Radio Quiet Pulsars with Ultra-Strong Magnetic Fields
  The notable absence of radio pulsars having measured magnetic dipole surface field strengths above $B_0\sim 3\times 10^{13}$ Gauss naturally raises the question of whether this forms an upper limit to pulsar magnetization. Recently there has been increasing evidence that neutron stars possessing higher dipole spin-down fields do in fact exist, including a growing list of anomalous X-ray pulsars (AXPs) with long periods and spinning down with high period derivatives, implying surface fields of $10^{14}$--$10^{15}$ Gauss. Furthermore, the recently reported X-ray period and period derivative for the Soft Gamma-ray Repeater (SGR) source SGR1806-20 suggest a surface field around $10^{15}$ Gauss. None of these high-field pulsars have yet been detected as radio pulsars. We propose that high-field pulsars should be radio-quiet because electron-positron pair production in their magnetospheres, thought to be essential for radio emission, is efficiently suppressed in ultra-strong fields ($B_0\gtrsim 4\times 10^{13}$ Gauss) by the action of photon splitting, a quantum electrodynamical process in which a photon splits into two. Our computed radio quiescence boundary in the radio pulsar $P-\dot P$ diagram, where photon splitting overtakes pair creation, is located just above the boundary of the known radio pulsar population, neatly dividing them from the AXPs. We thus identify a physical mechanism that defines a new class of high-field radio-quiet neutron stars that should be detectable by their pulsed emission at X-ray and perhaps $\gamma$-ray energies. 
Matthew G. Baring and Alice K. Harding