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Spectroscopic modes provide the most sensitive probe of the very weakinteractions responsible for the properties of the long-wavelength cycloid inthe multiferroic phase of \BF below $\TN \approx 640$ K. Three of the fourmodes measured by THz and Raman spectroscopies were recently identified using asimple microscopic model. While a Dzyaloshinskii-Moriya (DM) interaction $D$along $[-1,2,-1]$ induces the cycloid with wavevector $(2\pi /a)(0.5+\delta,0.5, 0.5-\delta)$ ($\delta \approx 0.0045$), easy-axis anisotropy $K$ along the$[1,1,1]$ direction of the electric polarization ${\bf P}$ induces higherharmonics of the cycloid, which split the $\Psi_1$ modes at 2.49 and 2.67 meVand activate the $\Phi_2$ mode at 3.38 meV. However, that model could notexplain the observed low-frequency mode at about 2.17 meV. We now demonstratethat an additional DM interaction $D'$ along $[1,1,1]$ not only produces theobserved weak ferromagnetic moment of the high-field phase above 18 T but alsoactivates the spectroscopic matrix elements of the nearly-degenerate,low-frequency $\Psi_0$ and $\Phi_1$ modes, although their scatteringintensities remain extremely weak. Even in the absence of easy-axis anisotropy,$D'$ produces cycloidal harmonics that split $\Psi_1 $ and activate $\Phi_2$.However, the observed mode frequencies and selection rules require that both$D'$ and $K$ are nonzero. This work also resolves an earlier disagreementbetween spectroscopic and inelastic neutron-scattering measurements.

The Spin State and Spectroscopic Modes of Multiferroic BiFeO3