Confirmation of some formulas related to spin coherence time

TL;DR

This paper verifies the correctness of two related formulas for spin coherence time in electric rings by demonstrating their consistency under specific conditions, confirming their validity for different oscillation regimes.

Contribution

It shows that two previously derived formulas for spin coherence time are consistent and valid across different oscillation periods, clarifying their applicability.

Findings

The formulas are valid for times much larger than the oscillation period.

A continuous transition exists between the two formulas.

Both formulas follow from the same fundamental equation under certain conditions.

Abstract

This paper considers two sets of formulas related to a Spin Coherence Time (SCT) case with only vertical oscillations in a purely electric ring, the first derived by the author for the field index m > 0 and the second by Ivan Koop for the field index m = 0. I argue that a continuous transition can exist from one set to the other (contrary to appearances), and assume that a necessary condition for the transition is that both sets of formulas follow from the same equation. I demonstrate that they do follow when one takes into account that the first set of formulas holds only for times much larger than the period of the vertical oscillations. This demonstration confirms the correctness of the formulas and the equation.