Question

What is the Dirac vector model in NMR?

Answer 1

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 `1/2, 3/2, . . . , 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,..., 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:

`color(blue)(v_0 = -1/(2pi)gammaB_0)`

where:

• `nu_0` is the frequency of the applied pulse in `"Hz"`. A possible value for a Bruker NMR is `"300 MHz"`.
• `gamma` is the gyromagnetic ratio in `"1/T"cdot"s"` or `"1/G"cdot"s"`, depending on what units you want to use.
• `B_0` is the applied magnetic field in either `"T"` (Tesla) or `"G"` (Gauss) for the magnetic field strength units; `"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.

That's about all you need to know, probably. You can read more about it here.

http://www-keeler.ch.cam.ac.uk/lectures/understanding/chapter_3.pdf