- #1
pmiranda
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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?
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?
