# NMR mathematics

1. Sep 28, 2010

### phygiks

Hello all,
I have questions regarding my understanding of NMR and the mathematics of it. If anyone would take the time to read this, I would very much appreciate it. Please bear with me, I am a beginner. Thank you!

First, there is the question on how to model this phenomenon. Up till now I have used the Bloch equations. Recently I have come across the NMR pictured with a general Hamiltonian with zeeman, quadrupole, internuclear, and electron-nucleus terms. I have unable to comprehend this, is there a good reference to help explain this? How can I use a Hamiltonian to model the precession?

I would also like to know if there are any references to NMR experiments dealing with problem of inhomogeneity of the static field, and offsets in excitation rotations (the spins are not rotated as expected since the field changes at different depths of the sample).

Thank you.

2. Sep 28, 2010

Malcolm H. Levitt "Spin Dynamics. Basics of Nuclear Magnetic Resonance" has this chapter:

Internal Spin Interactions 195 9.1 Chemical Shift 195 9.1.1 Chemical shift tensor 196 9.1.2 Principal axes 197 9.1.3 Principal values 198 9.1.4 Isotropic chemical shift 198 9.1.5 Chemical shift anisotropy (CSA) 198 9.1.6 Chemical shift for an arbitrary molecular orientation 200 9.1.7 Chemical shift frequency 201 9.1.8 Chemical shift interaction in isotropic liquids 201 9.1.9 Chemical shift interaction in anisotropic liquids 203 9.1.10 Chemical shift interaction in solids 204 9.1.11 Chemical shift interaction: summary 206 9.2 Electric Quadrupole Coupling 206 9.2.1 Electric field gradient tensor 207 9.2.2 Nuclear quadrupole Hamiltonian 208 9.2.3 Isotropic liquids 209 9.2.4 Anisotropic liquids 209 9.2.5 Solids 210 9.2.6 Quadrupole interaction: summary 210 9.3 Direct Dipole–Dipole Coupling 211 9.3.1 Secular dipole–dipole coupling 213 9.3.2 Dipole–dipole coupling in isotropic liquids 215• xii Contents 9.3.3 Dipole–dipole coupling in liquid crystals 216 9.3.4 Dipole–dipole coupling in solids 216 9.3.5 Dipole–dipole interaction: summary 217 9.4 J-Coupling 217 9.4.1 Isotropic J-coupling 219 9.4.2 Liquid crystals and solids 221 9.4.3 Mechanism of the J-coupling 222 9.4.4 J-coupling: summary 223 9.5 Spin–Rotation Interaction 223 9.6 Summary of the Spin Hamiltonian Terms

3. Oct 11, 2010

### phygiks

Thanks, I have referenced that book before, but it has very little mathematical explanation. I am trying to find out what happens to spins that are off resonance when a pulse sequence is applied. For example a pulse sequence of 90 will rotate all on-resonance spins into the XY plane, but what about spins not on resonance.

4. Oct 12, 2010

### DrDu

But that's a question you should be able to work out using the Bloch equations which you already understood.
The Bloch equations arise in the Heisenberg picture of QM from the equation of motion of a single spin under the influence of the Hamiltonian H:
$$i\hbar d \mathbf{\sigma}/dt=[H, \mathbf{\sigma}]$$
Using e.g. $$H=\Omega \sigma_z +a \cos(\omega t) \sigma_y$$
and working out the commutators of the spin matrices you should arive at the Bloch equations. If the hamiltonian also contains more complicated terms, like J-coupling, then you get a system of coupled differential equations for all the spins.

5. Oct 12, 2010

### phygiks

I am very new to QM, so I cannot make the connection so easily, but I now I know there is one.
Yes I should be able to find out what happens to off resonance spins with the Bloch equations, but I get a very ugly diff-eq. In fact, I am puzzled how the rotation matrices are even derived from the bloch equations. Solving the diff eq for on resonance spins gives a solution for the magnetization M = (0 sin wt cos wt)' or something along those lines, not the rotation matrix.

6. Oct 12, 2010

### DrDu

Ok, yes, I forgot to mention M. Once you know $$\mathbf{\sigma}(t)$$, you can also calculate $$\mathbf{M}\cdot \mathbf{\sigma}$$. Refering to these time dependent sigma matrices, the components of M are constant. However, you can express this vector also in terms of the sigmas at t=0. Then you have to transfer the time dependence to the Ms.

I am sorry, but I can only give you a draft how it works. If you are keen to learn about that stuff, you should try to work it out yourself or consider a book. You don't need more than $$[\sigma_x,\sigma_y]=i\sigma_z$$ and all the two similar relations which result from cyclic permutations of xyz. If you get stuck, you can always ask.

7. Oct 12, 2010

### DrDu

What's going on is the following. On resonance a spin (or magnetization) will be driven around the Bloch sphere until the pulse ends. Off resonance, it will be driven in one direction, then will slow down when it gets off resonance to be finally driven back to where it started when it is 180 degrees out of phase. If it is seriously out of phase, it will only perform tiny oscillations about its original position.

8. Oct 12, 2010

### phygiks

Is there a book you can recommend?