Collisions of paramagnetic molecules in magnetic fields: an analytic model based on Fraunhofer diffraction of matter waves
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Collisions of paramagnetic molecules in magnetic fields: an analyticmodel 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: October 24, 2018)We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisionsof open-shell molecules ( Σ, Σ, and Π) with closed-shell atoms. Our treatment makes use of theFraunhofer model of matter wave scattering and its recent extension to collisions in electric [M.Lemeshko and B. Friedrich, J. Chem. Phys. , 024301 (2008)] and radiative fields [M. Lemeshkoand B. Friedrich, Int. J. Mass. Spec. in press (2008)]. A magnetic field aligns the molecule in thespace-fixed frame and thereby alters the effective shape of the diffraction target. This significantlyaffects the differential and integral scattering cross sections. We exemplify our treatment by eval-uating the magnetic-field-dependent scattering characteristics of the He – CaH ( X Σ + ), He – O ( X Σ − ) and He – OH ( X Π Ω ) systems at thermal collision energies. Since the cross sections can beobtained 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 – O . While odd ∆ M transitions dominate the He – OH( J = 3 / , f → J ′ , e/f ) integral cross sections in a magnetic field parallel to the relative velocityvector, even ∆ M transitions prevail in the case of the He – CaH ( X Σ + ) and He – O ( X Σ − )collision systems. For the latter system, the magnetic field opens inelastic channels that are closedin 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.EzKeywords: Rotationally inelastic scattering, paramagnetic molecules, alignment and orientation, Zeemaneffect, 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 ofa recent date, having been prompted by the newfashioned techniques to magnetically manipulate,control and confine paramagnetic atoms and molecules. Theoretical accounts of molecular collisionsin magnetic fields are usually based on rigorous close-coupling treatments [4]. Analytic models ofsuch collisions are scarce, and limited to the Wigner regime, see, e.g., ref. [5]. Here we present ananalytic model of state-to-state rotationally inelastic collisions of closed-shell atoms with open-shellmolecules 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 toinclude collisions in electrostatic [9] and radiative [10] fields. The magnetic field affects the collisiondynamics by aligning the molecular axis with respect to the relative velocity vector, thereby changingthe effective shape of the diffraction target. We consider open-shell molecules in the Σ, Σ, and Πelectronic 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( X Σ + ), O ( X Σ − ), and OH( X Π Ω ) species.We take, as the closed-shell collision partner, a He atom. Helium is a favorite buffer gas, used tothermalize 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 Fraunhofermodel of matter-wave scattering. In Sections III, IV, and V, we extend the Fraunhofer modelto account for scattering of open-shell molecules with closed-shell atoms in magnetic fields: inSection III, we work out closed-form expressions for the partial and total differential and integralcross sections and the steric asymmetry of collisions between closed-shell atoms and paramagnetic Σ molecules, and apply them to the He–CaH( X Σ , J = 1 / → J ′ ) collision system; in Sec. IV wepresent the analytic theory for Σ molecules and apply it to the He–O ( X Σ , N = 0 , J = 1 → N ′ , J ′ )scattering; in Section V we develop the theory for collisions of Π molecules and exemplify the resultsby treating the He–OH( X Π , J = 3 / , f → J ′ , e/f ) inelastic scattering. Finally, in Section VI, wecompare the results obtained for the collisions of the different molecules with helium and drawconclusions 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 f i → f ( ϑ ) = h f | f ( ϑ ) | i i (1)for scattering into an angle ϑ from an initial, | i i , to a final, | f i , state by the elastic scatteringamplitude, f ( ϑ ). This is tantamount to the energy sudden approximation, which is valid when thecollision time is much smaller than the rotational period, as dictated by the inequality ξ ≪
1, where ξ = ∆ E rot kR E coll ≈ BkR E coll , (2)is the Massey parameter, see e.g. Refs. [11],[12]. Here ∆ E rot is the rotational level spacing, B therotational constant, E coll the collision energy, k ≡ (2 mE coll ) / / ~ the wavenumber, m the reducedmass of the collision system, and R the radius of the scatterer.The second approximation replaces the elastic scattering amplitude f ( ϑ ) in Eq. (1) by theamplitude for Fraunhofer diffraction by a sharp-edged, impenetrable obstacle as observed at a pointof radiusvector r from the scatterer, see Fig. 1. This amplitude is given by the integral f ( ϑ ) ≈ Z e − ikRϑ cos ϕ d R (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-freemolecular 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 rλ ≪ 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 MeVcollision energies as well as for molecular collisions at thermal and hyperthermal energies. In thelatter case, inequality (4) is fulfilled due to the compensation of the larger molecular size a by alarger de Broglie wavelength λ pertaining to thermal molecular velocities.For a nearly-circular scatterer, with a boundary R ( ϕ ) = R + δ ( ϕ ) in the XY plane, theFraunhofer integral of Eq. (3) can be evaluated and expanded in a power series in the deformation δ ( ϕ ), f ( ϑ ) = f ( ϑ ) + f ( ϑ, δ ) + f ( ϑ, δ ) + · · · (5)with f ( ϑ ) the amplitude for scattering by a disk of radius R f ( ϑ ) = i ( kR ) J ( kR ϑ )( kR ϑ ) (6)and f the lowest-order anisotropic amplitude, f ( ϑ ) = ik π Z π δ ( ϕ ) e − i ( kR ϑ ) cos ϕ dϕ (7)where J is a Bessel function of the first kind. Both Eqs. (6) and (7) are applicable at small valuesof ϑ . ◦ , i.e., within the validity of the approximation sin ϑ ≈ ϑ .The scatterer’s shape in the space fixed frame, see Fig. 1, is given by R ( α, β, γ ; θ, ϕ ) = X κνρ Ξ κν D κρν ( αβγ ) Y κρ ( θ, ϕ ) (8)where ( α, β, γ ) are the Euler angles through which the body-fixed frame is rotated relative to thespace-fixed frame, ( θ, ϕ ) are the polar and azimuthal angles in the space-fixed frame, D κρν ( αβγ ) arethe Wigner rotation matrices, and Ξ κν are the Legendre moments describing the scatterer’s shapein the body-fixed frame. Clearly, the term with κ = 0 corresponds to a disk of radius R , R ≈ Ξ √ π (9)Since of relevance is the shape of the target in the XY plane, we set θ = π in Eq. (8). As a result, δ ( ϕ ) = R ( α, β, γ ; π , ϕ ) − R = R ( ϕ ) − R = X κνρκ =0 Ξ κν D κρν ( αβγ ) Y κρ ( π , ϕ ) (10)By combining Eqs. (1), (7), and (10) we finally obtain f i → f ( ϑ ) ≈ h f | f + f | i i = h f | f | i i = ikR π X κνρκ =0 κ + ρ even Ξ κν h f | D κρν | i i F κρ J | ρ | ( kR ϑ ) (11)where F κρ = ( − ρ π (cid:0) κ +14 π (cid:1) ( − i ) κ √ ( κ + ρ )!( κ − ρ )!( κ + ρ )!!( κ − ρ )!! for κ + ρ even and κ ≥ ρ ρ , the factor ( − i ) κ is to be replaced by i κ . III. SCATTERING OF Σ MOLECULES BY CLOSED-SHELL ATOMS IN AMAGNETIC FIELDA. A Σ molecule in a magnetic field The field-free Hamiltonian of a rigid Σ molecule H = BN + γ N · S (13)is represented by a 2 × | N, J, M i . Here N and S are the rotational and (electronic) spin angular momenta, B is the rotational constant and γ thespin-rotation constant. Its eigenfunctionsΨ ± ( J, M ) = 1 √ (cid:20)(cid:12)(cid:12) S, (cid:11)(cid:12)(cid:12) J, Ω , M (cid:11) ± (cid:12)(cid:12) S, − (cid:11)(cid:12)(cid:12) J, − Ω , M (cid:11)(cid:21) , (14)are combinations of (electronic) spin functions | S, M S i with Hund’s case (a) (i.e., symmetric top)functions | J, Ω , M i pertaining to the total angular momentum J = N + S , whose projections on thespace- and body-fixed axes are M and Ω = ± , respectively. The Hund’s case (a) wavefunctionsare given by: | J, M, Ω i = r J + 14 π D J ∗ M Ω ( ϕ, θ, γ = 0) (15)The Ψ + and Ψ − states are conventionally designated as F and F states, for which therotational quantum number N = J − and N = J + , respectively. Equation (14) can be recastin terms of N instead of J : | Ψ ǫ ( N, M ) i = 1 √ (cid:20)(cid:12)(cid:12) S, (cid:11)(cid:12)(cid:12) N + ǫ , Ω , M (cid:11) + ǫ (cid:12)(cid:12) S, − (cid:11)(cid:12)(cid:12) N + ǫ , − Ω , M (cid:11)(cid:21) , (16)with ǫ = ± F and F are given by E + (cid:16) N + , M ; F (cid:17) = BN ( N + 1) + γ N (17) E − (cid:16) N − , M ; F (cid:17) = BN ( N + 1) − γ N + 1) , (18)whence we see that the spin-rotation interaction splits each rotational level into a doublet separatedby ∆ E ≡ E + − E − = γ ( N + ).In a static magnetic field, H , directed along the space-fixed Z axis, the Hamiltonian acquiresa magnetic dipole potential which is proportional to the projection, S Z , of S on the Z axis V m = S Z ω m B, (19)with ω m ≡ g S µ B H B (20)a dimensionless interaction parameter involving the electron gyromagnetic ratio g S ≃ . µ B , and the rotational constant B .The Zeeman eigenproperties of a Σ molecule can be readily obtained in closed form, sincethe V m operator couples states that differ in N by 0 or ± H = H + V m , factors into 2 × N : H = − ω m B − M N +1 + E − ω m B [1 − M ( N +1 / ] [1 − M ( N +1 / ] M N +1 + E + ω m B (21)As a result, the Zeeman eigenfunctions of a Σ molecule are given by a linear combination of thefield-free wavefunctions (16), ψ ( ˜ N , ˜ J, M ; ω m ) = a ( ω m ) (cid:12)(cid:12) Ψ − ( N, M ) (cid:11) + b ( ω m ) (cid:12)(cid:12) Ψ + ( N, M ) (cid:11) , (22)with the hybridization coefficients a ( ω m ) and b ( ω m ) obtained by diagonalizing Hamiltonian (21).Although N and J are no longer good quantum numbers in the magnetic field, they can be employedas adiabatic labels of the states: we use ˜ N and ˜ J to denote the angular momentum quantum numbersof the field-free state that adiabatically correlates with the given state in the field. Since the Zeemaneigenfunction comprises rotational states with either N even or N odd, the parity of the eigenstatesremains definite even in the presence of the magnetic field; it is given by ( − ˜ N .The degree of mixing of the Hund’s case (b) states that make up a Σ Zeeman eigenfunctionis determined by the splitting of the rotational levels measured in terms of the rotational constant,∆
E/B : for ω m ≤ ∆ E/B the mixing (hybridization) is incomplete, while it is perfect in the high-field limit, ω m ≫ ∆ E/B . We note that in the high-field limit, the eignevectors can be foundfrom matrix (21) with E ± /ω m B →
0. As an example, Table I lists the values of the hybridizationcoefficients a ( ω m ) and b ( ω m ) for the N = 2 , J = , M states of the CaH molecule in the high-fieldlimit, which is attained at ω m ≫ . h cos θ i , which,in the Σ case, can be obtained in closed form. To the best of our knowledge, this result has notbeen presented in the literature before; therefore, we give it in Appendix C. The dependence of thealignment cosine on the magnetic field strength parameter ω m is shown in Fig. 2 for the two lowest N states of the CaH molecule. One can see that for ω m ≫ ∆ E/B , the alignment cosine smoothlyapproaches a constant value, corresponding to as good an alignment as the uncertainty principleallows.
B. The field-dependent scattering amplitude
In what follows, we consider scattering from the N = 0 , J = 1 / N ′ , J ′ statein a magnetic field. Since the N = 0 state of a Σ molecule is not aligned, the effects of the magneticfield 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 initialwave vector k , we introduce a field-fixed coordinate system X ♯ Y ♯ Z ♯ , whose Z ♯ -axis is defined bythe direction of the electric field vector ε . By making use of the relation D J ∗ M Ω ( ϕ ♯ , θ ♯ ,
0) = X ξ D JξM ( ϕ ε , θ ε , D J ∗ ξ Ω ( ϕ, θ,
0) (23)we transform the wavefunctions (22) to the space-fixed frame. For the initial and the final states wehave: | i ( N, M ) i = 1 √ π X ξ ( a ( ω m ) √ N D N − ξM ( ϕ ε , θ ε , (cid:20) D N − ∗ ξ Ω ( ϕ, θ, − D N − ∗ ξ − Ω ( ϕ, θ, (cid:21) + b ( ω m ) √ N + 1 D N + 12 ξM ( ϕ ε , θ ε , (cid:20) D N + 12 ∗ ξ Ω ( ϕ, θ, − D N + 12 ∗ ξ − Ω ( ϕ, θ, (cid:21)) (24) h f ( N ′ , M ′ ) | = 1 √ π X ξ ′ ( a ′ ( ω m ) √ N ′ D N ′ − ξ ′ M ′ ( ϕ ε , θ ε , (cid:20) D N ′ − ∗ ξ ′ Ω ( ϕ, θ, − D N ′ − ∗ ξ ′ − Ω ( ϕ, θ, (cid:21) + b ′ ( ω m ) √ N ′ + 1 D N ′ + 12 ξ ′ M ′ ( ϕ ε , θ ε , (cid:20) D N ′ + 12 ∗ ξ ′ Ω ( ϕ, θ, − D N ′ + 12 ∗ ξ ′ − Ω ( ϕ, θ, (cid:21)) (25)where Ω = for a Σ molecule.By substituting from Eqs. (24) and (25) into Eq. (11), we finally obtain the scatteringamplitude for inelastic collisions of Σ molecules with closed-shell atoms in a magnetic field: f ω m i → f ( ϑ ) = ikR π X κρκ =0 κ + ρ even Ξ κ D κ ∗− ρ, ∆ M ( ϕ ε , θ ε , F κρ J | ρ | ( kR ϑ ) " ( − κ + ( − ∆ N (cid:21) × ( a ( ω m ) a ′ ( ω m ) r NN ′ C (cid:16) N − , κ, N ′ − ; Ω0Ω (cid:17) C (cid:16) N − , κ, N ′ − ; M ∆ M M ′ (cid:17) + a ( ω m ) b ′ ( ω m ) r NN ′ + 1 C (cid:16) N − , κ, N ′ + ; Ω0Ω (cid:17) C (cid:16) N − , κ, N ′ + ; M ∆ M M ′ (cid:17) + a ′ ( ω m ) b ( ω m ) r N + 1 N ′ C (cid:16) N + , κ, N ′ − ; Ω0Ω (cid:17) C (cid:16) N + , κ, N ′ − ; M ∆ M M ′ (cid:17) + b ( ω m ) b ′ ( ω m ) r N + 1 N ′ + 1 C (cid:16) N + , κ, N ′ + ; Ω0Ω (cid:17) C (cid:16) N + , κ, N ′ + ; M ∆ M M ′ (cid:17)) (26)As noted above, there is no hybridization of the initial state for the N = 0 , J = → N ′ , J ′ collisions,i.e., a ( ω m ) = 0, b ( ω m ) = 1 in Eq. (26). By making use of the properties of the Clebsch-Gordancoefficients [25],[26], the expression for the scattering amplitude from the N = 0 , J = , M = ± state to an N ′ , J ′ , M ′ state simplifies to f ω m , , ± → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π Ξ N ′ N ′ + 1 X ρρ + N ′ even d N ′ − ρ, ∆ M ( θ ε ) F N ′ ρ J | ρ | ( kR ϑ ) × " ± a ′ ( ω m ) q N ′ ∓ M ′ + + b ′ ( ω m ) q N ′ ± M ′ + (27)The amplitude is seen to be directly proportional to the Ξ N ′ Legendre moment. We note thatthe 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]. C. Results for He – CaH ( X Σ , J = 1 / → J ′ ) scattering in a magnetic field Here we apply the analytic model scattering to the He – CaH( Σ + , J = → J ′ ) collisionsystem. The CaH molecule, employed previously in thermalization experiments with a He buffergas [15], [16], has a rotational constant B = 4 . − and a spin-rotational interaction parameter γ = 0 . − [27]. Such values of molecular constants result in an essentially perfect mixing(and alignment) of the molecular states for field strengths H ≥ . − . − . Such a weak attractive well can be neglected at a collision energy as lowas 200 cm − (which corresponds to a wave number k = 6 .
58 ˚A − ). The corresponding value of theMassey parameter, ξ ≈ .
5, warrants the validity of the sudden approximation to the He – CaHcollision system from this collision energy on. The “hard shell” of the potential energy surface wasfound by a fit to Eq. (8) for κ ≤
8, and is shown in Fig. 3. The coefficients Ξ κ obtained from thefit are listed in Table II. According to Eq. (9), the Ξ coefficient determines the hard-sphere radius R , which is responsible for elastic scattering.
1. Differential cross sections
The state-to-state differential cross sections for scattering in a field parallel ( k ) and perpen-dicular ( ⊥ ) to k are given by I ω m , ( k , ⊥ )0 → J ′ ( ϑ ) = X M ′ I ω m , ( k , ⊥ )0 , → J ′ ,M ′ ( ϑ ) (28)with I ω m , ( k , ⊥ )0 , → J ′ ,M ′ ( ϑ ) = (cid:12)(cid:12)(cid:12) f ω m , ( k , ⊥ )0 , → ˜ J ′ ,M ′ ( ϑ ) (cid:12)(cid:12)(cid:12) (29)They are presented in Figs. 4, 5 for He–CaH collisions at zero field, ω m = 0, as well as at high field, ω m = 0 . H =2.75 T for CaH), where the hybridization and alignment are ascomplete 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 Ξ N ′ Legendre moment. According to Table II, theLegendre expansion of the He–CaH potential energy surface is dominated by Ξ . Therefore, thetransition 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 ′ ( ω m ) and b ′ ( ω m ), whose values cannot affect the angular dependence, as this is determined solelyby the Bessel functions, J | ρ | ( kR ϑ ). 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 havefor large angles such that ϑ ≫ πρ/ kR : J | ρ | ( kR ϑ ) ∼ cos (cid:0) kR ϑ − π (cid:1) for ρ evensin (cid:0) kR ϑ − π (cid:1) for ρ odd (30)For the He – CaH system, the phase shift between the J and J Bessel functions, which contributeto the N = 0 → N ′ = 1 , ◦ . Therefore thereis no field-induced phase shift, neither in the parallel nor in the perpendicular case, as illustrated byFigs. 4, 5.Figs. 4 and 5 show that the magnetic field induces only small changes in the amplitudes ofthe cross sections, without shifting their oscillations. The amplitude variation is so small becausethe magnetic field fails to mix contributions from the different Ξ κ, Legendre moments, in contrastto scattering in electrostatic [9] and radiative [10] fields. The changes in the amplitudes of thedifferential cross sections are closely related to the field dependence of the partial integral crosssections, which are analyzed next.
2. Integral cross sections
The angular range, ϑ . ◦ , where the Fraunhofer approximation applies the best, comprisesthe 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 integratingthe Fraunhofer differential cross sections, Eq. (28) and (29), over the solid angle sin ϑdϑdϕ , with0 ≤ ϑ ≤ π and 0 ≤ ϕ ≤ π .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 thecross sections for the N = 0 , J = → N ′ , J ′ transitions is that, in the parallel field geometry, theyincrease for the F final states and decrease for the F states, while it is the other way around forthe perpendicular geometry.In order to make sense of these trends in the field dependence of the M -averagedcross sections, let us take a closer look at the partial, M -resolved cross sections for the N = 0 , J = , 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 k . In this case, the real Wignermatrices reduce to the Kronecker delta functions, d N ′ − ρ, ∆ M (0) = δ − ρ, ∆ M , and the scatteringamplitude (27) becomes: f ω m , k , , ± → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π Ξ N ′ N ′ + 1 F N ′ , − ∆ M J | ∆ M | ( kR ϑ ) × " ± a ′ ( ω m ) q N ′ ∓ M ′ + + b ′ ( ω m ) q N ′ ± M ′ + (31)Eq. (31) allows to readily interpret the dependences presented in Fig. 6. First, we see that the F N ′ , − ∆ M coefficients, defined by eq. (12), lead to a selection rule, namely that the cross sectionsvanish 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 arecontributions from the transitions leading to the F states with M = ± J , since these states exhibitno 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 / → N ′ , J ′ transitions is a result of a competition among the partial M -resolved cross sections. Thereforewe 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 = , M → N ′ = 2 , J ′ = , M ′ . Substitutingthe coefficients from Table I into Eq. (31), we see that the term in the square brackets vanishesfor M = − , M ′ = − and for M = − , M ′ = , but equals √ M = , M ′ = and M = , M ′ = − . In addition, taking into account that F , > F , , we see that the M -averagedcross section must go up with increasing field strength.More generally, the dependence of the cross sections on the magnetic field is containedin the two hybridization coefficients a ′ ( ω m ) and b ′ ( ω m ). In the field-free case, a ′ ( ω m ) = 0 and b ′ ( ω m ) = 1 for collisions leading to F states, whereas a ′ ( ω m ) = 1, b ′ ( ω m ) = 0 for collisions thatproduce F states. In a magnetic field, the a ′ ( ω m ) and b ′ ( ω m ) coefficients assume values rangingbetween − a ′ ( ω m ) and b ′ ( ω m ) coefficients have the same signs forthe F states and opposite signs for the F states. Clearly, then, for an F state, | a ′ ( ω m ) | increaseswith the field strength, while | b ′ ( ω m ) | decreases. Hence the factor in the square brackets of Eq. (31)increases with ω m for M = and decreases for M = − , because of the opposite sign of the a ′ ( ω m )coefficient.This is reversed for the final F states, i.e., the cross sections increase for M = − , anddecrease for M = .(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k . In this case, Eq. (27)takes the form: f ω m , ⊥ , , ± → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π Ξ N ′ N ′ + 1 X ρρ + N ′ even d N ′ − ρ, ∆ M ( π ) F N ′ ρ J | ρ | ( kR ϑ ) × " ± a ′ ( ω m ) q N ′ ∓ M ′ + + b ′ ( ω m ) q N ′ ± M ′ + (32)which is more involved than for the parallel case, although the field-dependent coefficients a ′ ( ω m )and b ′ ( ω m ) remain outside of the summation. The difference between the parallel and perpendicularcases is related to the values of the real Wigner d -matrices mixed by eq. (32). For instance, aninspection of the coefficients from Table I, along with the d N ′ − ρ, ∆ M matrices, reveals that the M -averaged cross sections for the N = 0 , J = → N ′ = 2 , J ′ = transition (i.e., transition to an F final state) will decrease with ω m , 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 theexpression S i → f = σ k − σ ⊥ σ k + σ ⊥ (33)where the integral cross sections σ k , ⊥ correspond, respectively, to H k k and H ⊥ k . The fielddependence of the steric asymmetry for the He – CaH collisions is presented in Fig. 8. One cansee that a particularly pronounced asymmetry obtains for the scattering into the N ′ = 1 , J ′ = , final states, while it is smaller for the N = 0 → N ′ = 2 channels. Moreover, the steric asymmetryexhibits a sign alternation: it is positive for F final states and negative for F final states. Thisbehavior is a reflection of the alternation in the trends of the integral cross sections for the F and F final states, discussed in the previous subsection, cf. the corresponding M -dependent integralcross 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 dependson the radius R only: a sphere looks the same from any direction.0 IV. SCATTERING OF Σ MOLECULES BY CLOSED-SHELL ATOMS IN AMAGNETIC FIELDA. A Σ molecule in a magnetic field The field-free Hamiltonian of a Σ electronic state consists of rotational, spin-rotation, andspin-spin terms H = BN + γ N · S + λ (3 S z − S ) (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 × J values(except for J =0). The matrix elements of Hamiltonian (34) can be found, e.g., in ref. [30] (seealso [14] and [24]). The eigenenergies of H are (in units of the rotational constant B ): E ( J ) /B = J ( J + 1) + 1 − γ ′ − λ ′ − XE ( J ) /B = J ( J + 1) − γ ′ + 2 λ ′ E ( J ) /B = J ( J + 1) + 1 − γ ′ − λ ′ X (35)with X ≡ " J ( J + 1)( γ ′ − + (cid:18) γ ′ + 2 λ ′ − (cid:19) / γ ′ ≡ γ/Bλ ′ ≡ λ/B The eigenenergies (35) correspond to the three ways of combining rotational and electronic spinangular momenta N and S for S = 1 into a total angular momentum J ; the total angular momentumquantum number takes values J = N + 1, J = N , and J = N − F , F , and F , respectively. For the case when N = 1, J = 0, the sign of the X termshould be reversed [32]. The parity of the states is ( − N .The interaction of a Σ molecule with a magnetic field H is given by: V m = S Z ω m B, (36)where ω m ≡ g S µ B H 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 i = c NJ | J, , M i + c NJ | J, , M i + c − NJ | J, − , M i (38)using the matrix elements of the S Z operator given in Appendix A.1The Zeeman eigenfunctions are hybrids of the Hund’s case (b) basis functions (38): (cid:12)(cid:12)(cid:12) ˜ N, ˜ J, M ; ω m E = X NJ a ˜ N ˜ JNJ ( ω m ) | N, J, M i (39)and are labeled by ˜ N and ˜ J , which are the angular momentum quantum numbers of the field-freestate that adiabatically correlates with a given state in the field. Since V m 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 ( − ˜ N . However, the Zeeman matrix for a Σ molecule is no longer finite, unlikethe 2 × Σ state.Using the Hund’s case (b) rather than Hund’s case (a) basis set makes it possible to directlyrelate the field-free states and the Zeeman states, via the hybridization coefficients a ˜ N ˜ JNJ .The alignment cosine, h cos θ i , of the Zeeman states can be evaluated from the matrixelements of Appendix B. The dependence of h cos θ i on the magnetic field strength parameter ω m is exemplified in Fig. 9 for ˜ N = 1 , O molecule. B. The field-dependent scattering amplitude
We consider scattering from an initial
N, J state to a final N ′ , J ′ state. We transform thewavefunctions (39) to the space-fixed frame by making use of Eq. (23) – cf. Section III B. As aresult, the initial and final states become: | i i ≡ (cid:12)(cid:12)(cid:12) ˜ N , ˜ J, M, ω m E = X NJ r J + 14 π a ˜ N ˜ JNJ ( ω m ) X Ω c Ω NJ X ξ D JξM ( ϕ ε , θ ε , D J ∗ ξ Ω ( ϕ, θ,
0) (40) h f | ≡ h ˜ N ′ , ˜ J ′ , M ′ , ω m | = X N ′ J ′ r J ′ + 14 π b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ) X Ω ′ c Ω ′ N ′ J ′ X ξ ′ D J ′ ξ ′ M ′ ( ϕ ε , θ ε , D J ′ ∗ ξ ′ Ω ′ ( ϕ, θ,
0) (41)On substituting from Eqs. (40) and (41) into Eq. (11) and some angular momentum algebra, weobtain a general expression for the scattering amplitude: f ω m i → f ( ϑ ) = ikR π X κρκ =0 κ + ρ even Ξ κ D κ ∗− ρ, ∆ M ( ϕ ε , θ ε , F κρ J | ρ | ( kR ϑ ) × X NJN ′ J ′ r J + 12 J ′ + 1 a ˜ N ˜ JNJ ( ω m ) b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ) C ( JκJ ′ ; M ∆ M M ′ ) X Ω c Ω NJ c Ω N ′ J ′ C ( JκJ ′ ; Ω0Ω) (42) C. Results for He–O ( X Σ , N = 0 , J = 1 → N ′ , J ′ ) scattering in a magnetic field The O ( Σ − ) molecule has a rotational constant B = 1 . − , a spin-rotation con-stant γ = − . − , and a spin-spin constant λ = 1 . − [33]. According to Ref. [34],the ground state He – O potential energy surface has a global minimum of − .
90 cm − , whichcan be neglected at a collision energy 200 cm − (corresponding to a wave number k = 6 .
49 ˚A − ).A small value of the Massey parameter, ξ ≈ .
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 the2Legendre moments Ξ κ , obtained from a fit to the potential energy surface of Ref. [34], are listed inTable II. Since the He – O potential is of D symmetry, only even Legendre moments are nonzero.Furthermore, since the nuclear spin of O is zero and the electronic ground state antisym-metric (a Σ − g state), only rotational states with an odd rotational quantum number N are allowed.We will assume that the O molecule is initially in its rotational ground state, | N = 1 , J = 0 , M = 0 i .Expression (42) for the scattering amplitude further simplifies for particular geometries. Inwhat follows, we will consider two such geometries.(i) Magnetic field parallel to the initial wave vector, H k k , in which case the scatteringamplitude becomes: f ω m , k , , → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π J | M ′ | ( kR ϑ ) X κ =0 κ even Ξ κ F κM ′ × X NJN ′ J ′ r J + 12 J ′ + 1 a NJ ( ω m ) b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ) C ( JκJ ′ ; 0 M ′ M ′ ) X Ω c Ω NJ c Ω N ′ J ′ C ( JκJ ′ ; Ω0Ω) (43)(ii) Magnetic field perpendicular to the initial wave vector, H ⊥ k , in which case Eq. (42)simplifies to: f ω m , ⊥ , , → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π X κ,ρ even κ =0 Ξ κ d κ − ρ,M ′ ( π ) F κρ J | ρ | ( kR ϑ ) × X NJN ′ J ′ r J + 12 J ′ + 1 a NJ ( ω m ) b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ) C ( JκJ ′ ; 0 M ′ M ′ ) X Ω c Ω NJ c Ω N ′ 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 particularlyeasy to see in the H k k case, where the F κM ′ coefficients vanish for M ′ odd as F κρ vanishes forodd κ + ρ . In the H ⊥ k case, a summation over ρ arises. Since for κ even and M ′ odd the realWigner matrices obey the relation d κ − ρ,M ′ (cid:0) π (cid:1) = − d κρ,M ′ (cid:0) π (cid:1) , the sum over ρ vanishes and so do thepartial cross sections for M ′ odd.
1. Differential cross sections
The differential cross sections of the He – O ( N = 1 , J = 0 → N ′ , J ′ ) collisions, calculatedfrom 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-freecase, ω m = 0, as well as for ω m = 5, which for O 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 ′ | ( kR ϑ ), and is not affected by the magnetic field. Since onlyeven- κ terms contribute to the sum in the scattering amplitude and the coefficients F κρ vanish for κ + ρ odd, the differential cross sections are given solely by even Bessel functions. This is the casefor both parallel and perpendicular geometries. As the elastic scattering amplitude is given by anodd Bessel function, Eq. (6), the elastic and rotationally inelastic differential cross sections oscillatewith an opposite phase.3According to the general properties of Bessel functions [29], at large angles the phase shiftbetween different even Bessel functions disappears and their asymptotic form is given by Eq. (30).For the system under consideration, the phase shift between J ( kR ϑ ) and J ( kR ϑ ) functionsbecomes negligibly small at angles about 40 ◦ , while the shift between J ( kR ϑ ) and either J ( kR ϑ )or J ( kR ϑ ) can only be neglected at angles about 120 ◦ . Therefore, if the cross section is comprisedonly of J and J contributions, it will not be shifted with respect to the field-free case, whilethere may appear a field-iduced phase shift if the J Bessel function also contributes. Indeed, in theparallel case, the 1 , → , J | ρ | ( kR ϑ ) for a range of ρ ’s are mixed, see eq. (44), whichresults in a field-induced phase shift for both the 1 , → , , → , ◦ , since this angular range dominates theintegral 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 sectionsis the onset of inelastic scattering for channels that are closed in the absence of the field: these involvethe transitions N = 1 , J = 0 → N ′ , J ′ with J ′ = N ′ . That these channels are closed in the field-freecase can be gleaned from the scattering amplitude (43) for ω m = 0, which reduces to f F F , , → N ′ ,J ′ ,M ′ ( ϑ ) = ikR π J | M ′ | ( kR ϑ ) Ξ J ′ √ J ′ + 1 F J ′ M ′ c N ′ J ′ (45)This field-free amplitude vanishes because the c N ′ J ′ coefficients are zero for N ′ = J ′ , as can be shownby the diagonalization of the field-free Hamiltonian (34). As a result, the field-free cross sectionsfor the transitions to 1 , , c ± N ′ J ′ which are nonvanishing for N ′ = J ′ . The feature manifests itself in the integralcross sections as well.
2. Integral cross sections
The integral cross sections for the He – O ( N = 1 , J = 0 , M = 0 → N ′ , J ′ , M ′ ) scatteringare shown in Figs. 12 and 13 for H k k and H ⊥ k , respectively.First we note that since the expansion of the He – O potential is dominated by Ξ κ , seeTable II, the sum in Eqs. (43) and (44) over κ can be approximated by the κ = 2 term. In that casethe Clebsch-Gordan coefficient C ( J, κ, J ′ , J and J ′ to terms with J ′ = J ; J ±
2. In the field-free case, this selection rule is only satisfiedfor 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. 12and 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 thevariation of the partial N = 1 , J = 0 , M = 0 → N ′ , J ′ , M ′ contributions. Therefore, we willtake a look at how the M -resolved integral cross sections vary with the magnetic field. The fielddependence of the partial N = 1 , J = 0 , M = 0 → M ′ , J ′ , M ′ cross sections is contained in thehybridization coefficients a NJ ( ω m ) and b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ), both for H k k and H ⊥ k . The a NJ and b ˜ N ′ ˜ J ′ N ′ J ′ coefficients for various values of ˜ N ′ , ˜ J ′ are presented in Figs. 14, 15, and 16 for M = 0 ,
2, and − N = ˜ N , J = ˜ J and are zerootherwise, see Figs. 14–16 (a). Once the field is applied, the “distributions” of the a NJ ( ω m ) and b ˜ N ′ ˜ J ′ N ′ J ′ ( ω m ) coefficients undergo a broadening, as presented in Figs. 14–16 (b)-(d).Such a broadening enhances the mixing of the 1 , , , , , , N = 1 , J = 0 → N ′ = 1 , J ′ = 1 cross section, see Figs. 12 and 13 (b). Onthe other hand, the broadening of the “distribution” of the coefficients of the 1 , , , , , , → , , , , → , , , , , , , , κ = 2 is taken into account), but turning on the field enhances themixing of the 3 , , , , ω m , the spread of coefficients becomes so large that the products a NJ b N ′ J ′ , corresponding to theselection rule J ′ = J ; J ±
2, become very small, cf. Fig. 14 (d). As a result, the cross section for the1 , , → , , M ′ = ± M ′ . These are due to thechanging overlap of the hybridization coefficients, just as for the 3 , , M ′ = 2 and M ′ = − , , − , , − , , → , , − M = 2, there is little mixing of the 3 , , , , , , , , → , , M ′ = ± 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) revealsthat the integral cross sections for the 1 , → , H ⊥ k due to thecoefficients d κ − ρ, ( π ). A similar argument holds in the case of scattering in electrostatic [9] andradiative [10] fields.
3. Frontal-versus-lateral steric asymmetry
Fig. 17 shows the steric asymmetry for the He – O ( N = 1 , J = 0 → N ′ , J ′ ) collisions as afunction 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 fielddependence of the integral cross sections, see Figs. 12 and 13. V. SCATTERING OF Π MOLECULES BY CLOSED-SHELL ATOMS IN MAGNETICFIELDSA. The Π molecule in magnetic field In this Section, we consider a Hund’s case molecule, equivalent to a linear symmetric top. Agood example of such a molecules is the OH radical in its electronic ground state, X Π Ω , whose elec-tronic spin and orbital angular momenta are strongly coupled to the molecular axis. Each rotationalstate within the Π Ω ground state is equivalent to a symmetric-top state | J, Ω , M i with projections5Ω and M of the total angular momentum J on the body- and space-fixed axes, respectively. Dueto a coupling of the Π state with a nearby Σ state [37], the levels with the same Ω are split intonearly-degenerate doublets whose members have opposite parities. The Ω doubling of the Π stateof OH increases as J , whereas that of the Π 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 i = 1 √ (cid:20) | J, M, Ω i + ǫ | J, M, − Ω i (cid:21) (46)where the symmetry index ǫ distinguishes between the members of a given Ω doublet. Here andbelow we use the definition Ω ≡ | Ω | . The symmetry index takes the value of +1 or − e or f levels, respectively. The parity of wave function (46) is equal to ǫ ( − J − [35]. The rotationalenergy level structure of the OH radical in its X Π Ω 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 V m = J Z ω m B (47)with J Z the Z component of the total angular momentum (apart from nuclear spin), J , and ω m ≡ ( g L Λ + g S Σ) µ B H /B (48)Here Λ and Σ are projections of the orbital, L , and spin, S , electronic angular momenta on themolecular axis, g L = 1 and g S ≃ . µ B 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 h J ′ M ′ ǫ ′ | V m | JM ǫ i = ω m B ǫǫ ′ ( − J + J ′ +2Ω ! ( − J + J ′ + M − / × p (2 J + 1)(2 J ′ + 1) j j ′ − Ω 0 Ω ′ ! j ′ jM − M ′ ! (49)where the last two factors are 3 j -symbols [25], [26]. The matrix elements (49) are a generalization ofEqs. (A1)–(A6), and were presented, e.g., in Ref. [39]. For an OH molecule in its ground Π / state,the parity factor, (1 + ǫǫ ′ ( − J + J ′ +2Ω ) /
2, reduces to δ ǫǫ ′ , which means that the Zeeman interactionpreserves parity. The Zeeman eigenstates are hybrids of the field-free states (46) (cid:12)(cid:12)(cid:12) ˜ J, M, Ω , ǫ ; ω m E = X J a ˜ JJM ( ω m ) | J, M, Ω , ǫ i (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 a ˜ JJM ( ω m ) can be obtained by thediagonalization of the Hamiltonian (47) in the basis (46).The dependence of the alignment cosine, h cos θ i , on the field strength parameter ω m isshown in Fig. 18 for the 3 / , f and 5 / , f states of the OH molecule. The matrix elements of the h cos θ i operator are listed in Appendix B. B. The field-dependent scattering amplitude
We consider scattering from the initial J = 3 / , e state to some J ′ , e/f state. As in theprevious Sections, we use Eq. (23) to transform the wavefunctions (46). Considering only the Ω-6conserving transitions (Ω ′ = Ω), the initial and final states are: | i i ≡ (cid:12)(cid:12)(cid:12) ˜ J, M, Ω , ǫ, ω m E = 1 √ X J r J + 14 π a ˜ JJM ( ω m ) X ξ D JξM ( ϕ ε , θ ε , (cid:2) D J ∗ ξ Ω ( ϕ, θ,
0) + ǫ D J ∗ ξ − Ω ( ϕ, θ, (cid:3) (51) h f | ≡ D ˜ J ′ , M ′ , Ω , ǫ ′ , ω m (cid:12)(cid:12)(cid:12) = 1 √ X J ′ r J ′ + 14 π b ˜ J ′ J ′ M ′ ( ω m ) X ξ ′ D J ′ ∗ ξ ′ M ′ ( ϕ ε , θ ε , h D J ′ ξ ′ Ω ′ ( ϕ, θ,
0) + ǫ ′ D J ′ ξ ′ − Ω ′ ( ϕ, θ, i (52)By substituting Eqs. (51) and (52) into Eq. (11), we obtain a closed expression for the scatteringamplitude: f ω m i → f ( ϑ ) = ikR π X κρκ =0 κ + ρ even Ξ κ D κ ∗− ρ, ∆ M ( ϕ ε , θ ε , F κρ J | ρ | ( kR ϑ ) × X JJ ′ r J + 12 J ′ + 1 a ˜ JJM ( ω m ) b ˜ J ′ J ′ M ′ ( ω m ) C ( JκJ ′ ; M ∆ M M ′ ) C ( JκJ ′ ; Ω0Ω) (cid:2) ǫǫ ′ ( − κ +∆ J (cid:3) (53)Eq. (53) simplifies for parallel or perpendicular orientations of the magnetic field with respect to therelative velocity vector.(i) For H k k we have f ω m , k i → f ( ϑ ) = ikR π J | ∆ M | ( kR ϑ ) X κ =0 κ +∆ M even Ξ κ F κ ∆ M × X JJ ′ r J + 12 J ′ + 1 a ˜ JJM ( ω m ) b ˜ J ′ J ′ M ′ ( ω m ) C ( JκJ ′ ; M ∆ M M ′ ) C ( JκJ ′ ; Ω0Ω) (cid:2) ǫǫ ′ ( − κ +∆ J (cid:3) (54)(ii) and for H ⊥ k we obtain f ω m , ⊥ i → f ( ϑ ) = ikR π X κρκ =0 κ + ρ even Ξ κ d κ ∗− ρ, ∆ M ( π ) F κρ J | ρ | ( kR ϑ ) × X JJ ′ r J + 12 J ′ + 1 a ˜ JJM ( ω m ) b ˜ J ′ J ′ M ′ ( ω m ) C ( JκJ ′ ; M ∆ M M ′ ) C ( JκJ ′ ; Ω0Ω) (cid:2) ǫǫ ′ ( − κ +∆ J (cid:3) (55) C. Results for He–OH ( X Π , J = , 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 − .
02 cm − , which could be considered negligible with respect to a collision energy onthe order of 100 cm − , as for the He – CaH and He – O systems treated above. However, theOH molecule has a large rotational constant, B = 18 . − , and so the Massey parameter (2)becomes significantly smaller than unity only at higher energies. Therefore, in order to ensurethe validity of the sudden approximation, we had to work with a collision energy of 1000 cm − ( k = 13 .
86 ˚A − ; Massey parameter ξ ≈ . Π) potential energy surface is shown in Fig. 3, and the Legendre moments, Ξ κ , obtained by fittingthe surface, are listed in Table II.Because of the negative spin-orbit constant, A = − .
051 cm − [37], the Ω doublet ofthe OH( X Π Ω ) molecule is inverted, with the paramagnetic Π / 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) couplingscheme. In what follows, we consider OH ( Π) radicals prepared in the v = 0 , Ω = , J = , f stateby hexapole state selection, like, e.g., in ref. [41]. The molecules enter a magnetic field region wherethey collide with He atoms. The scattered molecules are state-sensitively detected in a field-freeregion.
1. Differential cross sections
We note that due to a large rotational constant, the Zeeman effect in the case of the OHradical is very weak, and so are the field-induced changes of the scattering. The differential crosssections for the He – OH collisions, as obtained from Eqs. (28) and (29), are shown in Figs. 19and 20, together with the elastic scattering cross section obtained from Eq. (6). The differentialcross sections are presented for the field-free case, ω m = 0, as well as for ω m = 5, which for the OHradical corresponds to an extreme field strength of H =99.2 T. First, let us consider the field-freescattering amplitude f w =0 i → f ( ϑ ) = r J + 12 J ′ + 1 J | ∆ M | ( kR ϑ ) × X κ =0 κ +∆ M even Ξ κ F κ, ∆ M C ( JκJ ′ ; M ∆ M M ′ ) C ( JκJ ′ ; Ω0Ω) (cid:2) ǫǫ ′ ( − κ +∆ J (cid:3) (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 F κ, ∆ M coefficient provide a selection rule: ∆ M + ∆ J mustbe 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 withthe 3 / , f → / , f and 3 / , f → / , e cross sections, but out of phase with 3 / , f → / , e and3 / , f → / , f cross sections.For a magnetic field parallel to the relative velocity, H k k , the angular dependence is givenexplicitly by J | ∆ M | ( kR ϑ ), and is seen to be independent of the field, cf. Eq. (54). Therefore, asFig. 19 shows, no field-induced phase shift of the differential cross sections takes place. For H ⊥ k ,a mix of Bessel functions, J | ρ | ( kR ϑ ), contribute to the sum. However, since the Zeeman effect isso weak for the OH molecule, it is the a ˜ JJM ( ω m ), b ˜ J ′ J ′ M ′ ( ω m ) hybridization coefficients with J = ˜ J which provide the main contribution to the sum, even at ω m ≈
5. As a result, no contributions fromhigher Bessel functions are drawn in, and so no significant field-induced phase shift is observed forthe perpendicular case either.
2. Integral cross sections
The integral cross sections for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions are presentedin Fig. 21 for J ′ = 5 / , / M -averaged integral cross sections (here 3 / , f → J ′ , e/f )on the magnetic field can be traced to the field-dependences of the partial M - and M ′ -dependentcontributions. Since for OH ( J = 3 / , f ), the initial state comprises four M values and the final statesix or eight M ′ values, a discussion of how the M - and M ′ -averaged cross sections come about wouldbe rather involved. Therefore, we resort to considering an example, namely how the 3 / , f → / , f cross section arises from the 3 / , f, M → / , f, M ′ components, shown in Fig. 22 for a magneticfield parallel to the relative velocity. From Table II, we see that the He – OH potential is dominatedby odd Ξ κ terms – in contrast to the He – CaH and He – O potentials. Therefore, odd ∆ M values will yield the main contribution to the scattering amplitude (54), because of the selectionrule which dictates that κ + ∆ M be even. This effect can be clearly discerned in Fig. 22. In thefield-free case, even ∆ M transitions have very small amplitudes, cf. Eq. (56). When the field is on,the corresponding cross sections increase with ω m due to the increasing overlap of the a ˜ JJM ( ω m ) and b ˜ J ′ J ′ M ′ ( ω m ) coefficients. This is a situation analogous to the one described in detail in Sec. IV C 2 forthe O – He system (see also Figs. 15, 16).
3. Frontal-versus-lateral steric asymmetry
The steric asymmetry for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions, calculated by meansof Eq. (33), is presented in Fig. 23. The most pronounced asymmetry is observed for the 5 / , e and7 / , f channels, while the asymmetry for the J = 3 / , f → / , f and J = 3 / , f → / , e channelsis almost flat, especially at the feasible magnetic field strengths of up to 20 T. The difference betweenthe scattering for parallel and perpendicular orientations of the magnetic field with respect to theinitial velocity is contained in the d κ ∗− ρ, ∆ M ( π ) matrices, appearing in Eq. (55), and the observedtrends can be gleaned from Eqs. (54) and (55). VI. CONCLUSIONS
We extended the Fraunhofer theory of matter wave scattering to tackle rotationally inelasticcollisions of paramagnetic, open shell molecules with closed-shell atoms in magnetic fields. Thedescription is inherently quantum and, therefore, capable of accounting for interference and othernon-classical effects. The effect of the magnetic field enters the model via the directional propertiesof the molecular states, which exhibit alignment of the molecular axis induced by the magnetic field.We applied the model to the He–CaH( X Σ , J = 1 / → J ′ ) and He–O ( X Σ , N = 0 , J = 1 → N ′ , J ′ )scattering at a collision energy of 200 cm − , as well as to the He–OH( X Π , J = 3 / , f → J ′ , e/f )collisions at an energy of 1000 cm − .In this Section, we mull over the results for the three collision systems and point out whatthey 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 scatteringare due to the Zeeman effect of the final state. The magnetic eigenproperties of a Σ molecule canbe obtained in closed form, by diagonalizing a 2 × Σ case, the Hamiltonian matrices of the Σ and Π molecules in amagnetic field are in principle infinite. In practice, the Zeeman interaction couples a range ofrotational states which is limited by the strength of the interaction to less than ten for ω m ≤ collision system exhibits a dramatic feature: in the absence of a magneticfield, the scattering vanishes for channels leading to the F manifold, i.e., final states with J ′ = N ′ .However, in the presence of the field, such transitions become allowed, and, although weak, should beobservable. Also, some of the He – O integral cross sections are non-monotonous – unlike the crosssections of the He – CaH and He – OH systems. They exhibit maxima or minima, which dependcharacteristically on the sign of M ′ , see Figs. 12 and 13. This contributes, for some channels, tothe strong dependence of the He – O cross sections on the orientation of the magnetic field withrespect 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 bythe magnetic field, see Fig. 18. As a result, the field-induced changes of the scattering cross sectionsare tiny, Figs. 19 – 21, and so is the variation with field of the steric asymmetry, Fig. 23. Unlike theHe – CaH and He – O systems, the equipotential line on the He – OH potential energy surface isdominated by odd Legendre moments, see Table II. This gives rise to scattering features which arequalitatively different from those of the other systems. For instance, as described in Sec. V C 2, it isthe odd ∆ M transitions that dominate the He – OH ( J = 3 / , f → J ′ , e/f ) integral cross sectionsin 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 – O 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 geometricaleffects and to qualitatively explain the resulting scattering features. These include the angularoscillations in the state-to-state differential cross sections or the rotational-state dependent variationof the integral cross sections as functions of the magnetic field. We hope that the model will inspirenew collisional experiments that make use either of crossed molecular beams or of a combination ofa magnetic trap with a hot beam of atoms. Acknowledgments
Our special thanks are due to Gerard Meijer for discussions and support, and to ElenaDashevskaya and Evgueni Nikitin for their most helpful comments. We greatly enjoyed discussionswith Bas van de Meerakker, Ludwig Scharfenberg, Joop Gilijamse, and Steven Hoekstra.
APPENDIX A: MATRIX ELEMENTS OF THE J Z OPERATOR
In general, the Zeeman operator is proportional to the projection, J Z , of the total electronicangular momentum, J , on the space-fixed field axis, Z , see e.g. eq. (47). In this Appendix we presentthe matrix elements of the J Z operator, employed in this work. For Σ electronic states, J Z reducesto S Z .0We transform the angular momentum projection operator from the body-fixed to the space-fixed coordinates using the direction cosines operator, Φ: J Z = (cid:0) Φ + Z J − + Φ − Z J + (cid:1) + Φ zZ J z (A1)The matrix elements of the body-fixed spin operator in the Hund’s case (a) basis, | J, Ω , M i , aregiven by the standard relations [42]: h J, Ω , M | J ± | J, Ω ∓ , M i = p ( J ± Ω)( J ∓ Ω + 1) (A2) h J, Ω , M | J z | J, Ω , M i = 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-vanishingmatrix elements for M ′ = M : h J ′ , Ω , M | Φ zZ | J, Ω , M i = Ω MJ ( J +1) J ′ = J √ ( J +Ω+1)( J − Ω+1)( J + M +1)( J − M +1)( J +1) √ (2 J +1)(2 J +3) J ′ = J + 1 √ ( J +Ω)( J − Ω)( J + M )( J − M ) J √ (2 J +1)(2 J − J ′ = J − h J ′ , Ω − , M | Φ + Z | J, Ω , M i = 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) √ (2 J +1)(2 J +3) J ′ = J + 1 − √ ( J +Ω)( J +Ω − J + M )( J − M ) J √ (2 J +1)(2 J − J ′ = J − h J ′ , Ω + 1 , M | Φ − Z | J, Ω , M i = 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) √ (2 J +1)(2 J +3) J ′ = J + 1 √ ( J − Ω)( J − Ω − J + M )( J − M ) J √ (2 J +1)(2 J − J ′ = J − APPENDIX B: MATRIX ELEMENTS OF THE (Φ zZ ) OPERATOR
The matrix elements of the alignment cosine can be reduced to the matrix elements of theΦ zZ operator by means of the full basis set: h cos θ i = h J ′ , Ω , M | (Φ zZ ) | J, Ω , M i = X J ′′ h J ′ , Ω , M | Φ zZ | J ′′ , Ω , M ih J ′′ , Ω , M | Φ zZ | J, Ω , M i (B1)By taking into account that the matrix elements (A4) are nonzero only for ∆ J = 0 , ±
1, we obtainthe matrix elements of the direction cosine operator: h J, Ω , M | (Φ zZ ) | J, Ω , M i = |h J − , Ω , M | Φ zZ | J, Ω , M i| + |h J, Ω , M | Φ zZ | J, Ω , M i| + |h J + 1 , Ω , M | Φ zZ | J, Ω , M i| (B2)1 h J − , Ω , M | (Φ zZ ) | J, Ω , M i = h J − , Ω , M | Φ zZ | J, Ω , M i (cid:26) h J, Ω , M | Φ zZ | J, Ω , M i + h J − , Ω , M | Φ zZ | J − , Ω , M i (cid:27) (B3) h J +1 , Ω , M | (Φ zZ ) | J, Ω , M i = h J +1 , Ω , M | Φ zZ | J, Ω , M i (cid:26) h J, Ω , M | Φ zZ | J, Ω , M i + h J +1 , Ω , M | Φ zZ | J +1 , Ω , M i (cid:27) (B4) h J − , Ω , M | (Φ zZ ) | J, Ω , M i = h J − , Ω , M | Φ zZ | J − , Ω , M ih J − , Ω , M | Φ zZ | J, Ω , M i (B5) h J + 2 , Ω , M | (Φ zZ ) | J, Ω , M i = h J + 2 , Ω , M | Φ zZ | J + 1 , Ω , M ih J + 1 , Ω , M | Φ zZ | J, Ω , M i (B6) APPENDIX C: THE ALIGNMENT COSINE OF THE Σ MOLECULE IN A MAGNETICFIELD
Within the Hund’s (b) basis functions, eq. (16), the expectation value of the alignmentcosine takes the form: h cos θ i = a ( ω m ) (cid:10) N − , Ω , M (cid:12)(cid:12) cos θ (cid:12)(cid:12) N − , Ω , M (cid:11) + b ( ω m ) (cid:10) N + , Ω , M (cid:12)(cid:12) cos θ (cid:12)(cid:12) N + , Ω , M (cid:11) + 2 a ( ω m ) b ( ω m ) (cid:10) N − , Ω , M (cid:12)(cid:12) cos θ (cid:12)(cid:12) N + , Ω , M (cid:11) , (C1)where the matrix elements of the cos θ operator in the Hund’s case (a) basis can be obtainedfrom (B2) and (B4): (cid:10) J, Ω , M (cid:12)(cid:12) cos θ (cid:12)(cid:12) J, Ω , M (cid:11) = 13 + 23 (cid:2) J ( J + 1) − M (cid:3) (cid:2) J ( J + 1) − (cid:3) J ( J + 1)(2 J − J + 3) (C2) (cid:10) J, Ω , M (cid:12)(cid:12) cos θ (cid:12)(cid:12) J + 1 , Ω , M (cid:11) = 2Ω M p [( J + 1) − M ] [( J + 1) − Ω ] J ( J + 1)( J + 2) p (2 J + 1)(2 J + 3) (C3)The coefficients a ( ω m ) and b ( ω m ) are given by the solution of the Zeeman problem, Eqs. (21)–(22). [1] Special issue on cold molecules , Eur. Phys. J. D , 149 (2004)[2] Special issue on coherent control , J. Phys. B , 074001 (2008)[3] Special issue on stereodynamics , Eur. Phys. J. D , 3 (2006)[4] R. V. Krems, A. Dalgarno, J. Chem. Phys , 2296 (2004).[5] H. R. Sadeghpour, J. L. Bohn, M. J. Cavagnero, B. D. Esry, I. I. Fabrikant, J. H. Macek, andA. R. P. Rau, J. Phys. B. , R93 (2000).[6] S. I. Drozdov, Soviet. Phys. JETP. , 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. , 5559 (1984).[9] M. Lemeshko, B. Friedrich, J. Chem. Phys., , 024301 (2008).[10] M. Lemeshko, B. Friedrich, Int. J. Mass. Spec., in press; see arXiv:0804.4845 (2008). [11] A. I. Maergoiz, E. E. Nikitin, J. Troe, V. G. Ushakov, J. Chem. Phys. , 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. , 419 (2000).[14] A. Boca, B. Friedrich, J. Chem. Phys. , 3609 (2000).[15] J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich, and J. M. Doyle, Nature(London) , 148(1998).[16] B. Friedrich, J. D. Weinstein, R. deCarvalho, and J. M. Doyle, J. Chem. Phys. , 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 , 107-117 (1979).[19] A. Ichimura, M. Nakamura, Phys. Rev. A , 022716 (2004);M. Nakamura, A. Ichimura, Phys. Rev. A , 062701 (2005).[20] A. J. Marks, J. Chem. Soc. Far. Trans. , 2857 (1994).[21] S. Bosanac, Phys. Rev. A , 282 (1982).[22] A. Gijsbertsen, H. Linnartz, C. A. Taatjes, and S. Stolte, J. Am. Chem. Soc , 8777 (2006);C. A. Taatjes, A. Gijsbertsen, M. J. L. de Lange, and S. Stolte, J. Chem. Phys , 7631 (2007).[23] M. Born and E. Wolf, Principles of optics , 7th ed., Cambridge University Press (1997).[24] H. Lefebvre-Brion, R. W. Field,
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Quantum Mechanics, Non-Relativistic Theory , Oxford: Pergamon (1977). TABLE I: The hybridization coefficients a ( ω m ) and b ( ω m ) for the N = 2 , J = , M state in the high-fieldlimit, ω m ≫ ∆ E/B , which arises for ω m ≫ .
025 for the N = 2 level of the CaH( X Σ + ) molecule. See text. M a ( ω m ) b ( ω m ) q q - q q √ √ -
32 2 √ √ ± TABLE II: Hard-shell Legendre moments Ξ κ for He – CaH ( X Σ + ) and He – O ( X Σ − ) potential energysurfaces at a collision energy of 200 cm − , and for He – OH ( X Π ) potential at 1000 cm − .Ξ κ (˚A) κ He–CaH He–O He–OH0 13.3207 9.5987 7.79411 -0.4397 0 0.13802 1.0140 0.5672 0.16253 0.6147 0 0.09614 0.0337 -0.1320 0.017895 -0.1475 0 -0.00326 -0.0653 0.0250 -0.00347 0.0265 0 -0.00088 0.0277 -0.0060 0.0002 FIG. 1: Schematic of Fraunhofer diffraction by an impenetrable, sharp-edged obstacle as observed at apoint 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. (a)
1, 3/2, -1/21, 3/2, ±3/2
1, 1/2, 1/2 (b)
1, 1/2, -1/2
2, 5/2, 1/2 (c)
2, 5/2, -1/22, 5/2, ±5/22, 5/2, 3/22, 5/2, -3/2 (d)
2, 3/2, -1/22, 3/2, 3/22, 3/2, -3/2 ! m A li gn m e n t c o s i n e H (Tesla) FIG. 2: Expectation values of the alignment cosine h cos θ i for the Zeeman states of CaH( X Σ + ) as afunction of the magnetic field strength parameter ω m . States are labeled as ˜ N , ˜ J, M , see text. o o o o o o o o o o R ( Å ) CaHOHO FIG. 3: Equipotential lines R ( θ ) for the He – CaH ( X Σ + ) and He – O ( X Σ − ) potential energy surfacesat a collision energy of 200 cm − , and for the He – OH ( X Π Ω ) potential at 1000 cm − . The Legendremoments for the potential energy surfaces are listed in Table II. (a) 0,1/2 (b) 1,3/2 (c) 1,1/2 (d) 2,5/2 D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) o o o o o o o (e) 2,3/2 ! FIG. 4: Differential cross sections for the He – CaH ( N = 0 , J = → N ′ , J ′ ) collisions in a magnetic field ω m = 0 . H k k . The field-free cross sections areshown by the green solid line. The dashed vertical line serves to guide the eye in discerning the angularshifts of the partial cross sections. (a) 0,1/2 (b) 1,3/2 (c) 1,1/2 (d) 2,5/2 D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) o o o o o o o (e) 2,3/2 ! FIG. 5: Differential cross sections for the He – CaH ( N = 0 , J = → N ′ , J ′ ) collisions in a magneticfield ω m = 0 . H ⊥ k . The field-free crosssections are shown by the green solid line. The dashed vertical line serves to guide the eye in discerning theangular shifts of the partial cross sections. M = M' = M = -1/2, M' = -3/2 N' = J' = (a) M = M' = -1/2 M = -1/2, M' = N' = J' = M = M' = -1/2 M = -1/2, M' = (b) M = M' = -3/2 N' = J' = (c) M = -1/2, M' = M = M' = M = -1/2, M' = -1/2 M = M' = M = -1/2, M' = -5/2 M = M' = -3/2 N' = J' = (d) M = -1/2, M' = M = M' = M = -1/2, M' = -1/2 ! m P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) FIG. 6: Partial integral cross sections for the He – CaH ( N = 0 , J = , M → N ′ , J ′ , M ′ ) collisions in amagnetic field parallel to the initial wave vector, H k k . The red solid lines show the M ′ -averaged crosssections for the ( N = 0 , J = → N ′ , J ′ ) collisions. N' = J' = (a) M = M' = M = -1/2, M' = -1/2 N' = J' = M = M' = M = -1/2, M' = -1/2 (b) M = M' = -3/2 N' = J' = (c) M = -1/2, M' = M = M' = M = -1/2, M' = -1/2 M = M' = M = -1/2, M' = -5/2 M = M' = -3/2 N' = J' = (d) M = -1/2, M' = M = M' = M = -1/2, M' = -1/2 ! m P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) FIG. 7: Partial integral cross sections for the He – CaH ( N = 0 , J = , M → N ′ , J ′ , M ′ ) collisions in amagnetic field perpendicular to the initial wave vector, H ⊥ k . The red solid lines show the M ′ -averagedcross sections for the ( N = 0 , J = → N ′ , J ′ ) collisions. -0.4-0.200.20.4 0 0.1 0.2 0.3 N' = J' = ! m H (Tesla) S t e r i c a s y mm e t r y FIG. 8: Steric asymmetry, as defined by Eq. (33), for the He – CaH ( N = 0 , J = → N ′ , J ′ ) collisions.Curves are labeled by N ′ , J ′ . ! m A li gn m e n t c o s i n e H (Tesla) FIG. 9: Expectation values of the alignment cosine h cos θ i for the lowest Zeeman states of O ( X Σ − ) asa function of the magnetic field strength parameter ω m . States are labeled by ˜ N, ˜ J, M . (a) 1,0 (b) 1,2 (c) 1,1 (d) 3,2 D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) (e) 3,4 -6 -5 o o o o o o o (f) 3,3 ! FIG. 10: Differential cross sections for the He – O ( N = 1 , J = 0 → N ′ , J ′ ) collisions in a magnetic field ω m = 5 (red dashed line) parallel to the relative velocity vector, H k k . The field-free cross sections areshown by the green solid line. The dashed vertical line serves to guide the eye in discerning the angularshifts of the partial cross sections. The field-free cross sections for the scattering to final states with J ′ = N ′ vanish, see text. (a) 1,0 (b) 1,2 (c) 1,1 (d) 3,2 D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) (e) 3,4 -6 -5 o o o o o o o (f) 3,3 ! FIG. 11: Differential cross sections for the He – O ( N = 1 , J = 0 → N ′ , J ′ ) collisions in a magnetic field ω m = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k . The field-free cross sectionsare shown by the green solid line. The dashed vertical line serves to guide the eye in discerning the angularshifts of the partial cross sections. The field-free cross sections for the scattering to final states with J ′ = N ′ vanish, see text. P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) (a) (b) (c) (d) ! m (e) FIG. 12: Partial integral cross sections for the He – O ( N = 1 , J = 0 , M = 0 → N ′ , J ′ , M ′ ) collisions in amagnetic field parallel to the initial wave vector, H k k . Curves are labeled by N ′ , J ′ , M ′ . The red solidlines show the M ′ -averaged cross sections. The partial cross sections, corresponding to negative M ′ , areshown by dashed lines. P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) (a) (b) (c) (d) (e) ! m FIG. 13: Partial integral cross sections for the He – O ( N = 1 , J = 0 , M = 0 → N ′ , J ′ , M ′ ) collisions in amagnetic field perpendicular to the initial wave vector, H ⊥ k . Curves are labeled by N ′ , J ′ , M ′ . The redsolid lines show the M ′ -averaged cross sections. The partial cross sections, corresponding to negative M ′ ,are shown by dashed lines. (a) ! = m N,J (b) ! = m (c) ! = m (d) ! = m N,J
FIG. 14: Absolute values of the hybridization coefficients a NJ ( ω m ) (black dashed line, full circles) and b ˜ N ˜ JNJ ( ω m ) for different values of the interaction parameter ω m , for the O ( X Σ − ) molecule. The following b ˜ N ˜ JNJ ( ω m ) coefficients are presented: ˜ N = 1 , ˜ J = 2 (red solid line, full diamonds), ˜ N = 1 , ˜ J = 1 (blue dashedline, full squares), ˜ N = 3 , ˜ J = 2 (green solid line, empty circles), ˜ N = 3 , ˜ J = 4 (orange dashed line, emptydiamonds), ˜ N = 3 , ˜ J = 3 (light blue solid line, empty squares). For all coefficients M = 0. See text. (a) ! = m N,J (b) ! = m (c) ! = m (d) ! = m N,J
FIG. 15: Absolute values of the hybridization coefficients b ˜ N ˜ JNJ ( ω m ) for different values of the interactionparameter ω m , for the O ( X Σ − ) molecule. The following b ˜ N ˜ JNJ ( ω m ) 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 dashedline, empty diamonds), ˜ N = 3 , ˜ J = 3 (light blue solid line, empty squares). For all coefficients M = 2. Seetext. (a) ! = m N,J (b) ! = m (c) ! = m (d) ! = m N,J
FIG. 16: Absolute values of the hybridization coefficients b ˜ N ˜ JNJ ( ω m ) for different values of the interactionparameter ω m , for the O ( X Σ − ) molecule. The following b ˜ N ˜ JNJ ( ω m ) 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 dashedline, empty diamonds), ˜ N = 3 , ˜ J = 3 (light blue solid line, empty squares). For all coefficients M = −
2. Seetext. -1-0.500.51 0 1 2 3 4 5 6 70 1 2 3 4 5 ! m H (Tesla) S t e r i c a s y mm e t r y FIG. 17: Steric asymmetry, as defined by Eq. (33), for the He – O ( N = 1 , J = 0 → N ′ , J ′ ) collisions.Curves are labeled by N ′ , J ′ . (a) M = M = ! M = M = ! (b) M = M = ! M = M = ! M = M = ! " m A li gn m e n t c o s i n e H (Tesla) FIG. 18: Expectation values of the alignment cosine h cos θ i for the 3 / , f (a) and 5 / , f (b) states of theOH molecule, as a function of the magnetic field strength parameter ω m . (a) 3/2, f -5 (b) 5/2, f -5 (c) 5/2, e D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) -5 (d) 7/2, f -5 o o o o o o o (e) 7/2, e ! FIG. 19: Differential cross sections for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions in a magnetic field ω m = 5 (red dashed line) parallel to the relative velocity vector, H k k . The field-free cross sections areshown by the green solid line. The dashed vertical line serves to guide the eye in discerning the angularshifts of the partial cross sections. See text. (a) 3/2, f -5 (b) 5/2, f -5 (c) 5/2, e D i ff e r e n ti a l c r o ss - s ec ti on ( Å / s t e r a d ) -5 (d) 7/2, f -5 o o o o o o o (e) 7/2, e ! FIG. 20: Differential cross sections for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions in a magnetic field ω m = 5 (red dashed line) perpendicular to the relative velocity vector, H ⊥ k . The field-free cross sectionsare shown by the green solid line. The dashed vertical line serves to guide the eye in discerning the angularshifts of the partial cross sections. See text. P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) (a) (b) ! m FIG. 21: Partial integral cross sections for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions in a magnetic field(a) parallel, H k k , and (b) perpendicular, H ⊥ k , to the initial wave vector. Curves are labeled by J ′ , e/f . P a r ti a l i n t e g r a l c r o ss s ec ti on s ( Å ) H (Tesla) -5 (a) M' = M' = ! M' = M' = ! M' = ! M' = -5 (b) M' = M' = ! M' = M' = ! M' = ! M' = -5 M' = M' = ! M' = M' = ! M' = ! M' = (c) -5 (d) M' = M' = ! M' = M' = ! " m FIG. 22: Logarithm of the partial integral cross sections for the He – OH ( J = 3 / , f, M → J ′ = 5 / , f, M ′ )collisions in a magnetic field parallel to the initial wave vector, H k k . The panels correspond to differentinitial states: M = 1 / M = − / M = 3 / M = − / -0.2-0.100.10.2 0 20 40 60 800 1 2 3 4 5 ! m H (Tesla) S t e r i c a s y mm e t r y FIG. 23: Steric asymmetry, as defined by Eq. (33), for the He – OH ( J = 3 / , f → J ′ , e/f ) collisions. Curvesare labeled by J ′ , e/f, e/f