The Bloch equations are a set of differential equations that describe the dynamics of nuclear magnetization under the influence of external magnetic fields, specifically in the context of nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). They provide a mathematical framework for understanding how spins of particles like nuclei evolve over time due to applied magnetic fields and interactions.
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The Bloch equations describe an ensemble magnetization with Larmor precession plus phenomenological recovery and decay:Here are model parameters, not a microscopic explanation of relaxation. The precession term follows from the torque on a magnetic moment; the two dissipative terms prescribe exponential decay. Bloch explicitly introduced this macroscopic relaxation description as a simplifying approximation.[ref]
Nuclear Induction by Felix Bloch. §4, pp. 467–468, equations (32), (37) and (38): longitudinal recovery and assumed exponential transverse decay
For positive and no transverse drive, rotates as : its signed rotation is negative about , although the resonance angular frequency is positive. This distinction prevents a hidden sign change when choosing a rotating frame.