# What is the Dirac vector model in NMR?

Jan 30, 2016

In Nuclear Magnetic Resonance Spectroscopy, the nucleus of every atom in the sample has a magnetic moment, giving it a nuclear spin.

The nuclear spin depends on the number of protons ${p}^{+}$ and neutrons ${n}^{0}$ in the nucleus.

• If the number of ${p}^{+}$ and ${n}^{0}$ are EACH even, then the nuclear spin is $0$.
• If the number of ${p}^{+}$ and ${n}^{0}$ SUM to be odd, then the nuclear spin is $\frac{1}{2} , \frac{3}{2} , . . . , \frac{n}{2}$ where $n$ is a positive odd integer.
• If the number of ${p}^{+}$ and ${n}^{0}$ are EACH odd, then the nuclear spin is $1 , 2 , 3 , \ldots , n$ where $n$ is a positive integer.

The magnetic moment interacts with an applied magnetic field ${B}_{0}$, and the bulk magnetization (the magnetization of the entire sample by the same magnetic field) is such that the net magnetic field ${B}_{z}$ is in the same direction as the applied magnetic field ${B}_{0}$.

The net magnetization can be represented as a single magnetization vector:

When a pulse of frequency ${v}_{0}$ in the radio frequency range is applied to the magnetic field, it tilts it away from the $z$-axis by some angle we can call $\beta$. This tilting is called a Larmor precession.

While the magnetization vector is tilted, it rotates in the direction of the magnetic field.

Using the right-hand-rule and noting that the precession frequency is negative, the vector rotates clockwise (instead of counterclockwise like the right-hand rule would predict for a positive precession frequency).

The Larmor precession ${\omega}_{0}$ can be converted into the frequency ${v}_{0}$, resulting in the relationship:

$\textcolor{b l u e}{{v}_{0} = - \frac{1}{2 \pi} \gamma {B}_{0}}$

where:

• ${\nu}_{0}$ is the frequency of the applied pulse in $\text{Hz}$. A possible value for a Bruker NMR is $\text{300 MHz}$.
• $\gamma$ is the gyromagnetic ratio in $\text{1/T"cdot"s}$ or $\text{1/G"cdot"s}$, depending on what units you want to use.
• ${B}_{0}$ is the applied magnetic field in either $\text{T}$ (Tesla) or $\text{G}$ (Gauss) for the magnetic field strength units; $\text{1 G = 10"^(-4) "T}$.

When you place a small coil of wire on the x-axis, it basically detects the x component of the Larmor precession, taking in a current induced by the magnetic field.

(This is like the induced current you can get when you send a magnetic field through a solenoid.)

If we suppose the magnitude of the vector is ${M}_{0}$, then the projection on the x-axis is shown below:

This induced current is essentially amplified and encoded into an NMR signal.