Gradient zeroth moment ($m_0$)

The gradient zeroth moment is the signed time integral of an effective magnetic field gradient waveform along a specified encoding axis:Its units are tesla seconds per metre. Effective means that the sign convention accounts for any refocusing pulses in the nuclear magnetic resonance sequence; it is not necessarily the unsigned physical pulse area.

For a stationary spin at position $x_0$, choose the phase convention $\varphi =\gamma \int G(t)x_{0}dt$. Then $\varphi =\gamma x_0{m}_0$, where $\gamma$ is the angular gyromagnetic ratio. A balanced waveform with $m_0=0$ cancels this position-dependent contribution within the model. It does not assert that every possible background contribution to phase vanishes.

As an original rectangular-pulse example, $5mT/m$ for $2\mathrm{ms}$ has $m_0=10^{-5}\mathrm{T}s/m$. Adding $-10mT/m$ for $1\mathrm{ms}$ cancels it. Equal areas, rather than equal pulse heights, produce the balance.

For a moving spin with $x(t)=x_0+vt$, the remaining contribution can be $\gamma v{m}_1$, involving the gradient first moment. That separation explains how magnetic resonance velocity encoding can suppress sensitivity to stationary position while preserving motion information. Early phase-sensitive flow imaging supplied the experimental motivation for extracting such information from resonance phase.^[ref]