Exact analytical solutions of the Bloch equation for the hyperbolic-secant and chirp pulses.

ABSTRACT:

Purpose

To improve the accuracy and generality of analytical solutions of the Bloch equation for the hyperbolic-secant (HS1) and chirp pulses in order to facilitate application to truncated and composite pulses and use in quantitative methods.

Theory and methods

Previous analytical solutions of the Bloch equation during an HS1 pulse driving function are refined and extended in this exact solution for arbitrary initial magnetization and pulse parameters including asymmetrical truncation. An unapproximated general solution during the chirp pulse is derived in a non-spinor formulation for the first time. The solution on the extended complex plane for the square pulse is included for completeness.

Results

The exact solutions for the HS1, chirp, and square pulses demonstra

SUBMITTER: Smith RHB 

PROVIDER: S-EPMC12393199 | biostudies-literature | 2025 Nov

REPOSITORIES: biostudies-literature