# What is Rabi nutation in NMR?

What is the mathematical and physical significance of Rabi Nutation in terms of NMR?

## 1 Answer

Here's my understanding based on my QM class and a lab I did a while back. I've tried to set it up in an intuitive way. This is my first answer so I'm sure there will be room for improvement but hopefully it gives you a general idea how it works:

Say we apply a constant magnetic field to a sample of hydrogen nuclei along the z direction, the Hamiltonian is given by: $H=-m \cdot B$. The quantized energy states of an individual proton nuleus would be $\pm \frac{g_n u_N B}{2}$.

For a sample of N of these nuclei, the populations of each energy state at a given temperature is found from the Boltzmann distribution and the density matrix . When one speaks of Rabi nutation, they are referring to the time evolution of these populations when a time dependent magnetic field is applied.

So let's say we also apply a time dependent magnetic field in the x direction (RF and about 15Mhz for this example): $$B=B_1cos(\omega t) \hat{i}$$ Our Schrö­din­ger equa­tion is : $$i\frac{d}{dt}\left(\begin{array}{cc} a(t)e^{-iw_+t} \\ b(t)e^{-iw_-t} \end{array} \right) = \left( \begin{array}{cc} \omega_+ & \frac{\omega_R}{2}(e^{-i\omega t}+e^{i\omega t} \\ \frac{\omega_R}{2}(e^{-i\omega t}+e^{i\omega t}) & \omega_-\\ \end{array} \right) \left( \begin{array}{cc} a(t)e^{-iw_+t} \\ b(t)e^{-iw_-t} \\ \end{array} \right)$$ Where $\omega_R$ is the Rabi frequency and $\omega_\pm$ are the original energy eigenvalues divided by $\hbar$. Now the idea with NMR is to tune the magnetic field frequency to $\omega_+-\omega_-$ (the energy spacing$/\hbar$) of the original system. If you can set up your experiment to (roughly) do that, (and solve the DE's with a Rotating Wave approximation) you get these results for the time dependence of the samples' population:

$a(t)=cos(\frac{\omega_R t}{2})$ and $b(t)=isin(\frac{\omega_R t}{2})$

So the population of the two states will oscillate at half the rabi frequency. Knowing this frequency we can control the bulk magnetization by applying timed pulses. For example, by applying a $\frac{\pi}{2}$ pulse followed by a $\pi$ pulse with the right timing, (see Spin Echo) you can get the characteristic relaxation time of the sample, which is the useful information used in MRI's.