Nuclear Magnetic Resonance (NMR and MRI)

Nuclear Magnetic Resonance is an important tool in chemical analysis. As the name implies, it uses the spin magnetic moments of nuclei (particularly hydrogen) and resonant excitation. Magnetic Resonance Imaging uses the same principle to get an image (of the inside of the body for example).

In basic NMR, a strong static B field is applied. A spin \bgroup\color{black}${1\over 2}$\egroup proton in a hydrogen nucleus then has two energy eigenstates. After some time, most of the protons fall into the lower of the two states. We now use an electromagnetic wave (RF pulse) to excite some of the protons back into the higher energy state. Surprisingly, we can calculate this process already. The proton's magnetic moment interacts with the oscillating B field of the EM wave.


As we derived, the Hamiltonian is

\begin{displaymath}\bgroup\color{black}H=-\vec{\mu}\cdot \vec{B}=-{g_pe\over 2m_...
=-{g_p\over 2}\mu_N\vec{\sigma}\cdot\vec{B}\egroup\end{displaymath}

Note that the gyromagnetic ratio of the proton is about +5.6. The magnetic moment is 2.79 \bgroup\color{black}$\mu_N$\egroup (Nuclear Magnetons). Different nuclei will have different gyromagnetic ratios, giving us more tools to work with. Let's choose our strong static B field to be in the z direction and the polarization on our oscillating EM wave so that the B field points in the x direction. The EM wave has (angular) frequency \bgroup\color{black}$\omega$\egroup.

\begin{displaymath}\bgroup\color{black}H=-{g_p\over 2}\mu_N \left(B_z\sigma_z+B_...
...matrix{B_z & B_x\cos\omega t\cr B_x\cos\omega t & -B_z} \egroup\end{displaymath}

Now we apply the time dependent Schrödinger equation.

i\hbar{d\chi\over dt} & = & H\chi \\
...os\omega t\cr \omega_1\cos\omega t & -\omega_0}\pmatrix{a\cr b}

The solution of these equations represents and early example of time dependent perturbation theory.

\begin{displaymath}\bgroup\color{black}{d\over dt}{(be^{i\omega_0t})}
= {i\ome...
...2}(e^{i(\omega+2\omega_0)t} + e^{-i(\omega-2\omega_0t)})\egroup\end{displaymath}

Terms that oscillate rapidly will average to zero. The first term oscillates very rapidly. The second term will only cause significant transitions if \bgroup\color{black}$\omega\approx 2\omega_0$\egroup. Note that this is exactly the condition that requires the energy of the photons in the EM field \bgroup\color{black}$E=\hbar\omega$\egroup to be equal to the energy difference between the two spin states \bgroup\color{black}$\Delta E=2\hbar\omega_0$\egroup. The conservation of energy condition must be satisfied well enough to get a significant transition rate. Actually we will find later that for rapid transitions, energy conservation does not have to be exact.

So we have proven that we should set the frequency \bgroup\color{black}$\omega$\egroup of our EM wave according to the energy difference between the two spin states. This allows us to cause transitions to the higher energy state. In NMR, we observe the transitions back to the lower energy state. These emit EM radiation at the same frequency and we can detect it after the stronger input pulse ends (or by more complex methods). We don't yet know why the higher energy state will spontaneously decay to the lower energy state. To calculate this, we will have to quantize the field. But we already see that the energy terms \bgroup\color{black}$e^{-iEt/\hbar}$\egroup of standard wave mechanics will require energy conservation with photon energies of \bgroup\color{black}$E=\hbar\omega$\egroup.

NMR is a powerful tool in chemical analysis because the molecular field adds to the external B field so that the resonant frequency depends on the molecule as well as the nucleus. We can learn about molecular fields or just use NMR to see what molecules are present in a sample.

In MRI, we typically concentrate on one nucleus like hydrogen. We can put a gradient in \bgroup\color{black}$B_z$\egroup so that only a thin slice of the material has \bgroup\color{black}$\omega$\egroup tuned to the resonant frequency. Therefore we can excite transitions to the higher energy state in only a slice of the sample. If we vary (in the orthogonal direction!) the B field during the decay of the excited state, we can get a two dimensional picture. If we vary B as a function of time during the decay, we can get to 3D. While there are more complex methods used in MRI, we now understand the basis of the technique. MRIs are a very safe way to examine the inside of the body. All the field variation takes some time though. Ultimately, a very powerful tool for scanning materials (a la Star Trek) is possible.

Jim Branson 2013-04-22