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## Nuclear SpinOne of the noteworthy properties of a proton and a neutron is that each has intrinsic The magnitude of the angular momentum, and thus the magnitude of the magnetic moment, is characterized by the |

For ^{4}He there are two protons and two neutrons. The two protons are paired in the same orbital, with their
magnetic moments (north and south poles) opposing and canceling each other. The same is true for the neutrons, leaving ^{4}He with no net spin (*I* = 0)
and thus no magnetic moment. Similarly, all of the protons and neutrons are paired in ^{12}C, and thus is also has no magnetic moment,
but there is one unpaired proton in ^{13}C, leading to *I* = ½. There is one unpaired proton and one unpaired neutron in ^{14}N and thus *I* = 1. Note that protons and neutrons are distinguishable particles
and thus there are separate sets of orbitals for protons and for neutrons.

The magnitude of the angular moment depends upon the particle's spin:

|**p**| = ℏ [*I* (*I* + 1)]^{½}

The physical constant ℏ equals *h*/2π, where *h* = 6.626 x 10^{-34} J sec is the Planck constant.

The magnetic moment is directly portional to the angular moment

**μ** = γ **p**

The proportionality constant (γ) is called the *magnetogyric ratio*. Each nucleus has an unique value of *γ*, which has units of
sec^{-1} T^{-1}. T = Tesla, a unit of magnetic field strength.

Nucleus | Spin | 10^{-7} γ (sec^{-1} T^{-1}) | |

^{1}H | ½ | 26.7522 | |

^{13}C | ½ | 6.72828 | |

^{14}N | 1 | 1.93378 | |

^{15}N | ½ | -2.71262 | |

^{31}P | ½ | 10.8394 |

Notice that the magnetic moment of ^{1}H is four times larger than that of ^{13}C, even though both are spin ½ nuclei. For this reason,
proton NMR is inherently four times more sensitive than ^{13}C NMR. There are also other factors that affect sensitivity. Almost all naturally occurring hydrogen
is ^{1}H. Only 1% of naturally occurring carbon is ^{13}C.

If you obtain a pair of bar magnets and explore how they interact, you will quickly observe that the north pole of one magnet is attracted to the south pole of the other. Imagine the large blue magnet in the image at left has a powerful magnetic field ( The orientations of the magnets in the image at the right is the state with lowest energy. Once the system reaches this configuration, it is at equilibrium and no additional change will occur. |

There is a profound difference, however, between the behavior of the macroscopic magnets illustrated above and the magnetic moment of a nucleus. If a nucleus
is placed in a stationary magnetic field (**B**) that is aligned along the *z* axis, the nucleus will attempt to align its magnetic moment against that of the
magnetic field, similar to the bar magnets depicted above. Unlike the bar magnets, however, the alignment with the magnetic field is constrained to discrete levels.

The component of the angular moment (and hence magnetic moment) that can be aligned along the *z* axis (that is, aligned along **B**) is given by the quantum
number *m _{I}*, which has possible values of -

For a spin ½ nucleus like ^{1}H, the permissable values of *m _{I}* are ± ½.

μ_{z} = γ *p _{z}* = ℏ γ

The consequence of this quantitization is that the nuclear magnetic moment can never be perfectly aligned with **B** in the way the bar magnets can become perfectly
aligned. There is always a residual component of the magnetic moment in the *xy* plane and thus there is always a non-zero torque on the nucleus in the *xy* plane.

Joseph Larmor worked out the mathematical description of this persistent torque on the nuclear magnetic moment. The torque causes the nuclear magnetic moment
to precess or rotate about the *z* axis at a constant rate, as illustrated in the animations shown below. This behavior is called *Larmor precession*.

For a spin ½ nucleus, *m _{I}* = +½ or -½, with +½ being the lower energy state. Both possible states are shown below.

Start the animation and observe the behavior of the nuclear magnetic moment (**μ**), which is shown by the red and green arrows.
In particular, observe the following behaviors.

- What happens to the
*z*-component of the magnetic moment (*μ*) as time passes?_{z} - How does
*μ*depend upon the quantum number_{z}*m*?_{z} - What is the effect of the torque on the orientation
**μ**?

The frequency of precession on NMR spectrometers is in the MHz range, with some instruments as high as 1 GHz. Thus the time required
for one full rotation is a few nsec. Thus actual Larmor precession is approximately a *billion times faster* than the simulation shown below.

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Bulk Magnetization

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