= Gradient zeroth moment
{title2=$m_0$}
The <gradient zeroth moment> is the signed time <integral> of an effective <magnetic field gradient> waveform along a specified encoding axis:
$$
m_0=\int_{t_a}^{t_b}G(t)\,dt.
$$
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 $\phi=\gamma\int G(t)x_0\,dt$. Then $\phi=\gamma x_0m_0$, 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, $5\,\mathrm{mT/m}$ for $2\,\mathrm{ms}$ has $m_0=10^{-5}\,\mathrm{T\,s/m}$. Adding $-10\,\mathrm{mT/m}$ 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 $\gamma vm_1$, 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.<A flow velocity zeugmatographic interlace for NMR imaging in humans by P. R. Moran>[Primary abstract; displayed moment relations are explicit idealized derivations]{ref}
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