Effective RF Fields Created by a PIP

A PIP is composed of a series of back-to-back pulses of an equal width Ar as shown in Fig. 2. In addition to an initial phase cp0 assigned to all the steps, a phase increment A p (— 180° A p 180°) is implemented in the following way 

For a non-interacting spin-1/2 system subjected to a PIP of an arbitrary shape, the Hamiltonian in the rotating frame with a RF carrier frequency /rf 

According to the definition, a(t) in Eq. (3) is a periodic function of / with a period of Ar as shown in Fig. 3. Through the above procedure, the phase p(t), as it appeared in the exponential in Eq. (2), is now separated from the product of (f t)Iz. Consequently, both terms of e ,(t) and e a( which are the function of can be expanded into a Fourier series of complex exponentials51 

These jump points, however, will not contribute to the evolution of the spin system since they have infinitesimal durations and therefore have no accumulative effects. 

According to Eqs. (8) and (9), the effective RF fields are scaled unsymmet-rically (in respect to the sideband number n) by the scaling factor Xn, in response to the unsymmetrical excitation profile. When K 0, the nth effective RF field becomes negative, which corresponds to a sign change of the operator Ix from positive to negative, or equivalently, a 180° phase shift is introduced to the nth effective RF field. Consequently, a phase inversion occurs in the transverse magnetization of the nth excitation band. 

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All the effective RF fields created by a PIP are shifted from the carrier frequency fa. To calculate the rath excitation band for example, a transformation from the rotating frame to a new one (or the second rotating frame) is often needed, where the rath field plays the role of a new carrier. This transformation can be achieved by a unitary operator of1... 

As any RF pulse, each effective RF field created by a PIP dissipates power to the probe. The power by the rath effective field for the kth step,... 

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