Cancellation of D2 line transitions of alkali-metal atoms by magnetic-field values
Abstract
In a previous work the π transitions of the D1 line of alkali-metal atoms were shown to cancel at magnetic-field values given by a single closed-form expression. The D2 line has until now resisted the same treatment, because the fixed-m Hamiltonian blocks of the 2P3/2 manifold reach the dimension 4×4, 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=±1, 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 D2 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 D1 line. All the magnetic-field values canceling π, σ+ and σ- transitions of 23Na, 39K, 40K, 41K, 85Rb, 87Rb and 133Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the D1 line, the σ± transitions of the D2 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.
Related
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- Nuclear slowing-down factors in alkali-metal vapors
BibTeX
@misc{aleksanyan2026cancellation,
title = {{Cancellation of $D_2$ line transitions of alkali-metal atoms by magnetic-field values}},
author = {Artur Aleksanyan and Susanna Petrosyan and Emil Gazazyan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29700},
eprint = {2608.29700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29700},
}
RIS
TY - UNPB
TI - Cancellation of D_2 line transitions of alkali-metal atoms by magnetic-field values
AU - Artur Aleksanyan
AU - Susanna Petrosyan
AU - Emil Gazazyan
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29700
UR - https://arxiv.org/abs/2608.29700
AB - 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 4×4, 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=±1, 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 ^23Na, ^39K, ^40K, ^41K, ^85Rb, ^87Rb and ^133Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the D_1 line, the σ^± 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.
ER -