# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Bloch equations

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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. ^{[1]} Sometimes they are called the equations of motion of nuclear magnetization.

## Contents

## Bloch equations in laboratory (stationary) frame of reference[edit]

Let **M**(*t*) = (*M _{x}*(

*t*),

*M*(

_{y}*t*),

*M*(

_{z}*t*)) be the nuclear magnetization. Then the Bloch equations read:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac {d M_x(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _x - \frac {M_x(t)} {T_2}}**

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[edit]

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,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\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**.

### Bloch equations are macroscopic equations[edit]

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[edit]

The above form is simplified assuming

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

- .

where

- .

The real and imaginary parts of *M _{xy}* correspond to

*M*and

_{x}*M*respectively.

_{y}*M*is sometimes called

_{xy}**transverse nuclear magnetization**.

## Bloch equations in rotating frame of reference (outline)[edit]

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

### Solution of Bloch equations with *T*_{1}, *T*_{2} → ∞[edit]

Assume that:

- at
*t*= 0 the transverse nuclear magnetization*M*_{xy}(0) experiences a constant magnetic flux density**B**(*t*) = (0, 0,*B*_{0}); *B*_{0}is positive;- there are no longitudinal and transverse relaxations (that is
*T*_{1}and*T*_{2}→ ∞).

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 ω_{0} = γ*B*_{0} in counterclockwise direction (this 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[edit]

This is the conclusion of the previous section: in a constant magnetic flux density *B*_{0} along *z* axis the transverse magnetization *M*_{xy} rotates around the this axis in counterclockwise direction with angular frequency ω_{0}. If the observer were around the same axis in counterclockwise direction with angular frequency Ω, *M*_{xy} it would appear to him

It is often 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*_{0} (given in the **laboratory reference system** (*x*, *y*, *z*)) with angular frequency Ω. This is equivalent to assuming that:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle M_{z}(t) = M_{z}'}**

What are the equations of motion of *M _{xy}*(

*t*)′ and

*M*(

_{z}*t*)′?

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac {d M_{xy}'(t)} {d t} = \frac {d M_{xy}(t) e^{+i \Omega t}} {d t} = e^{+i \Omega t} \frac {d M_{xy}(t) } {d t} + i \Omega e^{+i \Omega t} M_{xy} = e^{+i \Omega t} \frac {d M_{xy}(t) } {d t} + i \Omega M_{xy}' }**

Substitute from the Bloch equation in laboratory frame of reference:

## Simple solutions of Bloch equations (outline)[edit]

### Relaxation of transverse nuclear magnetization *M*_{xy}[edit]

_{xy}

### Relaxation of longitudinal nuclear magnetization *M*_{z}[edit]

_{z}

### 90 and 180° RF pulses[edit]

## References[edit]

- ↑ F Bloch,
*Nuclear Induction*, Physics Review**70**, 460-473 (1946)