General Analysis of ¹⁴N (l = 1) Electron Spin Echo Envelope Modulation

The analysis methods described to date for ¹⁴N electron spin echo envelope modulation (ESEEM) mostly deal with isotropic g- and ¹⁴N hyperfine coupling tensors. However, many cases of rhombic tensors are encountered. In the present report we present general equations for analyzing orientation-selective ESEEM and illustrate their use. (i) We present general equations for the nuclear interactions in an electron spin system where the EPR signal arises from an isolated Kramers doublet, then give the nuclear (electron-nuclear double resonance) frequencies for I = 1 associated with such a system, (ii) These are incorporated into equations for single-crystal ESEEM amplitudes, which in turn are incorporated into general equations for the orientation-selective ESEEM that arises when the EPR envelope of a frozen-solution (powder) sample is determined by g anisotropy. (iii) This development is first used in the simplest limit of an isotropic g-tensor and leads to a more general picture of the response of the I = 1 modulation amplitude to variations in the nuclear hyperfine and quadrupole coupling constants, relative to the nuclear Zeeman interaction, than had been presented previously. We find that strong modulation occurs not only in the well-known regime where the "exact/near cancellation" condition (A/2 ≃ ν_N) is satisfied, but also when the nuclear hyperfine interaction is much larger than the nuclear Zeeman interaction (A/ν_N > 3) with A/K = 4 ≃ 5. (iv) We then describe the orientation-selective ¹⁴N ESEEM frequency-domain patterns (g vs frequency) in the presence of anisotropic (rhombic) hyperfine and electron Zeeman interactions for both coaxial and noncoaxial cases. We derive analytical solutions when the g-, hyperfine, and nuclear quadrupole tensors are coaxial. (v) The method is applied to the ESEEM of the nitrogenase MoFe protein (A ν 1) to determine the full hyperfine and nuclear quadrupole tensors of ¹⁴N nuclei interacting with the S = 3/2 FeMo-cofactor (Fe₇S₈Mo: homocitrate).