The Bloch Equations in High-Gradient Magnetic Resonance Force Microscopy: Theory and Experiment

We report theory and observations of paramagnetic resonance in a measured field gradient of 44,000 T per meter by the technique of magnetic resonance force microscopy (MRFM). Resonance was induced in a dilute solid solution of diphenylpicrylhydrazyl in polystyrene at 77 and 10 K by an amplitude-modulated microwave field. This modulated the force between resonant sample spins and a micrometer-scale SmCo magnetic tip on a force microscope cantilever. The force signals were typically of order 10 fN, and were detected above a thermal noise floor of 80 aN per root hertz at 10 K, equivalent to a magnetic moment noise of 200 μ_B per root hertz of bandwidth. Resonance saturation was readily observed. Starting with the Bloch equations, we derived simple analytic expressions for the predicted cantilever signal amplitudes and T₁-dependent phase lags, valid at low microwave power levels. For power levels below saturation, the data were in good agreement with the Bloch equation predictions, while above saturation the measured force increased more slowly with power than predicted. Several ESR mechanisms which might lead to non-Bloch dynamics in the MRFM environment are reviewed. Spin-relaxation mechanisms are also reviewed. A detailed description of the experimental apparatus is offered.