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At equilibrium, the bulk magnetization (**M**) is oriented along the *z* axis. Although all of the individual nuclei retain magnetization in the *xy*
plane, their μ_{x} and μ_{y} components are randomly distributed and thus perfectly cancel, with the result that *M _{x}* =

The μ_{z} components of the various nuclei are also randomly distributed, but in this case the various alignments do not perfectly cancel.
There is a slight difference in energies between the different *m _{I}* states. For

The random distribution of nuclei between states of different energies is described by the *Boltzmann distribution*. In this distribution, the difference in
energies between two states is the critical value. The energy for the interaction of a nuclear magnetic moment (**μ**) with the static magnetic field (**B**)
is the dot product of the two vectors:

*E* = - **μ** ⋅ **B** = - μ_{z} *B* = - ℏ γ *m _{I}*

The difference between two adjacent energy levels is

Δ*E* = - ℏ γ Δ*m _{I}*

Note that the difference between higher and lower energy states depends upon Δ*m _{I}* = (-½) - (½) = -1. Take note that
the energy difference is related to the

The difference in energy for NMR processes is very small. For example, if *B* = 10.0 T, then Δ*E* = 0.170 J/mol, which is a million times less than the
energy of a covalent bond. More to the point, it is also much less than the energy available from the thermal motion of molecules at room temperature (about 2.5 kJ/mol).
Thus there are *almost* the same number of nuclei in the +½ and -½ states.

The Boltzmann distribution is described by the following equation, in which *n _{0}* and

*n _{1}*/

For the example of ^{1}H is a 10.0 T field, *n _{1}*/

Let's suppose the bulk magnetization is not initially at equilibrium. How quickly does **M** relax to its equilibrium state?

In 1946 Felix Bloch proposed that the relaxation of **M** to its equilibrium state follows first-order kinetics. Moreover, he described the relaxation of
*M _{z}* and of

The relaxation of *M _{z}* to its equilibrium value of

The relaxation of *M _{x}* and

The differential equations describing the way the bulk magnetization changes with time are called the *Bloch equations*, which contain terms for the
first-order rate law for relaxation and terms for the precession.

*d M _{z}*/

*d M _{x}*/

*d M _{y}*/

The solution of the Bloch equations is illustrated in the simulation shown below. The bulk magnetization is artificially positioned along the *y* axis
at the outset of the experiment. The resulting signal is called a *free induction decay* (FID).

Run the simulation for various combinations of *f*, *T _{1}*, and

As with other simulations in this series, simulation occurs **much** more slowly than real precession. The simulation has been
slowed to permit visualization of the process.

- How does
*f*affect the simulation? How does it affect the FID? - How does
*T*affect the simulation? How does it affect the FID?_{1} - How does
*T*affect the simulation? How does it affect the FID?_{2} - Is the magnitude of the bulk magnetization (
*M*) a constant or does it vary during the experiment? Explain this behavior.

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