
0o 10o 20o 30o 40o 50o 60o
(f) 3,3
!
FIG. 11: Differential cross sections for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions in a magnetic field
ωm = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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. The field-free cross sections for the scattering to final states with J ′ = N ′
vanish, see text.
0.001
0.1
10
(a) 1,0
0.001
0.01
0.1
(b) 1,2
0.001
0.01
0.1 (c) 1,1
0.001
0.01
0.1
(d) 3,2
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
0.0001
0.001
0.01 (e) 3,4
10-6
10-5
0.0001
0.001
0o 10o 20o 30o 40o 50o 60o
(f) 3,3
!
FIG. 11: Differential cross sections for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions in a magnetic field
ωm = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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. The field-free cross sections for the scattering to final states with J ′ = N ′
vanish, see text.
Page 36
36
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7
(a)
1,2,0
1,2,2
1,2,-2
1,2
0
0.01
0.02
(b)
1,1,0
0
0.05
0.1
0.15
0.2 (c)
3,2,0
3,2,2
3,2,-2
3,2
0
0.01
0.02
(d)
3,4,0
3,4,2 3,4,-2
3,4
3,4,4
3,4,-4
0
0.001
0.002
0.003
0 1 2 3 4 5
3,3,0 3,3,2
3,3,-2
3,3
!m
(e)
FIG. 12: Partial integral cross sections for the He – O2 (N = 1, J = 0,M = 0 → N ′, J ′,M ′) collisions in a
magnetic field parallel to the initial wave vector, H ‖ k. Curves are labeled by N ′, J ′,M ′. The red solid
lines show the M ′-averaged cross sections. The partial cross sections, corresponding to negative M ′, are
shown by dashed lines.
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7
(a)
1,2,0
1,2,2
1,2,-2
1,2
0
0.01
0.02
(b)
1,1,0
0
0.05
0.1
0.15
0.2 (c)
3,2,0
3,2,2
3,2,-2
3,2
0
0.01
0.02
(d)
3,4,0
3,4,2 3,4,-2
3,4
3,4,4
3,4,-4
0
0.001
0.002
0.003
0 1 2 3 4 5
3,3,0 3,3,2
3,3,-2
3,3
!m
(e)
FIG. 12: Partial integral cross sections for the He – O2 (N = 1, J = 0,M = 0 → N ′, J ′,M ′) collisions in a
magnetic field parallel to the initial wave vector, H ‖ k. Curves are labeled by N ′, J ′,M ′. The red solid
lines show the M ′-averaged cross sections. The partial cross sections, corresponding to negative M ′, are
shown by dashed lines.
Page 38
38
0.25
0.5
0.75
1
(a) ! = 0
m
1,0 1,2 1,1 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,0 1,2 1,1 3,2 3,4 3,3
N,J
FIG. 14: Absolute values of the hybridization coefficients a10NJ (ωm) (black dashed line, full circles) and
bN˜J˜NJ (ωm) for different values of the interaction parameter ωm, for the O2(X3Σ−) molecule. The following
bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2 (red solid line, full diamonds), N˜ = 1, J˜ = 1 (blue dashed
line, full squares),N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed line, empty
diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = 0. See text.
0.25
0.5
0.75
1
(a) ! = 0
m
1,0 1,2 1,1 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,0 1,2 1,1 3,2 3,4 3,3
N,J
FIG. 14: Absolute values of the hybridization coefficients a10NJ (ωm) (black dashed line, full circles) and
bN˜J˜NJ (ωm) for different values of the interaction parameter ωm, for the O2(X3Σ−) molecule. The following
bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2 (red solid line, full diamonds), N˜ = 1, J˜ = 1 (blue dashed
line, full squares),N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed line, empty
diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = 0. See text.
Page 39
39
0.25
0.5
0.75
1
(a) ! = 0m
1,2 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,2 3,2 3,4 3,3N,J
FIG. 15: Absolute values of the hybridization coefficients bN˜J˜NJ (ωm) for different values of the interaction
parameter ωm, for the O2(X3Σ−) molecule. The following bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2
(red solid line, full diamonds), N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed
line, empty diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = 2. See
text.
0.25
0.5
0.75
1
(a) ! = 0m
1,2 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,2 3,2 3,4 3,3N,J
FIG. 15: Absolute values of the hybridization coefficients bN˜J˜NJ (ωm) for different values of the interaction
parameter ωm, for the O2(X3Σ−) molecule. The following bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2
(red solid line, full diamonds), N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed
line, empty diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = 2. See
text.
Page 40
40
0.25
0.5
0.75
1
(a) ! = 0m
1,2 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,2 3,2 3,4 3,3N,J
FIG. 16: Absolute values of the hybridization coefficients bN˜J˜NJ (ωm) for different values of the interaction
parameter ωm, for the O2(X3Σ−) molecule. The following bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2
(red solid line, full diamonds), N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed
line, empty diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = −2. See
text.
0.25
0.5
0.75
1
(a) ! = 0m
1,2 3,2 3,4 3,3
N,J
0.25
0.5
0.75
1
(b) ! = 1
m
0.25
0.5
0.75
1
(c) ! = 3
m
0.25
0.5
0.75
1
(d) ! = 5
m
1,2 3,2 3,4 3,3N,J
FIG. 16: Absolute values of the hybridization coefficients bN˜J˜NJ (ωm) for different values of the interaction
parameter ωm, for the O2(X3Σ−) molecule. The following bN˜J˜NJ (ωm) coefficients are presented: N˜ = 1, J˜ = 2
(red solid line, full diamonds), N˜ = 3, J˜ = 2 (green solid line, empty circles), N˜ = 3, J˜ = 4 (orange dashed
line, empty diamonds), N˜ = 3, J˜ = 3 (light blue solid line, empty squares). For all coefficients M = −2. See
text.
Page 41
41
-1
-0.5
0
0.5
1 0 1 2 3 4 5 6 7
0 1 2 3 4 5
1,1
1,2
3,2
!m
H (Tesla)
St
er
ic
a
sy
m
m
et
ry
3,3
3,4
FIG. 17: Steric asymmetry, as defined by Eq. (33), for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions.
Curves are labeled by N ′, J ′.
-1
-0.5
0
0.5
1 0 1 2 3 4 5 6 7
0 1 2 3 4 5
1,1
1,2
3,2
!m
H (Tesla)
St
er
ic
a
sy
m
m
et
ry
3,3
3,4
FIG. 17: Steric asymmetry, as defined by Eq. (33), for the He – O2 (N = 1, J = 0 → N ′, J ′) collisions.
Curves are labeled by N ′, J ′.
Page 42
42
0.15
0.25
0.35
0.45
0 20 40 60 80
(a)
M=1/2
M=!1/2
M=3/2
M=!3/2
0.26
0.3
0.34
0 1 2 3 4 5
(b)
M=1/2
M=!1/2
M=3/2
M=!3/2
M=5/2
M=!5/2
"m
A
lig
nm
en
t c
os
in
e
H (Tesla)
FIG. 18: Expectation values of the alignment cosine 〈cos2 θ〉 for the 3/2, f (a) and 5/2, f (b) states of the
OH molecule, as a function of the magnetic field strength parameter ωm.
0.15
0.25
0.35
0.45
0 20 40 60 80
(a)
M=1/2
M=!1/2
M=3/2
M=!3/2
0.26
0.3
0.34
0 1 2 3 4 5
(b)
M=1/2
M=!1/2
M=3/2
M=!3/2
M=5/2
M=!5/2
"m
A
lig
nm
en
t c
os
in
e
H (Tesla)
FIG. 18: Expectation values of the alignment cosine 〈cos2 θ〉 for the 3/2, f (a) and 5/2, f (b) states of the
OH molecule, as a function of the magnetic field strength parameter ωm.
Page 43
43
0.001
0.1
10
1000
(a) 3/2, f
10-5
0.001
0.1 (b) 5/2, f
10-5
0.001
0.1 (c) 5/2, e
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
10-5
0.001
0.1 (d) 7/2, f
10-5
0.001
0o 10o 20o 30o 40o 50o 60o
(e) 7/2, e
!
FIG. 19: Differential cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
ωm = 5 (red dashed line) parallel to the relative velocity vector, H ‖ k. 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. See text.
0.001
0.1
10
1000
(a) 3/2, f
10-5
0.001
0.1 (b) 5/2, f
10-5
0.001
0.1 (c) 5/2, e
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
10-5
0.001
0.1 (d) 7/2, f
10-5
0.001
0o 10o 20o 30o 40o 50o 60o
(e) 7/2, e
!
FIG. 19: Differential cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
ωm = 5 (red dashed line) parallel to the relative velocity vector, H ‖ k. 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. See text.
Page 44
44
0.001
0.1
10
1000
(a) 3/2, f
10-5
0.001
0.1 (b) 5/2, f
10-5
0.001
0.1 (c) 5/2, e
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
10-5
0.001
0.1 (d) 7/2, f
10-5
0.001
0o 10o 20o 30o 40o 50o 60o
(e) 7/2, e
!
FIG. 20: Differential cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
ωm = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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. See text.
0.001
0.1
10
1000
(a) 3/2, f
10-5
0.001
0.1 (b) 5/2, f
10-5
0.001
0.1 (c) 5/2, e
D
if
fe
re
nt
ia
l c
ro
ss
-s
ec
tio
n
(Å
2 /
st
er
ad
)
10-5
0.001
0.1 (d) 7/2, f
10-5
0.001
0o 10o 20o 30o 40o 50o 60o
(e) 7/2, e
!
FIG. 20: Differential cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
ωm = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k. 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. See text.
Page 45
45
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
0
0.04
0.08
0.12
0 20 40 60 80
(a)
5/2,f
5/2,e
7/2,f
7/2,e
0
0.04
0.08
0.12
0 1 2 3 4 5
(b)
5/2,f
5/2,e
7/2,f
7/2,e
!m
FIG. 21: Partial integral cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
(a) parallel, H ‖ k, and (b) perpendicular, H ⊥ k, to the initial wave vector. Curves are labeled by J ′, e/f .
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
0
0.04
0.08
0.12
0 20 40 60 80
(a)
5/2,f
5/2,e
7/2,f
7/2,e
0
0.04
0.08
0.12
0 1 2 3 4 5
(b)
5/2,f
5/2,e
7/2,f
7/2,e
!m
FIG. 21: Partial integral cross sections for the He – OH (J = 3/2, f → J ′, e/f) collisions in a magnetic field
(a) parallel, H ‖ k, and (b) perpendicular, H ⊥ k, to the initial wave vector. Curves are labeled by J ′, e/f .
Page 46
46
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
10-5
0.001
0.1
0 20 40 60 80
(a)
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
10-5
0.001
0.1
(b)
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
10-5
0.001
0.1
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
(c)
10-5
0.001
0.1
0 1 2 3 4 5
(d)
M'=1/2
M'=!1/2
M'=3/2
M'=!5/2
"m
FIG. 22: Logarithm of the partial integral cross sections for the He – OH (J = 3/2, f,M → J ′ = 5/2, f,M ′)
collisions in a magnetic field parallel to the initial wave vector, H ‖ k. The panels correspond to different
initial states: M = 1/2 (a), M = −1/2 (b), M = 3/2 (c), M = −3/2 (d). All partial cross sections are
non-vanishing, but the puny ones are not shown. See text.
Pa
rti
al
in
te
gr
al
c
ro
ss
se
ct
io
ns
(Å
2 )
H (Tesla)
10-5
0.001
0.1
0 20 40 60 80
(a)
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
10-5
0.001
0.1
(b)
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
10-5
0.001
0.1
M'=1/2
M'=!1/2
M'=3/2
M'=!3/2
M'=!5/2
M'=5/2
(c)
10-5
0.001
0.1
0 1 2 3 4 5
(d)
M'=1/2
M'=!1/2
M'=3/2
M'=!5/2
"m
FIG. 22: Logarithm of the partial integral cross sections for the He – OH (J = 3/2, f,M → J ′ = 5/2, f,M ′)
collisions in a magnetic field parallel to the initial wave vector, H ‖ k. The panels correspond to different
initial states: M = 1/2 (a), M = −1/2 (b), M = 3/2 (c), M = −3/2 (d). All partial cross sections are
non-vanishing, but the puny ones are not shown. See text.
Page 47
47
-0.2
-0.1
0
0.1
0.2 0 20 40 60 80
0 1 2 3 4 5
!m
H (Tesla)
St
er
ic
a
sy
m
m
et
ry
5/2,f
5/2,e
7/2,f
7/2,e
FIG. 23: Steric asymmetry, as defined by Eq. (33), for the He – OH (J = 3/2, f → J ′, e/f) collisions. Curves
are labeled by J ′, e/f .
-0.2
-0.1
0
0.1
0.2 0 20 40 60 80
0 1 2 3 4 5
!m
H (Tesla)
St
er
ic
a
sy
m
m
et
ry
5/2,f
5/2,e
7/2,f
7/2,e
FIG. 23: Steric asymmetry, as defined by Eq. (33), for the He – OH (J = 3/2, f → J ′, e/f) collisions. Curves
are labeled by J ′, e/f .
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