Each slice has a different center frequency Fc determined by its position (z) along the slice-select gradient (Gss), given by
Fc = γ(Bo + z•Gss) = fo + γ•z•Gss
where Bo is the main magnetic field strength and fo is its corresponding Larmor frequency.
Each slice has a finite width (Δz) and so it contains a range of frequencies (ΔF) centered around Fc. These quantities are related by the equation
ΔF = γ • Gss • Δz
In common practice ΔF is held constant (on the order of 1000-2000 Hz) and slice thickness Δz is varied by adjusting Gss. Stronger gradients produce thinner slices, and vice versa.
Mathematically it can be shown that the ideal RF-excitation to achieve this frequency profile is a so-called sinc pulse. The sinc pulse is an amplitude-modulated sine wave with base frequency Fc whose equation is given below
Here the range of frequencies (ΔF) subtending a slice equals the transmit bandwidth of the pulse. Transmitter bandwidth should be distinguished from receiver bandwidth (discussed in a prior Q&A). Receiver BW is function of the digitization rate of the recorded MR signal and has nothing to do with slice selection.
Today most RF-pulses cannot be described by mathematically simple waveforms, having eclectic shapes based on an computer-based polynomial design process known as the Shinnar-Le Roux (SLR) algorithm. SLR-generated pulses provide more careful control of RF phase properties, allowing the designer to trade off analytically several parameters describing pulse performance including frequency profile, ripple, duration, and energy deposition.
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The length of "standard" RF-pulses in is on the order of 2-3 msec, leaving time for no more than one pair of SINC or SLR side lobes. Some scanners offer options for longer or shorter pulses.
Extended length pulses may have durations of 4-5 msec. This allows time for more complex shapes and side lobes with better slice profiles and selectivity. SAR is also lower since the pulses are spread out over a longer period of time and their peak amplitudes are lower. The trade-off is that the minimum allowed TE values are prolonged and hence number of slices for a given TR is reduced.
Conversely, some scanners offer a "fast RF" pulses of lengths 1.5 msec or less. These are typically gaussian-shaped with compromised non-rectangular slice profiles and nonlinear phase shifts. These pulses are typically used for rapid imaging applications, such as HASTE, FIESTA, or TurboFLASH. Pulse amplitudes are higher and so are SAR's. The short pulse duration allows for shorter TE's and shorter echo spacings. Susceptibility artifacts are also reduced.
Pauly J, Le Roux P, Nishimura D, Macovski A. Parameter relations for the Shinnar-Le Roux selective excitation pulse design algorithm. IEEE Transact Med Imaging 1991; 10:53-65.
Balchandani P, Pauly J, Spielman D. Designing adiabatic radiofrequency pulses using the Shinnar-Le Roux algorithm. Magn Reson Med 2010; 54:843-851.