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Concept
Bloch equations
Summary
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:
one, B0, is constant in time,
the other one, ΔBz(t), may be time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal.
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.
Magnetic resonance (quantum mechanics)
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.
Bloch equations are macroscopic: they describe the equations of motion of macroscopic nuclear magnetization that can be obtained by summing up all nuclear magnetic moment in the sample.
Opening the vector product brackets in the Bloch equations leads to:
The above form is further simplified assuming
where i = .
Official source
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