GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY

Abstract

The fundamental description of relaxation (T₁ and T₂) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T₁ and T₂ relaxation. The T₁ decay is stable for the range of delays tested (1 μsec to 200 μsec), while the T₂ relaxation in this extended model exhibits a critical delay (typically 100 μsec to 200 μsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.