Gradient first moment

The gradient first moment of an effective magnetic field gradient waveform $G(t)$ is its time-weighted integral,Here effective includes the sign changes caused by refocusing pulses. Its units are $\mathrm{T}{\mathrm{s}}^2/m$ when $G$ is in tesla per metre. The gradient zeroth moment is $m_0=\int G(t)dt$. For position $x(t)=x_0+vt$ with constant velocity, the accumulated nuclear magnetic resonance phase iswhere $\gamma$ is the angular gyromagnetic ratio. Thus $m_0=0$ removes the dependence on initial position while $m_1$ can retain sensitivity to velocity, the basis of an idealized phase-contrast magnetic resonance imaging encoding.^[ref]

For a teaching bipolar gradient, let the first lobe have signed area $A$ and time centroid $t_1$, and the second have signed area $-A$ and centroid $t_2$. ThenEqual and opposite lobe areas cancel stationary-spin phase, but their time separation prevents cancellation of the constant-velocity phase. For $A=10^{-5}\mathrm{T}s/m$ and $t_2-t_1=2\mathrm{ms}$, the first moment is $-2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$. These are illustrative waveform values, not measurements from the source.

The time origin is part of the definition: replacing $t$ by $t-t_0$ gives $m_1^{\prime }=m_1-t_0{m}_0$. Balanced $m_0=0$ makes the first moment independent of that choice. Two encodings with equal zeroth moments use their first-moment difference $\Delta m_1$, rather than either individual value, to infer velocity.