Bloch equations

underconstruction. In physics and chemistry, specifically in 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 "T"₁ and "T"₂ are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. [F Bloch, "Nuclear Induction", Physics Review 70, 460-473 (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"), "B"₀ + "B"_"z"(t)) is the magnetic flux density experienced by the nuclei.The "z" component of the magnetic flux density B is sometimes composed of two terms:
*one, "B"₀, is constant in time,
*the other one, "B"_"z"(t), is 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."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 flux density B.

The relaxation terms,

:$left\; (\; -frac\; \{M\_x\}\; \{T\_2\},\; -frac\; \{M\_y\}\; \{T\_2\},\; -frac\; \{M\_z\; -\; M\_0\}\; \{T\_1\}\; ight\; )$

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.

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 form of Bloch equations

The above form is simplified assuming

:$M\_\{xy\}\; =\; M\_x\; +\; iM\_y\; ext\{\; and\; \}\; B\_\{xy\}\; =\; B\_x\; +\; iB\_y$

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

:$frac\; \{d\; M\_\{xy\}(t)\}\; \{d\; t\}\; =\; -i\; gamma\; left\; (\; M\_\{xy\}\; (t)\; B\_z\; (t)\; -\; M\_z\; (t)\; B\_\{xy\}\; (t)\; ight\; )\; -frac\; \{M\_\{xy\; \{T\_2\}$.:$frac\; \{d\; M\_z(t)\}\; \{d\; t\}\; =\; i\; gamma\; left\; (\; M\_\{xy\}\; (t)\; overline\{B\_\{xy\}\; (t)\}\; -\; overline\; \{M\_\{xy\; (t)\; B\_\{xy\}\; (t)\; ight\; )-\; frac\; \{M\_z\; -\; M\_0\}\; \{T\_1\}$

where

:$overline\; \{M\_\{xy\; =\; M\_x\; -\; i\; M\_y$.

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 (outline)

In rotating frame of reference the it is easier to understand the behaviour of nuclear magnetization M. This is the motivation:

Assume that at "t" = 0 the transverse nuclear magnetization "M"_xy(0) experiences a constant magnetic flux density B("t") = (0, 0, "B"_"z"). Assume for a moment that there is no longitudinal relaxation (that is "T"₁ → ∞). Then the Bloch equations are simplified to:

:$frac\; \{d\; M\_\{xy\}(t)\}\; \{d\; t\}\; =\; -i\; gamma\; M\_\{xy\}\; (t)\; B\_z\; -frac\; \{M\_\{xy\; \{T\_2\}$,:$frac\; \{d\; M\_z(t)\}\; \{d\; t\}\; =\; 0$.

These are two (not coupled) linear differential equations. Their solution is:

:$M\_\{xy\}(t)\; =\; M\_\{xy\}\; (0)\; e^\{-i\; gamma\; B\_z\; t\}\; e^\{-t\; /\; T\_2\}$,:$M\_z(t)\; =\; M\_0\; =\; ext\{const\}$.

Thus the transverse magnetization, "M"_"xy",
#rotates around the "z" axis with angular frequency ω_"z" = γ"B"_z and
#its magnitude, |"M"_"xy"|, decays to zero with a rate "T"₂.The longitudinal magnetization, "M"_z remains constant in time.

It is sometime more convenient to describe the physics and mathematics of nuclear magnetization in a rotating frame of reference: Let ("x"′, "y"′, "z"′) be a Cartesian coordinate system that is rotating around the static magnetic field "B"_"z" (given in the laboratory reference system ("x", "y", "z")) with angular frequency ω_"z". This is equivalent to assuming that:

:$M\_\{xy\}(t)\; =\; e^\{-i\; omega\_z\; t\}\; M\_\{xy\}\text{'}$:$M\_\{z\}(t)\; =\; M\_\{z\}\text{'}$

What are the equations of motion of "M_xy"("t")′ and "M_z"("t")′?

:$frac\; \{d\; M\_\{xy\}\text{'}(t)\}\; \{d\; t\}\; =\; frac\; \{d\; M\_\{xy\}(t)\; e^\{+i\; omega\_z\; t\; \{d\; t\}\; =e^\{+i\; omega\_z\; t\}\; frac\; \{d\; M\_\{xy\}(t)\; \}\; \{d\; t\}\; +\; i\; omega\_z\; e^\{+i\; omega\_z\; t\}\; M\_\{xy\}\; =e^\{+i\; omega\_z\; t\}\; frac\; \{d\; M\_\{xy\}(t)\; \}\; \{d\; t\}\; +\; i\; omega\_z\; M\_\{xy\}\text{'}$

Substitute from the Bloch equation in laboratory frame of reference:

:

& = left [-i gamma left ( M_{xy} (t) e^{+i omega_z t} B_z (t) - M_z (t) B_{xy} (t) e^{+i omega_z t} ight ) -frac {M_{xy} e^{+i omega_z t} } {T_2} ight ] + i omega_z M_{xy}' = \

& = -i gamma left ( M_{xy}' (t) B_z' (t) - M_z' (t) B_{xy}' (t) ight ) + i omega_z M_{xy}' -frac {M_{xy}'} {T_2} =

end{align}

imple solutions of Bloch equations (outline)

Relaxation of transverse nuclear magnetization "M_xy"

Relaxation of longitudinal nuclear magnetization "M_z"

90 and 180° RF pulses

References

Further reading

* Charles Kittel, "Introduction to Solid State Physics", John Wiley & Sons, 8th Edition edition (2004), ISBN-13: 978-0471415268. Chapter 13 is on Magnetic Resonance.