0
$\begingroup$

Precession due to magnetic field occurs when the magnetic moment of a small coil is proportional to its angular momentum ($\vec{m}=g\vec{L}$). This is due to Euler momentum equation, which for such a system reads: $$\frac{d\vec{L}}{dt}=\vec{M}=\vec{m}\times \vec{B}=g\vec{L}\times\vec{B}$$and has the shape of a precession equation. In NMR theory we are told that a single proton can both precess around a magnetic field and align to it. This happens because of its spin, so in principle even in absence of orbital angular momentum. First of all: to justify the precession, do we need to generalize Euler equation and write that $d\vec{J}/dt=\vec{M}$ (where $\vec{J}=\vec{L}+\hbar\vec{S})$? Does this always hold? Secondly, what causes alignment? Is alignment just a statistical prediction? Does it always hold?

$\endgroup$

1 Answer 1

0
$\begingroup$

You will need to use the full angular momentum of the system considered, here $\vec{J}=\vec{L}+\vec{S}$ (note usually $\vec{S}$ is considered the spin, i.e. including the $\hbar$). This certainly holds for simple systems like this one, in more complex systems the specific type of spin-orbit coupling may need to be considered.

Alignment of protons in a magnetic field is caused by the interaction of the field with the proton's magnetic moment. The spin, and hence the moment, being parallel, or anti-parallel to the magnetic field vector result in different energy levels, therefore the proton's spin will show preference for the orientation with lower energy. At temperature $T=0$ all spins will settle in this orientation. At higher temperatures, thermal energy will cause a lot of the spins (up to 50%, or balanced, at very high temperatures) to switch to the non-preferred orientation; in this sense we're talking about a statistical phenomenon. In fact, at room temperature in easily achievable magnetic fields the relative imbalance for protons is only $10^{-10}$ .. $10^{-8}$, explaining why NMR signals are so small at room temperature.

The spin pointing up or down is terminology that neglects half of the picture. It only represents the z component of the spin (magnetic field assumed in z direction). The operators of the components of angular momentum do not commute, that means that with determined (certain) z component, the values of the x and y components are completely uncertain. In particular they are not zero, so, in a semi-classical picture, the spin never points just up or down. Continuing this picture, the residual misalignment results in a force on the spin causing precession. This precession is unsynchronised over the ensemble of spins, but can be synchronised by external RF pulses with the right frequency; the synchronisation lasts a while so that an echo can be detected. This is what we call NMR. A proper (more rigorous), but much less intuitive description of this can be given in terms of Bloch vectors on a Bloch sphere.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.