NMR flip angle - angle of nutation

[Chrysoperla]

TL;DR Summary

How to derive the angle of which magnetic dipole in static magnetic field flips after short oscillating pulse (angle of nutation).

Hi, I am interested in derivation of flip angle in nuclear magnetic resonance. We have magnetic dipole in static magnetic field. The magnetic dipole is precessing around the vector of this field with Larmore frequency. After applying square magnetic pulse with amplitude B1 oscillating on frequency w similar to Larmore frequency (=resonance) with duration T, the vector of magnetic dipole moment flips by an angle γB1T. I am curious how to derive this formula for the flip angle.

Probably it is obvious, but I can not see it now and also was not able to google it somehow. I would be grateful for any suggestions of literature which contains this derivation, available online. Thank you.

 

[marcusl]

Precession of the nuclear spin is governed by the Bloch equations, named after Felix Bloch who co-discovered NMR in 1947 and later shared the Nobel Prize with his competitor, the other co-discoverer Edward Purcell. These equations are nonlinear so it is not so simple in a strict sense, although linear approximations are commonly used to simplify the description. To answer your specific question, the circularly polarized B₁ field rotates at the Larmor frequency in synchrony with the spin system so it is most easily analyzed in a rotating coordinate system (the "rotating frame") whose axis is aligned with the magnetization M. B₁ is very weak compared to the main polarizing magnetic field B₀. In the rotating frame, the spins precess in the y-z plane at the secondary Larmor frequency $$\omega_1=\gamma B_1$$ where $$\gamma$$ is the same gyromagnetic ratio that determines the primary resonant frequency. The time for a π/2 pulse is one quarter of the secondary precession period.

I left the NMR field some decades ago and gave away my books and papers, so I'll give you references by memory. The classic books on NMR are Abragam, "The Principles of Nuclear Magnetic Resonance" and Slichter, "Principles of Magnetic Resonance." My former colleague and friend Ray Freeman wrote a nice book called "A Handbook of NMR" that reads like a concise encyclopedia, where different terms, in alphabetical order, are explained with a minimum of mathematics. You might find it useful. Sadly, Ray passed away recently. I imagine that there are many other books that have been published in the past 20 years.

A cursory google search turned up this article which seems, on a quick look, to be pretty good:
https://home.csulb.edu/~tgredig/phys545/nuclear-magnetic-resonance.html

EDIT: Hmm, what happened to Latex? There used to be a button, or one could use two dollar signs, to insert equations.
EDIT2: Hmm, it's rendering now! I'm fully confused. I guess you have to refresh the page to see the equations?

 

[Chrysoperla]

That's great! Thank you very much.