MRI: the Bloch Equations, the FID and k-Space Imaging

An MRI (magnetic resonance imaging) physics playground. The Bloch equations are solved analytically in the rotating frame: after a 90-degree pulse the magnetization precesses while the transverse component decays with $T_2$ and the longitudinal component recovers to $M_0$ with $T_1$, tracing a spiral on the Bloch sphere and producing the free induction decay whose Fourier transform is a Lorentzian. A brain-like phantom is imaged with the spin-echo equation $S \sim \rho (1 - e^{-TR/T_1}) e^{-TE/T_2}$ or the spoiled gradient-echo Ernst-angle equation; the image is transformed to k-space and reconstructed by a 2D inverse Fourier transform, and discarding the outer k-space lines blurs it. Panel A is the Bloch sphere, Panel B the FID and spectrum, Panel C the image and its k-space. The magnetisation magnitude is conserved under pure precession, $T_1$ and $T_2$ relaxation follow the Bloch laws, the spin-echo signal reaches its expected limits at the Ernst angle, and discarding the outer k-space lines blurs the image.