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 T₁ and T₂ are present. These are phenomenological equations that were introduced by Felix Bloch in 1946.^[1] 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) = (M_x(t), M_y(t), M_z(t)) be the nuclear magnetization. Then the Bloch equations read:

${\displaystyle {\frac {dM_{x}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{x}-{\frac {M_{x}(t)}{T_{2}}}}$

${\displaystyle {\frac {dM_{y}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{y}-{\frac {M_{y}(t)}{T_{2}}}}$

${\displaystyle {\frac {dM_{z}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{z}-{\frac {M_{z}(t)-M_{0}}{T_{1}}}}$

where γ is the gyromagnetic ratio and B(t) = (B_x(t), B_y(t), B₀ + ΔB_z(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, 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 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:

${\displaystyle {\frac {dM_{x}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{x}}$

${\displaystyle {\frac {dM_{y}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{y}}$

${\displaystyle {\frac {dM_{z}(t)}{dt}}=\gamma (\mathbf {M} (t)\times \mathbf {B} (t))_{z}}$

or, in vector notation:

${\displaystyle {\frac {d\mathbf {M} (t)}{dt}}=\gamma \mathbf {M} (t)\times \mathbf {B} (t)}$

This is the equation for Larmor precession of the nuclear magnetization M in an external magnetic field B.

The relaxation terms,

${\displaystyle \left(-{\frac {M_{x}}{T_{2}}},-{\frac {M_{y}}{T_{2}}},-{\frac {M_{z}-M_{0}}{T_{1}}}\right)}$

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. 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:

${\displaystyle {\frac {dM_{x}(t)}{dt}}=\gamma \left(M_{y}(t)B_{z}(t)-M_{z}(t)B_{y}(t)\right)-{\frac {M_{x}(t)}{T_{2}}}}$

${\displaystyle {\frac {dM_{y}(t)}{dt}}=\gamma \left(M_{z}(t)B_{x}(t)-M_{x}(t)B_{z}(t)\right)-{\frac {M_{y}(t)}{T_{2}}}}$Zdroj:https://en.wikipedia.org?pojem=Bloch_equations
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---