Shaffer Research Group

Atomic, Molecular, and Optical Physics at the University of Oklahoma
 

Technical Description of Stark Slowing

We want to test the feasibility of Stark slowing asymmetric molecules theoretically. We approximate an asymmetric polyatomic molecule as a rigid rotor with a permanent electric dipole moment, subject to an external applied electric field.

We then numerically calculate the electric field dependent eigenenergies and orientational probability distribution functions of the molecules' rotational quantum states. The Hamiltonian (=energy) of a single molecule in this approximation consists of two pieces: H=H_rot+H_stark, where H_rot is simply the rotational kinetic energy of the free asymmetric rigid rotor and H_stark is the interaction energy between the rotating molecule and the external dc electric field of ~0-100 kV/cm.

For an electric field pointing in the lab-fixed Z direction, H depends only on the three Euler angles Theta, Phi and Chi, giving the orientation of the molecule's body-fixed coordinate system (x,y,z) with respect to the lab-fixed coordinate system (X,Y,Z). Using the convention found in Zare (R. N. Zare, Angular Momentum (J. Wiley & Sons, New York, 1988).), we can choose Theta and Phi to be the familiar spherical coordinates, giving the direction of the molecules' z-axis in the lab frame. Chi then represents the orientation of the molecule with respect to rotation around it's own symmetry axis z.

The Stark shift energy curves are calculated using matrix diagonalization of the truncated Hamiltonian in the zero field asymmetric top basis. H is known analytically in the asymmetric top basis and has to be truncated at some high enough quantum numbers J, tau, M, since there is an infinity of eigenstates. At realistic electric fields, states up to about J_max=20 need to be included to get good accuracy for the lower states J=1..8.

The orientational probability distribution functions are then evaluated by expressing the asymmetric rotor "wavefunctions" in terms of the known symmetric rotor basis. "Wavefunctions" here are probability amplitudes Psi(Theta, Phi, Chi), the magnitude square of which gives the field-dependent probability to find the molecules body-fixed frame (x,y,z) at some orientation relative to the lab-fixed frame (X,Y,Z).

For applications in Stark slowing, weak field seeking states (WFS, positive Stark shift) are the only interesting ones, as it is impossible to create a minimum of the electric field in free space and so molecules have to be moved towards an electric field maximum.

We also calculate nonadiabatic transition probabilities, using a Landau-Zener type approach. These give the probability of making a transition between different Stark states due to a breakdown of the adiabatic approximation. We are considering typical nucleus velocities of molecules in a Stark slower and compare the Landau-Zener equation to results from numerical integration of the time-dependent Schroedinger equation.

Our first study dealt with Nitromethane and Acetaldehyde in a Stark slower. One of the results, the Nitromethane OPDF movies, showing orientational probability distributions along an avoided crossing of Stark energies, can be seen on this web-site. At such a crossing, the probability to make a nonadiabatic transition from the lower to the upper curve might get close to one, depending on the speed of the molecule, the size of the energy gap, the overlap between the two OPDFs and the electric fields in the Stark slower. Adiabatic following of such an avoided crossing will turn a WFS into a lost high-field seeker, limiting the maximum applicable fields.
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