Collisions of paramagnetic molecules in magnetic fields: An analytic model based on Fraunhofer diffraction of matter waves

We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions of open-shell molecules (${}^2\Sigma$, ${}^3\Sigma$, and ${}^2\Pi$) 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. 280, 19 (2009)]. 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 $\left(X{}^2\Sigma ^{+}\right)$, He-${\mathrm{O}}_2$ $\left(X{}^3\Sigma ^{–}\right)$, and He-OH $\left(X{}^2\Pi _{\Omega }\right)$ 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-${\mathrm{O}}_2$. While odd $\Delta M$ transitions dominate the He-OH $[J=3∕2,f\to J^{\prime },e/f]$ 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 $\left(X^2{\Sigma }^{+}\right)$ and He-${\mathrm{O}}_2$ $\left(X{}^3\Sigma ^{-}\right)$ 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 $N=1,J=0\to N^{\prime }$, $J^{\prime }$ with $J^{\prime }=N^{\prime }$.

Figure 1

(Color online) Schematic of Fraunhofer diffraction by an impenetrable, sharp-edged obstacle as observed at a point of radius vector $\mathbf{r}$ from the obstacle. Relevant is the shape of the obstacle in the $XY$ plane, perpendicular to the initial wave vector $\mathbf{k}$, itself directed along the $Z$ axis of the space-fixed system $XYZ$. The angle $\phi$ is the azimuthal angle of the radius vector $\mathbf{R}$ which traces the shape of the obstacle in the $X,Y$ plane and $\vartheta$ is the scattering angle (see the text).

Figure 2

(Color online) Expectation values of the alignment cosine $⟨{\cos }^2\theta ⟩$ for the Zeeman states of $\mathrm{Ca}\mathrm{H}\left(X{}^2\Sigma ^{+}\right)$ as a function of the magnetic field strength parameter ${\omega }_m$. States are labeled as $\tilde{N}$, $\tilde{J}$, $M$ (see the text).

Figure 3

(Color online) Equipotential lines $R\left(\theta \right)$ for the He-CaH $\left(X{}^2\Sigma ^{+}\right)$ and $\mathrm{He}\text{−}{\mathrm{O}}_2$ $\left(X{}^3\Sigma ^{-}\right)$ potential energy surfaces at a collision energy of $200{\mathrm{cm}}^{-1}$, and for the He-OH $\left(X{}^2\Pi _{\Omega }\right)$ potential at $1000{\mathrm{cm}}^{-1}$. The Legendre moments for the potential energy surfaces are listed in Table .

Figure 4

(Color online) Differential cross sections for the He-CaH $(N=0,J=\frac{1}{2}\to N^{\prime },J^{\prime })$ collisions in a magnetic field ${\omega }_m=0.3$ (blue dashed line) parallel to the relative velocity vector, $\mathcal{H}\parallel \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $200{\mathrm{cm}}^{-1}$. The dashed vertical line serves to guide the eye in discerning the angular shifts of the partial cross sections.

Figure 5

(Color online) Differential cross sections for the He-CaH $(N=0,J=\frac{1}{2}\to N^{\prime },J^{\prime })$ collisions in a magnetic field ${\omega }_m=0.3$ (blue dashed line) perpendicular to the relative velocity vector, $\mathcal{H}\perp \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $200{\mathrm{cm}}^{-1}$. The dashed vertical line serves to guide the eye in discerning the angular shifts of the partial cross sections.

Figure 6

(Color online) Partial integral cross sections for the He-CaH $(N=0,J=\frac{1}{2},M\to N^{\prime },J^{\prime },M^{\prime })$ collisions in a magnetic field parallel to the initial wave vector, $\mathcal{H}\parallel \mathbf{k}$. Elastic integral cross section ${\sigma }_{\mathrm{el}}=43.5{\mathrm{\AA }}^2$ does not change with the field. The collision energy is $200{\mathrm{cm}}^{-1}$. The red solid lines show the $M^{\prime }$ averaged cross sections for the $(N=0,J=\frac{1}{2}\to N^{\prime },J^{\prime })$ collisions.

Figure 7

(Color online) Partial integral cross sections for the He-CaH $(N=0,J=\frac{1}{2},M\to N^{\prime },J^{\prime },M^{\prime })$ collisions in a magnetic field perpendicular to the initial wave vector, $\mathcal{H}\perp \mathbf{k}$. Elastic integral cross section ${\sigma }_{\mathrm{el}}=43.5{\mathrm{\AA }}^2$ does not change with the field. The collision energy is $200{\mathrm{cm}}^{-1}$. The red solid lines show the $M^{\prime }$ averaged cross sections for the $(N=0,J=\frac{1}{2}\to N^{\prime },J^{\prime })$ collisions.

Figure 8

(Color online) Steric asymmetry, as defined by Eq. (34), for the He-CaH $(N=0,J=\frac{1}{2}\to N^{\prime },J^{\prime })$ collisions at the energy of $200{\mathrm{cm}}^{-1}$. Curves are labeled by $N^{\prime },J^{\prime }$.

Figure 9

(Color online) Expectation values of the alignment cosine $⟨{\cos }^2\theta ⟩$ for the lowest Zeeman states of ${}^{16}\mathrm{O}_2\left(X{}^3\Sigma ^{-}\right)$ as a function of the magnetic field strength parameter ${\omega }_m$. States are labeled by $\tilde{N},\tilde{J},M$.

Figure 10

(Color online) Differential cross sections for the $\mathrm{He}\text{−}{\mathrm{O}}_2$ $(N=1,J=0\to N^{\prime },J^{\prime })$ collisions in a magnetic field ${\omega }_m=5$ (red dashed line) parallel to the relative velocity vector, $\mathcal{H}\parallel \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $200{\mathrm{cm}}^{-1}$. 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 $J^{\prime }=N^{\prime }$ vanish (see the text).

Figure 11

(Color online) Differential cross sections for the $\mathrm{He}\text{−}{\mathrm{O}}_2$ $(N=1,J=0\to N^{\prime },J^{\prime })$ collisions in a magnetic field ${\omega }_m=5$ (red dashed line) perpendicular to the relative velocity vector, $\mathcal{H}\perp \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $200{\mathrm{cm}}^{-1}$. 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 $J^{\prime }=N^{\prime }$ vanish (see the text).

Figure 12

(Color online) Partial integral cross sections for the $\mathrm{He}\text{−}{\mathrm{O}}_2$ $(N=1,J=0,M=0\to N^{\prime },J^{\prime },M^{\prime })$ collisions in a magnetic field parallel to the initial wave vector, $\mathcal{H}\parallel \mathbf{k}$. Elastic integral cross section ${\sigma }_{\mathrm{el}}=22.4{\mathrm{\AA }}^2$ does not change with the field. Curves are labeled by $N^{\prime },J^{\prime },M^{\prime }$. The red solid lines show the $M^{\prime }$ averaged cross sections. The partial cross sections, corresponding to negative $M^{\prime }$, are shown by dashed lines. The collision energy is $200{\mathrm{cm}}^{-1}$.

Figure 13

(Color online) Partial integral cross sections for the $\mathrm{He}\text{−}{\mathrm{O}}_2$ $(N=1,J=0,M=0\to N^{\prime },J^{\prime },M^{\prime })$ collisions in a magnetic field perpendicular to the initial wave vector, $\mathcal{H}\perp \mathbf{k}$. Elastic integral cross section ${\sigma }_{\mathrm{el}}=22.4{\mathrm{\AA }}^2$ does not change with the field. Curves are labeled by $N^{\prime },J^{\prime },M^{\prime }$. The red solid lines show the $M^{\prime }$-averaged cross sections. The partial cross sections, corresponding to negative $M^{\prime }$, are shown by dashed lines. The collision energy is $200{\mathrm{cm}}^{-1}$.

Figure 14

(Color online) Absolute values of the hybridization coefficients $a_{NJ}^{10}\left({\omega }_m\right)$ (black dashed line, full circles) and $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ for different values of the interaction parameter ${\omega }_m$, for the ${\mathrm{O}}_2\left(X{}^3\Sigma ^{-}\right)$ molecule. The following $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ coefficients are presented: $\tilde{N}=1$, $\tilde{J}=2$ (red solid line, full diamonds), $\tilde{N}=1$, $\tilde{J}=1$ (blue dashed line, full squares), $\tilde{N}=3$, $\tilde{J}=2$ (green solid line, empty circles), $\tilde{N}=3$, $\tilde{J}=4$ (orange dashed line, empty diamonds), and $\tilde{N}=3$, $\tilde{J}=3$ (light blue solid line, empty squares). For all coefficients $M=0$ (see the text).

Figure 15

(Color online) Absolute values of the hybridization coefficients $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ for different values of the interaction parameter ${\omega }_m$, for the ${\mathrm{O}}_2\left(X{}^3\Sigma ^{-}\right)$ molecule. The following $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ coefficients are presented: $\tilde{N}=1$, $\tilde{J}=2$ (red solid line, full diamonds), $\tilde{N}=3$, $\tilde{J}=2$ (green solid line, empty circles), $\tilde{N}=3$, $\tilde{J}=4$ (orange dashed line, empty diamonds), and $\tilde{N}=3$, $\tilde{J}=3$ (light blue solid line, empty squares). For all coefficients $M=2$ (see the text).

Figure 16

(Color online) Absolute values of the hybridization coefficients $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ for different values of the interaction parameter ${\omega }_m$, for the ${\mathrm{O}}_2\left(X{}^3\Sigma ^{-}\right)$ molecule. The following $b_{NJ}^{\tilde{N}\tilde{J}}\left({\omega }_m\right)$ coefficients are presented: $\tilde{N}=1$, $\tilde{J}=2$ (red solid line, full diamonds), $\tilde{N}=3$, $\tilde{J}=2$ (green solid line, empty circles), $\tilde{N}=3$, $\tilde{J}=4$ (orange dashed line, empty diamonds), and $\tilde{N}=3$, $\tilde{J}=3$ (light blue solid line, empty squares). For all coefficients $M=-2$ (see the text).

Figure 17

(Color online) Steric asymmetry, as defined by Eq. (34), for the $\mathrm{He}\text{−}{\mathrm{O}}_2$ $(N=1,J=0\to N^{\prime },J^{\prime })$ collisions at the energy of $200{\mathrm{cm}}^{-1}$. Curves are labeled by $N^{\prime },J^{\prime }$.

Figure 18

(Color online) Expectation values of the alignment cosine $⟨{\cos }^2\theta ⟩$ 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 ${\omega }_m$.

Figure 19

(Color online) Differential cross sections for the He-OH $(J=3∕2,f\to J^{\prime },e∕f)$ collisions in a magnetic field ${\omega }_m=5$ (red dashed line) parallel to the relative velocity vector, $\mathcal{H}\parallel \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $1000{\mathrm{cm}}^{-1}$. The dashed vertical line serves to guide the eye in discerning the angular shifts of the partial cross sections (see the text).

Figure 20

(Color online) Differential cross sections for the He-OH $(J=3∕2,f\to J^{\prime },e∕f)$ collisions in a magnetic field ${\omega }_m=5$ (red dashed line) perpendicular to the relative velocity vector, $\mathcal{H}\perp \mathbf{k}$. The field-free cross sections are shown by the green solid line. The collision energy is $1000{\mathrm{cm}}^{-1}$. The dashed vertical line serves to guide the eye in discerning the angular shifts of the partial cross sections (see the text).

Figure 21

(Color online) Partial integral cross sections for the He-OH $(J=3∕2,f\to J^{\prime },e∕f)$ collisions in a magnetic field (a) parallel, $\mathcal{H}\parallel \mathbf{k}$, and (b) perpendicular, $\mathcal{H}\perp \mathbf{k}$, to the initial wave vector. The collision energy is $1000{\mathrm{cm}}^{-1}$. Curves are labeled by $J^{\prime },e∕f$. Elastic integral cross section ${\sigma }_{\mathrm{el}}=14.9{\mathrm{\AA }}^2$ does not change with the field.

Figure 22

(Color online) Logarithm of the partial integral cross sections for the He-OH $(J=3∕2,f,M\to J^{\prime }=5∕2,f,M^{\prime })$ collisions in a magnetic field parallel to the initial wave vector, $\mathcal{H}\parallel \mathbf{k}$. The collision energy is $1000{\mathrm{cm}}^{-1}$. The panels correspond to different initial states: $M=1∕2$ (a), $M=-1∕2$ (b), $M=3∕2$ (c), and $M=-3∕2$ (d). All partial cross sections are nonvanishing, but the puny ones are not shown (see the text).

Figure 23

(Color online) Steric asymmetry, as defined by Eq. (34), for the He-OH $(J=3∕2,f\to J^{\prime },e∕f)$ collisions at the energy of $1000{\mathrm{cm}}^{-1}$. Curves are labeled by $J^{\prime },e∕f$.