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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]

Magnetic resonance velocity encoding makes a component of motion measurable through the phase of a nuclear magnetic resonance signal. Controlled magnetic field gradients give different phase shifts to spins with different trajectories. Phase-contrast magnetic resonance imaging compares encodings to extract a velocity component, rather than interpreting ordinary image brightness as speed.^[ref]

In an original idealized model, assume constant velocity $v$ along one encoding axis and effective gradients with equal gradient zeroth moments. Then the encoding difference obeyswhere $\gamma$ is the angular gyromagnetic ratio and $\Delta m_1$ is the gradient first moment difference. For $\gamma =2\pi \cdot 42.58\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$, $\Delta m_1=2\times 10^{-8}\mathrm{T}{\mathrm{s}}^2/m$ and $v=0.10m/s$, this gives $\Delta \varphi \simeq 0.535\mathrm{rad}$. These stipulated values illustrate the conversion, rather than a particular scanner experiment.

Reversing the moment difference reverses the phase sign under this convention. Reducing its magnitude decreases sensitivity to a given velocity while increasing the velocity encoding limit. A measured voxel can contain a range of velocities, and background phase or phase aliasing can complicate reconstruction.

The encoded component also differs from volumetric flow rate: estimating transported volume requires combining the appropriate normal velocity with area and time, not merely recording one peak speed.

P. R. Moran authored a 1982 paper describing flow-sensitive phase modulation for magnetic resonance imaging. The method adds information about motion to spatial encoding: the abstract describes imaging a joint distribution of position and velocity, rather than treating the signal solely as a map of stationary material.^[ref]

The physical idea is magnetic resonance velocity encoding. Motion through a magnetic field gradient changes accumulated signal phase under a specified pulse sequence. Appropriate measurements can therefore distinguish moving spin populations that occupy similar positions. The cited abstract describes reconstructing flow-related quantities from complex image data; it does not establish a unique molecular or material identity from signal phase alone.

A synthetic model shows why a distribution of velocities can matter. At one location, suppose two equally weighted contributions with velocities $+v$ and $-v$ acquire phases $+\kappa v$ and $-\kappa v$. Their combined complex signal isIt is real even though neither population is stationary. For $\kappa v=\pi /3$, $S=1$, compared with $S=2$ for two stationary contributions in this model. A zero imaginary component alone consequently does not prove zero motion. This is an explanatory phase-summation example, not a numerical result reported in the 1982 paper.

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