Bloch equations

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 = ("M""x", "M""y", "M""z") as a function of time when relaxation times "T"1 and "T"2 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.
In the laboratory (stationary) frame of reference.
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") = ("B""x"("t"), "B""y"("t"), "B"0 + Δ"B""z"(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.
"M"0 is the steady state nuclear magnetization (that is, for example, when t → ∞); it is in the "z" direction.
Physical background.
With no relaxation (that is both "T"1 and "T"2 → ∞) 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.
As macroscopic equations.
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.
Alternative forms.
Opening the vector product brackets in the Bloch equations leads to:
The above form is further simplified assuming
where "i" = . After some algebra one obtains:
where
is the complex conjugate of "Mxy". The real and imaginary parts of "Mxy" correspond to "Mx" and "My" respectively.
"Mxy" is sometimes called transverse nuclear magnetization.
Matrix form.
The Bloch equations can be recast in matrix-vector notation:
In a rotating frame of reference.
In a rotating frame of reference, it is easier to understand the behaviour of the nuclear magnetization M. This is the motivation:
Solution of Bloch equations with "T"1, "T"2 → ∞.
Assume that:
Then the Bloch equations are simplified to:
These are two (not coupled) linear differential equations. Their solution is:
Thus the transverse magnetization, "M"xy, rotates around the "z" axis with angular frequency ω0 = γ"B"0 in clockwise direction (this is due to the negative sign in the exponent).
The longitudinal magnetization, "M"z remains constant in time. This is also how the transverse magnetization appears to an observer in the laboratory frame of reference (that is to a stationary observer).
"M"xy("t") is translated in the following way into observable quantities of "M"x("t") and "M"y("t"): Since
then
where Re("z") and Im("z") are functions that return the real and imaginary part of complex number "z". In this calculation it was assumed that "M"xy(0) is a real number.
Transformation to rotating frame of reference.
This is the conclusion of the previous section: in a constant magnetic field "B"0 along "z" axis the transverse magnetization "M"xy rotates around this axis in clockwise direction with angular frequency ω0. If the observer were rotating around the same axis in clockwise direction with angular frequency Ω, "M"xy it would appear to her or him rotating with angular frequency ω0 - Ω. Specifically, if the observer were rotating around the same axis in
clockwise direction with angular frequency ω0, the transverse magnetization "M"xy would appear to her or him stationary.
This can be expressed mathematically in the following way:
Obviously:
What is "Mxy"′("t")? Expressing the argument at the beginning of this section in a mathematical way:
Equation of motion of transverse magnetization in rotating frame of reference.
What is the equation of motion of "Mxy"′("t")?
Substitute from the Bloch equation in laboratory frame of reference:
But by assumption in the previous section: "B"z′("t") = "B"z("t") = "B"0 + Δ"B"z("t") and "M"z("t") = "Mz′("t"). Substituting into the equation above:
This is the meaning of terms on the right hand side of this equation:
Similarly, the equation of motion of "Mz" in the rotating frame of reference is:
Time independent form of the equations in the rotating frame of reference.
When the external field has the form:
We define:
and get (in the matrix-vector notation):
Simple solutions.
Relaxation of transverse nuclear magnetization "Mxy".
Assume that:
Then in the rotating frame of reference, the equation of motion for the transverse nuclear magnetization, "M"xy'("t") simplifies to:
This is a linear ordinary differential equation and its solution is
where "M"xy'(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, "M"xy("t") appears to be stationary.
Relaxation of longitudinal nuclear magnetization "Mz".
Assume that:
Then in the rotating frame of reference, the equation of motion for the longitudinal nuclear magnetization, "M"z("t") simplifies to:
This is a linear ordinary differential equation and its solution is
where "M"z(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:
Then for 0 ≤ "t" ≤ τ: