Effect of secondary beam neutrals on MSE: Theory

A standard calibration technique for Motional Stark Effect ({\sc mse}) diagnostics is to compare the polarization direction of Doppler-shifted H$\alpha$ emission from a diagnostic neutral beam ({\sc dnb}) that is fired into a gas-filled torus to the pitch angle inferred from known toroidal and vertical fields. However, the polarization direction of H$\alpha$ emission from `secondary' beam neutrals that ionize, gyrate about field lines, and then charge exchange a second time differs from the polarization direction of the `primary' beam neutrals and thus confuses the calibration results. We compute the ratio of secondary-to-primary H$\alpha$ emission, $I_s/I_p$, as a function of torus pressure for 50 keV hydrogen atoms in Alcator C-Mod. For helium gas, $I_s/I_p$ is about unity at P=1 mTorr for the {\sc dnb} in its recently re-oriented configuration ($7^{\circ}$ from perpendicular). The effect on the {\sc mse} calibration of H$\alpha$ emission from these secondary beam neutrals is calculated by adding the Stokes vectors for all secondary-beam gyro angles whose Doppler shift lies within the {\sc mse} filter passband. The computed calibration error increases linearly with torus pressure and has distinct dependencies on MSE viewing geometry and pitch angle which are in qualitative agreement with recent measurements.