Possible use of motional magnetic fields to flip spins in a Penning trap

The most precise determination of the magnetic moments of the electron and positron come from the University of Washington $g-2\mathrm{}$ experiments. A small distortion in the magnetic field is introduced in these experiments so as to provide a mechanism to flip and measure the spin. The experiments would be substantially improved if this inhomogeneous part of the magnetic field were removed. Driving the end caps produces an electric field at the frequency ${\omega }_d$ which is parallel to the magnetic field. The cyclotron motion of the electron at frequency ${\omega }_c$ transforms this electric field into a magnetic field which is perpendicular to the main magnetic field and which alternates at frequency ${\omega }_c+{\omega }_d$ in the particle's rest frame. One might think that such an alternating magnetic field would flip the spin, thereby removing the need for the field inhomogeneity. There are, however, subtle relativistic corrections to the spin motion. These nearly cancel the motional magnetic field effect and rule out this scheme of spin flipping.