Bloch equation
In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx, My, Mz) as a function of time when relaxation times T1 and T2 are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. Sometimes they are called the equations of motion of nuclear magnetization. They are analogous to the Maxwell–Bloch equations.
Let M(t) = (Mx(t), My(t), Mz(t)) be the nuclear magnetization. Then the Bloch equations read:
where γ is the gyromagnetic ratio and B(t) = (Bx(t), By(t), B0 + ΔBz(t)) is the magnetic field experienced by the nuclei. The z component of the magnetic field B is sometimes composed of two terms:
M(t) × B(t) is the cross product of these two vectors. M0 is the steady state nuclear magnetization (that is, for example, when t → ∞); it is in the z direction.
With no relaxation (that is both T1 and T2 → ∞) the above equations simplify to:
or, in vector notation:
This is the equation for Larmor precession of the nuclear magnetization M in an external magnetic field B.
The relaxation terms,
represent an established physical process of transverse and longitudinal relaxation of nuclear magnetization M.
These equations are not microscopic: they do not describe the equation of motion of individual nuclear magnetic moments. These are governed and described by laws of quantum mechanics.
...
Wikipedia