Understanding NMR: Exploring the Conflict With Quantum Physics

[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?

 

[marcusl]

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.

 

[simorgh]

see also below chapter one of below book:
"Basic one- and two-dimenstional NMR spectroscopy", Freibolian.
and for nmr below book is great:
"Spin Dynamics, Basics of Nuclear Magnetic Resonance" - Malcolm H. Levitt - John Wiley.