The Bloch-Siegert shift is a well-known perturbation to the rotating-wave approximation which becomes prominent when the nutation frequency of the driving rf field becomes comparable in magnitude with the Larmor frequency of the driven spins.
Consider the following scenario: we are equipped with a set of B1 coils at low-field (the longitudinal B0 field is~1.1 mT). Our untuned B1 coils are capable of broadband irradiation of spins with a transverse B1 field whose nutation frequency, ω1/(2π), can reach around ~2 kHz*.
A 2 kHz nutation frequency presents little food-for-thought for 1H spins (ω0/(2π) @ 1.1 mT ≈ 47 kHz) but really makes you reconsider your life choices if, like us, you have been working with 103Rh (ω0/(2π) @ 1.1 mT ≈ 1.5 kHz).
In this scenario it is necessary to calculate higher-order terms of the Bloch-Siegert shift Hamiltonian, which I have done to 12th (or 11th, depending on your convention) order.
*there is a subtle point here: with untuned coils operating in the high-inductance limit, the B1 field strength is proportional to 1/(ω0), whereas the nutation frequency is of course proportional to ω0. These factors cancel out as shown in pages 90-91 [Figs. 39-40 of my thesis], leading to a nutation frequency that is independent of frequency and hence γ:
