= Diffusion weighting b-value
{title2=$b=\int q(t)^2\,dt$}
The <diffusion weighting b-value> quantifies the encoding applied by a <magnetic field gradient> waveform in a <diffusion> measurement. For a refocused experiment in one spatial direction, define
$$
q(t)=\gamma\int_0^t G_{\mathrm{eff}}(u)\,du,\qquad b=\int_0^T q(t)^2\,dt.
$$
Here $\gamma$ is the angular <gyromagnetic ratio>, in radians per second per tesla, and $G_{\mathrm{eff}}$ includes the sign reversal caused by the refocusing pulse. The echo time is $T$, with $q(T)=0$ for refocusing of stationary spins. Thus $q$ has units $\mathrm{m^{-1}}$ and $b$ has units $\mathrm{s\,m^{-2}}$. The squared encoding history matters even though its final value is zero.<Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient by E. O. Stejskal and J. E. Tanner>[Pages289–290: time-dependent-gradient theory and refocusing]{ref}
For unrestricted, isotropic Gaussian <diffusion> with constant <diffusion coefficient> $D$, the otherwise matched signal ratio is $S/S_0=e^{-bD}$. Here $S_0$ uses the same pulse timing and detection conditions with the applied diffusion-encoding gradients removed. The <diffusion weighting b-value> describes the experiment's weighting; $D$ describes the assumed sample model.
For an original unit check, $1000\,\mathrm{s\,mm^{-2}}=10^9\,\mathrm{s\,m^{-2}}$. With $D=10^{-9}\,\mathrm{m^2\,s^{-1}}$, the ideal attenuation is $e^{-1}\approx0.368$. Quoting a numerical $b$ without its units can therefore cause a millionfold scaling error.
The <Stejskal-Tanner equation> gives $b=(\gamma g\delta)^2(\Delta-\delta/3)$ for its equal rectangular pulses. More general waveforms require their own <integral>. For direction-dependent <diffusion>, a scalar $b$ alone cannot specify the complete directional encoding and response; a <tensor> description is needed.
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