The spectroscopic modes of multiferroic BiFeO$_3$ provide detailed information about the very small anisotropy and Dzyaloshinskii-Moriya (DM) interactions responsible for the long-wavelength, distorted cycloid below $TN = 640$ K. A microscopic model that includes two DM interactions and easy-axis anisotropy predicts both the zero-field spectroscopic modes as well as their splitting and evolution in a magnetic field applied along a cubic axis. While only six modes are optically active in zero field, all modes at the cycloidal wavevector are activated by a magnetic field. The three magnetic domains of the cycloid are degenerate in zero field but one domain has lower energy than the other two in nonzero field. Measurements imply that the higher-energy domains are depopulated above about 6 T and have a maximum critical field of 16 T, below the critical field of 19 T for the lowest-energy domain. Despite the excellent agreement with the measured spectroscopic frequencies, some discrepancies with the measured spectroscopic intensities suggest that other weak interactions may be missing from the model.
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