Cancellation of $D_2$ line transitions of alkali-metal atoms by magnetic-field values

In a previous work the $π$ transitions of the $D_1$ line of alkali-metal atoms were shown to cancel at magnetic-field values given by a single closed-form expression. The $D_2$ line has until now resisted the same treatment, because the fixed-$m$ Hamiltonian blocks of the $^2P_{3/2}$ manifold reach the dimension $4\times4$, and the corresponding formulas were considered to be too heavy to be useful. In this work we show that this difficulty is only apparent. Since the Zeeman interaction couples only levels with $ΔF=\pm1$, every block, whatever its dimension, is tridiagonal; its characteristic polynomial therefore obeys a three-term recursion, and the components of its eigenvectors are polynomials in the eigenvalue. Using these two properties we obtain the eigenvalues in closed form by Ferrari's and Cardano's formulas, the eigenvectors without any further diagonalization, and finally a single relation which gives the magnetic-field value canceling a $D_2$ transition as an explicit function of the excited-state eigenvalue, the nuclear spin $I$, the ground-state hyperfine splitting and the magnetic quantum number. A necessary condition on $m$ and on the polarization is derived, which replaces the selection rule known for the $D_1$ line. All the magnetic-field values canceling $π$, $σ^+$ and $σ^-$ transitions of $^{23}$Na, $^{39}$K, $^{40}$K, $^{41}$K, $^{85}$Rb, $^{87}$Rb and $^{133}$Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the $D_1$ line, the $σ^{\pm}$ transitions of the $D_2$ line do cancel, and they account for the majority of the values found. The accuracy of the calculated $B$ values is limited only by the uncertainties of the excited-state hyperfine constants.