Vanishing electric dipole transition moment

We have discovered an interference which results in the vanishing of some electric dipole transition amplitudes. A transition-matrix element $T$ between certain $S$ and $P$ sublevels with the same value of $M_F$ vanishes when an applied magnetic field satisfies $X_s+X_p=-2\mathrm{}{M}_F$, where $X={\mu }_B{g}_J\frac{B}{A}$, and $A$ is the hfs constant. There must be an avoided crossing between the two states of the same $M_F$ which can only occur when one of the hyperfine manifolds has an inverted Zeeman effect. These conditions imply $M_F<0\mathrm{}$ and $I>J$. We observed the vanishing of $T$ in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ for one of the $\Delta M_F=0\mathrm{}$ optical transitions of the $3S_{\frac{1}{2}}\mathrm{}(F=1\mathrm{})\mathrm{}\to 3P_{\frac{1}{2}}\mathrm{}(F=1\mathrm{})\mathrm{}$ manifold. Absorption of cw laser light was monitored by observing total fluorescence perpendicular to both the atomic and laser beams, and absorption vanished near an applied field of 155 G. $T$ vanishes for a large number of cases including rf and microwave frequencies and is a rather general consequence of angular momentum selection rules and perturbation theory. This phenomenon may have application in optical pumping, Lamb-shift measurements, and atomic weak neutral-current experiments.