What is the Dirac vector model in NMR?

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