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Bloch equations

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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.

Contents

Relaxation of transverse nuclear magnetization Mxy

Assume that:

  • The nuclear magnetization is exposed to constant external magnetic field in the z direction Bz′(t) = Bz(t) = B0. Thus ω0 = γB0 and ΔBz(t) = 0.
  • There is no RF, that is Bxy' = 0.
  • The rotating frame of reference rotates with an angular frequency Ω = ω0.

    • Then in the rotating frame of reference, the equation of motion for the transverse nuclear magnetization, Mxy'(t) simplifies to:

    d M x y ′ ( t ) d t = − M x y ′ T 2

    This is a linear ordinary differential equation and its solution is

    M x y ′ ( t ) = M x y ′ ( 0 ) e − t / T 2 .

    where Mxy'(0) is the transverse nuclear magnetization in the rotating frame at time t = 0. This is the initial condition for the differential equation.

    Note that when the rotating frame of reference rotates exactly at the Larmor frequency (this is the physical meaning of the above assumption Ω = ω0), the vector of transverse nuclear magnetization, Mxy(t) appears to be stationary.

    Relaxation of longitudinal nuclear magnetization Mz

    Assume that:

  • The nuclear magnetization is exposed to constant external magnetic field in the z direction Bz′(t) = Bz(t) = B0. Thus ω0 = γB0 and ΔBz(t) = 0.
  • There is no RF, that is Bxy' = 0.
  • The rotating frame of reference rotates with an angular frequency Ω = ω0.

    • Then in the rotating frame of reference, the equation of motion for the longitudinal nuclear magnetization, Mz(t) simplifies to:

    d M z ( t ) d t = − M z ( t ) − M z , e q T 1

    This is a linear ordinary differential equation and its solution is

    M z ( t ) = M z , e q − [ M z , e q − M z ( 0 ) ] e − t / T 1

    where Mz(0) is the longitudinal nuclear magnetization in the rotating frame at time t = 0. This is the initial condition for the differential equation.

    90 and 180° RF pulses

    Assume that:

  • Nuclear magnetization is exposed to constant external magnetic field in z direction Bz′(t) = Bz(t) = B0. Thus ω0 = γB0 and ΔBz(t) = 0.
  • At t = 0 an RF pulse of constant amplitude and frequency ω0 is applied. That is B'xy(t) = B'xy is constant. Duration of this pulse is τ.
  • The rotating frame of reference rotates with an angular frequency Ω = ω0.
  • T1 and T2 → ∞. Practically this means that τ ≪ T1 and T2.

    • Then for 0 ≤ t ≤ τ:

    d M x y ′ ( t ) d t = i γ B x y ′ M z ( t ) d M z ( t ) d t = i γ 2 ( M x y ′ ( t ) B x y ′ ¯ − M x y ′ ¯ ( t ) B x y ′ )

    References

    Bloch equations Wikipedia
    (Text) CC BY-SA


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