# NMR: what happens at quantum level when the orientation of the magnetic filed changes

1. Mar 29, 2012

### pmiranda

Hello,

I am writing an appendix to my thesis where i try to explain how nmr works but i am having trouble understanding something.

Suppose that we have a sample with many nuclei with 1/2 spin precessing around an external magnetic field $\vec{B_0}$ aligned with the z axis. At Boltzmann equilibrium the net magnetization $\vec{M}$ of the sample would be along the positive direction of z.

Since the rate of change of individual magnetic moments is given by
$\frac{d\vec\mu}{dt}=\gamma\vec\mu\times \vec{B_0}$

the generalization to the net magnetization vector $\vec{M}$ would be
$\frac{d\vec{M}}{dt}=\gamma\vec{M}\times \vec{B_0}$

suppose that we now introduce a constant, non oscillating, magnetic field $\vec{B_1}$ orthogonally to $\vec{B_0}$. I guess it is safe to write that:
$\frac{d\vec{M}}{dt}=\gamma\vec{M}\times (\vec{B_0}+\vec{B_1})=\gamma\vec{M}\times (\vec{B_{eff}})$

This suggests that the magnetization vector would move in a circular path around $\vec{B_{eff}}$.

If everything i said until now is correct, i can't understand what happens at quantum level. I mean, although the macroscopic equation suggests that $\vec{M}$ would precess around $\vec{B_{eff}}$ what happens to the individual magnetic moments? I would think that a new Boltzmann equilibrium where they are randomly precessing around $\vec{B_{eff}}$ would be establish, however, that implies that $\vec{M}$ is static vector along that same direction!

I can't understand this conflict. There must be something wrong or something missing here. The only thing i can remember is that somehow the individual moments are no longer randomly distributed along the 2 cones but are now rotating in phase from where the excess of nuclei in the lower energy state would coincide with $\vec{M}$ but how did that happen?

My formation is quantum physics is not that advanced since i graduated in informatics and this is really killing me. Can someone help me understand what happens to the individual magnetic moments as soon as the second magnetic field is introduced?

2. Mar 29, 2012

### marcusl

Re: NMR: what happens at quantum level when the orientation of the magnetic filed cha

The QM view of NMR is different. The nuclear spins form a two-state system (spin aligned and anti-aligned with the field) that have an energy difference given by the quantum Zeeman splitting. Transitions between the states occur when a photon of correct energy (frequency) is absorbed or emitted--the quantum equivalent of the classical RF resonance condition.

There are many sites on the web that cover this, e.g.
http://users.fmrib.ox.ac.uk/~stuart/thesis/chapter_2/section2_2.html

or see any NMR text. A classic is Slichter, Principles of Nuclear Magnetism.

3. Apr 9, 2012

### simorgh

Re: NMR: what happens at quantum level when the orientation of the magnetic filed cha