The solution of the bloch equations in the presence of a varying B1 field—An approach to selective pulse analysis

• Published in: Journal of magnetic resonance (1969) 1979-01, Vol.35 (1), p.69-86
• Main Author: Hoult, D.I
• Format: Article
• Language: English
• ISSN: 0022-2364; 1557-8968
• DOI: 10.1016/0022-2364(79)90078-7

A method of solution of the Bloch equations in the presence of a varying B 1 field is proposed. The method, which uses perturbation theory and linear systems analysis, is applied to the situation where B 1 is in the form of a “selective pulse,” i.e., a pulse whose frequency spectrum is deliberately limited by suitable modulation of the amplitude of B 1 in a time ≪ T 2. The resulting system response is calculated as a function of frequency, and also as a function of time when a strong linear field gradient is applied. Although the method is general, the particular pulse shape chosen is rectangular in order to allow comparison with known results. In the field gradient case, the primary response of the system is shown to be a replica of the pulse. Thus, if the receiver is gated off during the pulse, the response may only be viewed by formation of an echo. The tertiary response, which becomes prominent for flip angles approaching 90°, is shown to be a parabolic decay following the rectangular pulse, and of duration equal to the pulse length. The frequency spectrum of this response, which is a manifestation of the nonlinearity of the system, is shown to bear only an indirect relationship to that of the original pulse.