Bloch equations


In physics and chemistry, specifically in nuclear magnetic resonance, magnetic resonance imaging, and electron spin resonance, the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization as a function of time when relaxation times and 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 be the nuclear magnetization. Then the Bloch equations read:
where is the gyromagnetic ratio and is the magnetic field experienced by the nuclei.
The component of the magnetic field is sometimes composed of two terms:
  • one,, is constant in time,
  • the other one,, may be time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal.
is the cross product of these two vectors.
is the steady state nuclear magnetization ; it is in the z direction.

Physical background

With no relaxation the above equations simplify to:
or, in vector notation:
This is the equation for Larmor precession of the nuclear magnetization in an external magnetic field.
The relaxation terms,
represent an established physical process of transverse and longitudinal relaxation of nuclear magnetization.

As macroscopic equations

These equations are not microscopic: they do not describe the equation of motion of individual nuclear magnetic moments. Those 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. After some algebra one obtains:
where
is the complex conjugate of. The real and imaginary parts of correspond to and respectively.
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:
  • at the transverse nuclear magnetization experiences a constant magnetic field ;
  • is positive;
  • there are no longitudinal and transverse relaxations.
Then the Bloch equations are simplified to:
These are two linear differential equations. Their solution is:
Thus the transverse magnetization,, rotates around the z axis with angular frequency in clockwise direction.
The longitudinal magnetization, remains constant in time. This is also how the transverse magnetization appears to an observer in the laboratory frame of reference.
is translated in the following way into observable quantities of and : Since
then
where and are functions that return the real and imaginary part of complex number. In this calculation it was assumed that is a real number.

Transformation to rotating frame of reference

This is the conclusion of the previous section: in a constant magnetic field along z axis the transverse magnetization rotates around this axis in clockwise direction with angular frequency. If the observer were rotating around the same axis in clockwise direction with angular frequency, it would appear to them rotating with angular frequency. Specifically, if the observer were rotating around the same axis in clockwise direction with angular frequency ω0, the transverse magnetization would appear to her or him stationary.
This can be expressed mathematically in the following way:
Obviously:
What is ? 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 ?
Substitute from the Bloch equation in laboratory frame of reference:
But by assumption in the previous section: and. Substituting into the equation above:
This is the meaning of terms on the right hand side of this equation:
  • is the Larmor term in the frame of reference rotating with angular frequency Ω. Note that it becomes zero when.
  • The term describes the effect of magnetic field inhomogeneity on the transverse nuclear magnetization; it is used to explain T2*. It is also the term that is behind MRI: it is generated by the gradient coil system.
  • The describes the effect of RF field on nuclear magnetization. For an example see below.
  • describes the loss of coherency of transverse magnetization.
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 :

Simple solutions

Relaxation of transverse nuclear magnetization ''Mxy''

Assume that:
  • The nuclear magnetization is exposed to constant external magnetic field in the z direction. Thus and.
  • There is no RF, that is.
  • The rotating frame of reference rotates with an angular frequency.
Then in the rotating frame of reference, the equation of motion for the transverse nuclear magnetization, simplifies to:
This is a linear ordinary differential equation and its solution is
where is the transverse nuclear magnetization in the rotating frame at time. This is the initial condition for the differential equation.
Note that when the rotating frame of reference rotates exactly at the Larmor frequency, the vector of transverse nuclear magnetization, 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. Thus and.
  • There is no RF, that is.
  • 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, simplifies to:
This is a linear ordinary differential equation and its solution is
where Mz 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. Thus and.
  • At an RF pulse of constant amplitude and frequency ω0 is applied. That is is constant. Duration of this pulse is τ.
  • The rotating frame of reference rotates with an angular frequency Ω = ω0.
  • and. Practically this means that and.
Then for :