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In physics and chemistry, specifically in NMR

NMR

NMR may refer to:Applications of Nuclear Magnetic Resonance:* Nuclear magnetic resonance* NMR spectroscopy* Solid-state nuclear magnetic resonance* Protein nuclear magnetic resonance spectroscopy* Proton NMR* Carbon-13 NMR...

(nuclear magnetic resonance) or MRI (magnetic resonance imaging), or ESR (electron spin resonance) 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

Relaxation (NMR)

In nuclear magnetic resonance spectroscopy and magnetic resonance imaging the term relaxation describes several processes by which nuclear magnetization prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins "forget" the...

T₁ and T₂ are present. These are phenomenological

Phenomenology (science)

The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...

equations that were introduced by Felix Bloch

Felix Bloch

Felix Bloch was a Swiss physicist, working mainly in the U.S.-Life and work:Bloch was born in Zürich, Switzerland to Jewish parents Gustav and Agnes Bloch. He was educated there and at the Eidgenössische Technische Hochschule, also in Zürich. Initially studying engineering he soon changed to physics...

in 1946.

Sometimes they are called the equations of motion of nuclear magnetization.

Bloch equations in laboratory (stationary) frame of reference

Let M(t) = (M_x(t), M_y(t), M_z(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), B₀ + ΔB_z(t)) is the magnetic field

Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

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, ΔB_z(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

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

of these two vectors.
M₀ 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 T₁ and T₂ → ∞) the above equations simplify to:

or, in vector notation:

This is the equation for Larmor precession

Larmor precession

In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...

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.

Bloch equations are 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

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

.

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 of Bloch equations

Opening the vector product brackets in the Bloch equations leads to:

The above form is further simplified assuming

where i = √(-1). After some algebra one obtains:

.

where

.

The real and imaginary parts of M_xy correspond to M_x and M_y respectively.
M_xy is sometimes called transverse nuclear magnetization.

Bloch equations in 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 T₁, T₂ → ∞

Assume that:

at t = 0 the transverse nuclear magnetization M_xy(0) experiences a constant magnetic field B(t) = (0, 0, B₀);
B₀ is positive;
there are no longitudinal and transverse relaxations (that is T₁ and T₂ → ∞).

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

Angular frequency

In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

ω₀ = γB₀ 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 B₀ 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 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 him stationary.

This can be expressed mathematically in the following way:

Let (x, y, z) the Cartesian coordinate system of the laboratory (or stationary) frame of reference, and
(x′, y′, z′) be a Cartesian coordinate system that is rotating around the z axis of the laboratory frame of reference with angular frequency Ω. This is called the rotating frame of reference. Physical variables in this frame of reference will be denoted by a prime.

Obviously:

.

What is M_xy′(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 (outline)

What is the equations of motion of M_xy′(t)?

Substitute from the Bloch equation in laboratory frame of reference:

But by assumption in the previous section: B_z′(t) = B_z(t) = B₀ + ΔB_z(t). Substituting into the equation above:

This is the meaning of terms on the right hand side of this equation:

i (Ω - ω) M_xy′(t) is the Larmor term in the frame of reference rotating with angular frequency Ω. Note that it becomes zero when Ω = ω₀.
The -i γ ΔB_z(t) M_xy′(t) term describes the effect of magnetic field inhomogeneity (as expressed by ΔB_z(t)) 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 i γ ΔB_xy′(t) M_z(t) describes the effect of RF field (the ΔB_xy′(t) factor) on nuclear magnetization. For an example see below.
- M_xy′(t) / T₂ describes the loss of coherency of transverse magnetization.

Relaxation of transverse nuclear magnetization M_xy

Assume that:

The nuclear magnetization is exposed to constant external magnetic field in the z direction B_z′(t) = B_z(t) = B₀. Thus ω₀ = γB₀ and ΔB_z(t) = 0.
There is no RF, that is B_xy' = 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, 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.

90 and 180° RF pulses

Assume that:

Nuclear magnetization is exposed to constant external magnetic field in z direction B_z′(t) = B_z(t) = B₀. Thus ω₀ = γB₀ and ΔB_z(t) = 0.
At t = 0 an RF pulse of constant amplitude and frequency ω₀ is applied.
The rotating frame of reference rotates with an angular frequency Ω = ω₀.

Bloch equations