Larmor Frequency

Why does the RF-field have to be applied at the Larmor frequency for resonance to occur?  

A fundamental property of resonance phenomena in any media (mechanical, electrical, acoustic, magnetic) is that the response is maximal at a specific frequency and requires energy input including that frequency.

Grandmother Mimi pushing my triplet daughters (Martha, Ellie, & Patti) on our backyard swing at precisely the resonant frequency. How did she know?

 
A familiar example is pushing a child on a swing.  The child-swing system, like a pendulum, has a natural resonance frequency that depends on the weight of the child and the length of the chains. Optimal swinging height is achieved when the frequency of pushing closely matches the natural frequency of the swing. Pushing faster or slower provides a less efficient transfer of energy.

The same situation holds for NMR.  Here the energy is supplied in the form of a rotating magnetic field, denoted B1, which is applied for a short time in the plane perpendicular to Bo.  To supply energy to the spin system and tip the net nuclear magnetization (M), the B1 field must rotate very near the resonance (precession) frequency of the protons. If B1 rotates at any other frequency then it would be alternately in and out of phase with the spins which produce M.  Only when the rotation rate of B1 closely matches the precession frequency can M and B1 remained locked together in the appropriate relation for tipping and energy exchange.  The absorption and exchange of energy constitute a resonance phenomenon, sharply tuned to the natural nuclear precession frequency.

The natural precession frequency of a spin system is also known as the Larmor frequency.

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Advanced Discussion (show/hide)»

It should be noted that the B₁ field does not have to applied exactly at the resonance frequency for some tipping of M to occur. It just has to be relatively "close" to the Larmor frequency (i.e, within 3-5%). Similarly, a grandmother pushing her child on a swing does not have to push at exactly the natural frequency to get motion to occur. It is just that the motion is maximal/optimal when applied precisely at that frequency.

An even more fascinating concept is that resonance may be induced by two or more RF excitations, each far away from the Larmor frequency! The only condition is that the photons supplied by each have to add up to the same energy as that at the Larmor frequency. This is called multiphoton magnetic resonance imaging and has great potential for reshaping the future of MRI. (See the paper by Han and Liu in the references).

As demonstrated in this paper, an RF-field applied along the z-axis by a separate coil or by oscillating the z-gradient transmits z-polarized photons into the subject. The second xy-gradient can be applied at a frequency shifted up or down from the standard Larmor frequency by the same amount, transmitting xy-polarized photons. The result is a two photon excitation of the system.

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References
      Eles PT, Michal CA. Two-photon two-color nuclear magnetic resonance. J Chem Phys 2004; 121:10167-73. [DOI link]
      Han V, Liu C. Multiphoton magnetic resonance in imaging: a classical description and implementation. Magn Reson Med 2020; 84:1184-1197. [DOI link] (A fascinating recent article demonstrating with imaging how two RF fields, each far away from the Larmor frequency, can actually induce resonance. See advanced discussion for further commentary).
     Rigden JS. Quantum states and precession: the two discoveries of NMR. Rev Mod Physics 1986; 58(2):433-448. [DOI Link]

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Related Questions
     Who was Larmor and how did he discover his famous frequency? 

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